**Quantization as Selection Rather than Eigenvalue Problem**

Peter Enders

22 Quantum Mechanics

542 Advances in Quantum Mechanics

1948.

2002.

[32] M.-J. Omero, M. Dzierzawa, M. Marsili, and Y.-C. Zhang. *J. Physique*, 7:1723, 1997.

(Princeton University Press, Princeton, 1955)], Berlin, 1932.

[33] C Shannon. A mathematical theory of communication. *Bell System Tech. Jour.*, 27:379–423,

[34] J. von Neumann. *Mathematische Grundlagen der Quantenmechanik*. Springer-Verlag [translated by R. T. Beyer as Mathematical Foundations of Quantum Mechanics

[35] James D. F. V. and P. G. Kwiat. Quantum state entanglement. *Los alamos Science*, 27:52–57,

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/53847

### **1. Introduction**

The experimental, in particular, spectroscopic results about atoms brought Bohr [3] to the following "principal assumptions" for a theory of atoms.


Assumption (1) states a strange contraposition of the conservation and of the changes of stationary states. Indeed, the conservation of energy - a generalization of Newton's Law 1 holds true quite general, while the change of stationary states along Newton's Law 2 is restricted to classical mechanics (CM). A smooth transition from representations of CM, which axiomatically fix not only the conditions of conservation, but also the manner of change of stationary states and the equation(s) of motion, respectively (Newton, Lagrange, Hamilton), is impossible, as observed also by Heisenberg [30] and Schrödinger [43]. However, in Leonhard Euler's representation of CM [24][25], only the conditions of conservation of stationary states are fixed, while their change has to be described according to the situation under consideration. This makes it suitable to serve as starting point for a smooth transition from classical to quantum mechanics (QM). Euler's principles of the change of stationary states of bodies will be generalized to classical conservative systems as well as to quantum systems. The latter will be used for deriving the time-dependent from the time-independent Schrödinger equation.

© 2013 Enders; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Enders; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In his pioneering papers 'Quantization as Eigenvalue Problem' [43], Schrödinger has posed four requirements.

*Axiom E1*: A body preserves its stationary state at rest, unless an external cause sets it in motion.

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545

*Axiom E2*: A body preserves its stationary state of straight uniform motion, unless an external

The stationary-state variable is the velocity vector, **v** (the mass of a body is always constant). The equation of stationary state reads **v**=**0** for the state at rest and **v**=*const* for the state of straight uniform motion. The position, **r**, is variable of the state of motion, but not of stationary states, because it changes during straight uniform motion, *ie*, in the absence of (external) causes for

Following [21], the changes of position, **r,** and velocity, **v**, of a body of mass *m* subject to the

The internal transformation, **U***int*, describes the internal change, *d***r**=**v***dt*, that is independent of the external force. The external transformation, **U***ext*, describes the external change, *d***v**=(**F**/*m*)*dt*, that depends on the external force. These matrices do not commute: **U***int***U***ext*≠**U***ext***U***int*. This means, that the internal and external transformations are not reducible onto another; the

CB1) The changes of stationary-state quantities (*d***v**) are external changes; they explicitly

CB2) The change of the stationary-state quantities (*d***v**) do not explicitly depend on the

CB3) The change of state-of-motion quantities (*d***r**) are internal changes; they explicitly depend

CB4) The change of stationary-state (*d***v**) and of state-of-motion quantities (*d***r**) are independent

CB5) As soon as the external causes (**F**) vanish, the body remains in the stationary state assumed

depend solely on external causes (**F**), but not on state-of-motion quantities (**r**);

**<sup>v</sup> <sup>F</sup> 00 v 0 1 F vF** *int ext* (1)

æö æ ö æ ö æ öæ ö æö æö æö ç÷ ç ÷ ç ÷ = = + º+ ç ÷ç ÷ ç÷ ç÷ ç÷ è ø è øè ø èø èø èø èø è ø

**<sup>r</sup> 01 r 0 r0 U U**

cause changes this state.

Thus, up to order *dt*,

each of another;

changing the stationary state [50].

**2.2. Eulerian principles of change of state for single bodies**

1 1

**v 0 0**

*m m*

internal and external changes are independent each of another.

at this moment: *Z*(*t*)=const=*Z*(*t*1)=**v**(*t*1) for *t*≥*t1*, if **F**(*t*)=**0** for *t*≥*t1*.

Accounting for *ddt*=0 and *d***F**=**0**, one obtains from eq. (1)

*d dt dt*

(external) force, **F**, during the time interval *dt* are

*dt*

stationary-state quantities (**v**) themselves;

solely on stationary-state quantities (**v**) ;

*dt*


Schrödinger's requirements 1 and 2 mean, that - contrary to the very title of the papers quantization is actually *not* an eigenvalue problem. For in the latter the discretization is imposed not by the differential equation itself, but by the boundary conditions, and this is the classical discretization for standing waves in organ pipes etc. I will fulfill all four requirements by treating quantization as a selection problem. The number of stationary states of a quantum system is smaller than that of a classical system [6]. I will describe, (i), the selection of quantum systems out of the set of all mechanical systems and, (ii), the selection of their stationary states out of the continuum of classical stationary states. Earlier arguing [19][10] is improved and extended in several essential points.

### **2. Elements of an Eulerian representation of classical mechanics**

#### **2.1. Euler's axioms**

Leonhard Euler [20-25] was the first to apply the calculus to all areas of mathematics and mechanics of his time, and he developed novel methods, such as the calculus of variations and topology. Moreover, he worked out an axiomatic of mechanics, where only Newton's Law (axiom) 1 concerning the conservation of stationary states is retained as an axiom, while Newton's Laws (axioms) 2 and 3 concerning the change of stationary states are treated as problems to be solved (for a detailed account, see [19][10][45]). This allows for introducing alternative equations of motion without loosing the contact to CM.

The existence of stationary states is postulated in the following axioms (as in Newton's axioms, rotatory motion is not considered).

*Axiom E0*: Every body is either resting or moving.

This means, that the subsequent axioms E1 and E2 are not independent of each other; they exclude each other and, at once, they are in harmony with each other [22].

*Axiom E1*: A body preserves its stationary state at rest, unless an external cause sets it in motion.

*Axiom E2*: A body preserves its stationary state of straight uniform motion, unless an external cause changes this state.

The stationary-state variable is the velocity vector, **v** (the mass of a body is always constant). The equation of stationary state reads **v**=**0** for the state at rest and **v**=*const* for the state of straight uniform motion. The position, **r**, is variable of the state of motion, but not of stationary states, because it changes during straight uniform motion, *ie*, in the absence of (external) causes for changing the stationary state [50].

#### **2.2. Eulerian principles of change of state for single bodies**

Following [21], the changes of position, **r,** and velocity, **v**, of a body of mass *m* subject to the (external) force, **F**, during the time interval *dt* are

$$d\begin{pmatrix} \mathbf{r} \\ \mathbf{v} \end{pmatrix} = \begin{pmatrix} \mathbf{v}dt \\ \frac{1}{m} \mathbf{F} dt \\ \frac{1}{m} \mathbf{F} dt \end{pmatrix} = \begin{pmatrix} \mathbf{0} & \mathbf{1} \\ \mathbf{0} & \mathbf{0} \end{pmatrix} \begin{pmatrix} \mathbf{r} \\ \mathbf{v} \end{pmatrix} dt + \begin{pmatrix} \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \frac{1}{m} \mathbf{1} \end{pmatrix} \begin{pmatrix} \mathbf{0} \\ \mathbf{F} \end{pmatrix} dt = \mathbf{U}\_{int} \begin{pmatrix} \mathbf{r} \\ \mathbf{v} \end{pmatrix} + \mathbf{U}\_{ext} \begin{pmatrix} \mathbf{0} \\ \mathbf{F} \end{pmatrix} \tag{1}$$

The internal transformation, **U***int*, describes the internal change, *d***r**=**v***dt*, that is independent of the external force. The external transformation, **U***ext*, describes the external change, *d***v**=(**F**/*m*)*dt*, that depends on the external force. These matrices do not commute: **U***int***U***ext*≠**U***ext***U***int*. This means, that the internal and external transformations are not reducible onto another; the internal and external changes are independent each of another.

Thus, up to order *dt*,

In his pioneering papers 'Quantization as Eigenvalue Problem' [43], Schrödinger has posed

**1.** The "quantum equation" should "carry the quantum conditions in itself" (Second

**2.** There should be a special mathematical method for solving the stationary Schrödinger equation, which accounts for the *non*-classical character of the quantization problem, *ie*, which is different from the classical eigenvalue methods for calculating the (eigen)modes

**3.** The derivation should uniquely decide, that the energy rather than the frequency values are discretized (*ibid*., pp. 511, 519), since only the former means true quantization, while

**4.** The use of the classical expressions for the kinetic and potential energies should be

Schrödinger's requirements 1 and 2 mean, that - contrary to the very title of the papers quantization is actually *not* an eigenvalue problem. For in the latter the discretization is imposed not by the differential equation itself, but by the boundary conditions, and this is the classical discretization for standing waves in organ pipes etc. I will fulfill all four requirements by treating quantization as a selection problem. The number of stationary states of a quantum system is smaller than that of a classical system [6]. I will describe, (i), the selection of quantum systems out of the set of all mechanical systems and, (ii), the selection of their stationary states out of the continuum of classical stationary states. Earlier arguing [19][10] is improved and

**2. Elements of an Eulerian representation of classical mechanics**

alternative equations of motion without loosing the contact to CM.

exclude each other and, at once, they are in harmony with each other [22].

Leonhard Euler [20-25] was the first to apply the calculus to all areas of mathematics and mechanics of his time, and he developed novel methods, such as the calculus of variations and topology. Moreover, he worked out an axiomatic of mechanics, where only Newton's Law (axiom) 1 concerning the conservation of stationary states is retained as an axiom, while Newton's Laws (axioms) 2 and 3 concerning the change of stationary states are treated as problems to be solved (for a detailed account, see [19][10][45]). This allows for introducing

The existence of stationary states is postulated in the following axioms (as in Newton's axioms,

This means, that the subsequent axioms E1 and E2 are not independent of each other; they

the latter corresponds to the classical discretization mentioned in requirement 2;

four requirements.

544 Advances in Quantum Mechanics

Commun., p. 511);

of strings, resonators and so on (*ibid*., p. 513);

justified (Fourth Commun., p. 113).

extended in several essential points.

rotatory motion is not considered).

*Axiom E0*: Every body is either resting or moving.

**2.1. Euler's axioms**

CB1) The changes of stationary-state quantities (*d***v**) are external changes; they explicitly depend solely on external causes (**F**), but not on state-of-motion quantities (**r**);

CB2) The change of the stationary-state quantities (*d***v**) do not explicitly depend on the stationary-state quantities (**v**) themselves;

CB3) The change of state-of-motion quantities (*d***r**) are internal changes; they explicitly depend solely on stationary-state quantities (**v**) ;

CB4) The change of stationary-state (*d***v**) and of state-of-motion quantities (*d***r**) are independent each of another;

CB5) As soon as the external causes (**F**) vanish, the body remains in the stationary state assumed at this moment: *Z*(*t*)=const=*Z*(*t*1)=**v**(*t*1) for *t*≥*t1*, if **F**(*t*)=**0** for *t*≥*t1*.

Accounting for *ddt*=0 and *d***F**=**0**, one obtains from eq. (1)

$$dd\begin{pmatrix}\mathbf{r} \\ \mathbf{v}\end{pmatrix} = \begin{pmatrix}\mathbf{0} & \mathbf{1} \\ \mathbf{0} & \mathbf{0}\end{pmatrix} d\begin{pmatrix}\mathbf{r} \\ \mathbf{v}\end{pmatrix} dt = \begin{pmatrix}\mathbf{0} & \mathbf{1} \\ \mathbf{0} & \mathbf{0}\end{pmatrix} \begin{pmatrix}\mathbf{0} & \mathbf{0} \\ \mathbf{0} & \frac{1}{m}\mathbf{1}\end{pmatrix} \begin{pmatrix}\mathbf{0} \\ \mathbf{F}\end{pmatrix} dt^2 = \begin{pmatrix}\frac{1}{m}\mathbf{F} \\ \mathbf{0}\end{pmatrix} dt^2\tag{2}$$

This equation is fulfilled, if

will be other ones, again.

**Schrödinger equation**

problem [43];

tation;

additional or novel assumptions, for instance,

equation (as in wave mechanics) [27].

; *Zd Z d a a dt dt*

of the system; (4) becoming Hamilton's equations of motion.

¶ ¶ <sup>=</sup> = - ¶ ¶

; *d H dH dt dt* ¶ ¶ <sup>=</sup> - = ¶ ¶ **p r**

**r p**

Compatibility with Newton's equation of motion yields *a*=1 and *Z*(**p**,**r**)=*H*(**p**,**r**), the Hamiltonian

It may thus be not too surprising that these principles can *cum grano salis* be applied also to quantum systems. Of course, the variables, which represent of stationary states and motion,

**3. Quantization as selection problem — I. Derivation of the stationary**

**•** to restrict the energy spectrum to the values *nhν* [41][6] or to (*n*/2)*hν* [3];

**•** to suppose the existence of *h* and to abandon the classical paths [30];

The usual foundations of QM consider CM to be not sufficient and, consequently, need

**•** to "distinguish" [31] or to "select" [39][36] the values *nħ* of the action integral,*∮pdq* (*<sup>n</sup>* -

**•** to suppose the existence of *h* and of a wave function being the solution of an eigenvalue

**•** to suppose the existence of a quantum logic [2][28] and of a Hilbert space for its represen‐

**•** to suppose the existence of transition probabilities obeying the Chapman-Kolmogorov

All these approaches have eventually resorted to CM in using the classical expressions *and* the interpretations of position, momentum, potential and kinetic energies, because 'it works'. In

contrast, I will present a concrete realization of Schrödinger's 4th requirement.

integer; in contrast to CM, the action integral is *not* subject to a variational principle);

**p r** (4)

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**r p** (5)

Thus, the principles CB1...CB5 are compatible with Newton's equation of motion (published first in [20]). For their relationship to Descartes' and Huygens' principles of motion, see [10][45].

#### **2.3. Eulerian principles of change of state for Hamiltonian systems**

According to Definition 2 and the axioms, or laws of motion (Laws 1 and 2, Corollary 3), the momentum is the stationary-state variable of a body in Newton's *Principia*. The total momen‐ tum "is not changed by the action of bodies on one another" (Corollary 3). The principles CB1...CB5 remain true, if the velocity, **v**, is replaced with the momentum, **p**. For this, I will use **p** rather than **v** in what follows.

For a free body, any function of the momentum, *Z*(**p**), is a conserved quantity. If a body is subject to an external force, its momentum is no longer conserved, but becomes a state-ofmotion variable like its position. Correspondingly, *Z*(**p**)≠*const*. Suppose, that there is never‐ theless a function, *Z*0(**p**,**r**), that is constant during the motion of the body. It describes the stationary states of the system body & force. External influences (additional forces) be described through a function *Zext*(**p**,**r**,*t*) such, that *Z*(**p**,**r**)=*Z*0(**p**,**r**)+*Zext*(**p**,**r**,*t*) takes over the role of the stationary-state function.

The following principles - a generalization of CB1...CB5 - will shown to be compatible with Hamilton's equations of motion. Up to order *dt*,

CS1) The change of stationary-state quantities (*dZ*) depends solely on the external influences (*Zext*), but not on state-of-motion quantities (**p**, **r**);

CS2) The change of the stationary-state quantities (*dZ*) is independent of the stationary-state quantities (*Z*) themselves;

CS3) The changes of state-of-motion quantities (*d***p**, *d***r**) directly depend solely on stationarystate quantities (*Z*); the external influences (*Zext*) affect the state-of-motion quantities (**p**, **r**) solely indirectly (via stationary-state quantities, *Z*);

CS4) The changes of stationary-state (*dZ*) and of state-of-motion quantities (*d***p**, *d***r**) are independent each of another;

CS5) As soon as the external influences (*Zext*) vanish, the system remains in the stationary state assumed at this moment: *Z*(*t*)=const=*Z*(*t*1) for *t*≥*t1*, if *Zext*=0 for *t*>*t1*.

These principles imply the equation of change of stationary state to read

$$d\mathbf{Z} = \frac{\partial \mathbf{Z}}{\partial \mathbf{p}} \cdot d\mathbf{p} + \frac{\partial \mathbf{Z}}{\partial \mathbf{r}} \cdot d\mathbf{r} + \frac{\partial \mathbf{Z}}{\partial t} dt = \frac{\partial \mathbf{Z}\_{ext}}{\partial t} dt \tag{3}$$

This equation is fulfilled, if

2 2 1

(2)

*dd d dt* 1 *dt dt m*

**v 00 v 00 F 0 1 <sup>0</sup>**

**2.3. Eulerian principles of change of state for Hamiltonian systems**

**p** rather than **v** in what follows.

546 Advances in Quantum Mechanics

of the stationary-state function.

quantities (*Z*) themselves;

independent each of another;

Hamilton's equations of motion. Up to order *dt*,

(*Zext*), but not on state-of-motion quantities (**p**, **r**);

solely indirectly (via stationary-state quantities, *Z*);

assumed at this moment: *Z*(*t*)=const=*Z*(*t*1) for *t*≥*t1*, if *Zext*=0 for *t*>*t1*.

These principles imply the equation of change of stationary state to read

æ ö æö æ ö æ öæ ö æ ö æö ç ÷ ç÷ ç ÷ ç ÷ç ÷ ç ÷ ç÷ == = è ø è øè ø è ø èø è ø èø

**0 0 r 01 r 01 0 <sup>F</sup>**

*m*

Thus, the principles CB1...CB5 are compatible with Newton's equation of motion (published first in [20]). For their relationship to Descartes' and Huygens' principles of motion, see [10][45].

According to Definition 2 and the axioms, or laws of motion (Laws 1 and 2, Corollary 3), the momentum is the stationary-state variable of a body in Newton's *Principia*. The total momen‐ tum "is not changed by the action of bodies on one another" (Corollary 3). The principles CB1...CB5 remain true, if the velocity, **v**, is replaced with the momentum, **p**. For this, I will use

For a free body, any function of the momentum, *Z*(**p**), is a conserved quantity. If a body is subject to an external force, its momentum is no longer conserved, but becomes a state-ofmotion variable like its position. Correspondingly, *Z*(**p**)≠*const*. Suppose, that there is never‐ theless a function, *Z*0(**p**,**r**), that is constant during the motion of the body. It describes the stationary states of the system body & force. External influences (additional forces) be described through a function *Zext*(**p**,**r**,*t*) such, that *Z*(**p**,**r**)=*Z*0(**p**,**r**)+*Zext*(**p**,**r**,*t*) takes over the role

The following principles - a generalization of CB1...CB5 - will shown to be compatible with

CS1) The change of stationary-state quantities (*dZ*) depends solely on the external influences

CS2) The change of the stationary-state quantities (*dZ*) is independent of the stationary-state

CS3) The changes of state-of-motion quantities (*d***p**, *d***r**) directly depend solely on stationarystate quantities (*Z*); the external influences (*Zext*) affect the state-of-motion quantities (**p**, **r**)

CS4) The changes of stationary-state (*dZ*) and of state-of-motion quantities (*d***p**, *d***r**) are

CS5) As soon as the external influences (*Zext*) vanish, the system remains in the stationary state

*t t*

**p r** (3)

*Z ZZ Zext dZ d d dt dt*

¶ ¶¶ ¶ = × + ×+ = ¶ ¶¶¶ **p r**

$$\frac{\partial \mathcal{Z}}{\partial \mathbf{p}} = a \frac{d\mathbf{r}}{dt}; \qquad \frac{\partial \mathcal{Z}}{\partial \mathbf{r}} = -a \frac{d\mathbf{p}}{dt} \tag{4}$$

Compatibility with Newton's equation of motion yields *a*=1 and *Z*(**p**,**r**)=*H*(**p**,**r**), the Hamiltonian of the system; (4) becoming Hamilton's equations of motion.

$$\frac{d\mathbf{p}}{dt} = -\frac{\partial H}{\partial \mathbf{r}}; \qquad \frac{d\mathbf{r}}{dt} = \frac{\partial H}{\partial \mathbf{p}}\tag{5}$$

It may thus be not too surprising that these principles can *cum grano salis* be applied also to quantum systems. Of course, the variables, which represent of stationary states and motion, will be other ones, again.

### **3. Quantization as selection problem — I. Derivation of the stationary Schrödinger equation**

The usual foundations of QM consider CM to be not sufficient and, consequently, need additional or novel assumptions, for instance,


All these approaches have eventually resorted to CM in using the classical expressions *and* the interpretations of position, momentum, potential and kinetic energies, because 'it works'. In contrast, I will present a concrete realization of Schrödinger's 4th requirement.

#### **3.1. The relationship between CM and non-CM as selection problem**

In his Nobel Award speech, Schrödinger ([44] p. 315) pointed to the logical aspect, which is central to the approach exposed here.

concluded, "that the set of possible energies of microscopic systems is smaller than that for

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549

Thus, the set of possible energies of a mechanical system is either continuous, or discrete.3

Einstein's alternative does not follow from purely mechanical arguing. For this, I continue the reasoning of subsection 3.1.1. The harmony between Newtonian and non-Newtonian CMs consists in that they both build an alternative to a non-CM mechanics, in which the set of allowed configurations comprises the *whole* configuration space, *Call* = *CNewton* ⋃ *Cnon-Newton* = {**r**}. Since the motions along paths in *CNewton* and *Cnon-Newton* are incompatible each to another, the

Thus, a mechanical system either moves along paths (CM), or it moves not along paths (non-CM). The configuration of a non-CM system can assume any element of *Call* even in the

Like Euler's axioms E1 and E2, these alternatives exclude each other and, at once, are "in harmony each with another". They dialectically determine each another in the sense of

*V*(**r**) [*T*(**p**)] is no longer the contribution of the (momentum) configuration **r** (**p**) to the total energy, *E*, since it is unbounded in the domain *Call* (*Pall*). For this, I define 'limiting factors',

For completeness I note that a system is either a mechanical, or a non-mechanical one.

The question thus is how a linear oscillator *without turning points* is to be described?

**3.2. Non-classical representation of the potential and kinetic energies**

() () () ; () ()() ;

*E E*

*E E*

through the restriction of the (momentum) configuration space.

ity, I chose the one-dimensional representation of unity and set

*FE*(**r**) and *GE*(**p**) are non-negative. *FE*(**r**)<0 would mean, that *VE*

*V*(**r**) is repulsive (attractive). *GE*(**p**)<0 would mean, that *TE*

*ncl all*

*V FV E C T GT E P* = £Î = £Î

**r rr r p pp p**

*ncl all*

The contribution of the (momentum) configuration **r** (**p**) of a non-classical system to its total

3 Because of the finite resolution of measurement apparatus, the set of rational numbers is physically equivalent to the

*ncl*(**p**)], depends on the energy, because the inequality is no longer realized

(6)

*ncl*(**r**) is attractive (repulsive), while

*ncl*(**p**) becomes negative. For simplic‐

*3.1.3. Selection problem between CM and non-CM in terms of allowed configurations*

motion of non-CM systems does not proceed along paths.

*3.1.4. Selection problem between mechanics and non-mechanics*

systems of our everyday experience."

stationary states.

Hegel [29].

energy, *VE*

set of real numbers.

*FE*(**r**) and *GE*(**p**), such, that

*ncl*(**r**) [*TE*

"We are faced here with the full force of the logical opposition between an

either – or (point mechanics)

and a

both – and (wave mechanics)

This would not matter much, if the old system were to be dropped entirely and to be *replaced* by the new."

This "logical opposition" consists in a hierarchy of selection problems.1

#### *3.1.1. Selection problem between Newtonian and non-Newtionian CM*

Consider a linear undamped oscillator. For each stationary state of total energy *E*, Newton's equation of motion confines its position, *x*, to the interval between and including the classical turning points: *xmin≤x≤xmax*. Its momentum is confined as *pmin≤p≤pmax*. More generally speaking, a system obeying Newton's equation of motion moves within the sets *CNewton*={**r**|*V*(**r**)*≤E*} and *PNewton*={**p**|*T*(**p**)*≤E*}.2

Alternatively, a classical, though non-Newtonian mechanics is conceivable, where *dp*/*dt*=-**F** and *E*=*V*-*T*. A linear oscillator would move beyond the turning points: *x≤xmin* or/and *x≥xmax*. In general, the set of possible configurations equals *Cnon-Newton*={**r**|*V*(**r**)*≥E*}. The momentum configuration is no longer limited: *Pnon-Newton*=*Pall*={**p**}.

Thus, a conservative CM system obeys either the laws of Newtonian CM, where *d***p**/*dt*=+**F** and *V*(**r**(*t*))≤*E*, or the laws of non-Newtonian CM, where *d***p**/*dt*=-**F** and *V*(**r**(*t*))≥*E*. In both cases, the system moves along paths, **r**(*t*).

#### *3.1.2. Einsteinian selection problem between CM and non-CM*

For both Newtonian and non-Newtonian classical systems, the set of possible energies (the energy spectrum) is continuous. Einstein [6] has observed that this leads to a temperatureindependent specific heat (Dulong-Petit's law) for an ensemble of classical oscillators. In contrast, the discrete set of possible energies of a Planck oscillator yields a specific heat that decreases with decreasing temperature, in agreement with then recent experiments. He

<sup>1</sup> I will deviate from the exposition in [19][10] to make it shorter, though clearer and to correct few statements about the momentum configurations.

<sup>2</sup> V(r)≥0, since it equals the "disposable work storage of a system" [33]. 'r' stays for all configuration variables, 'p' stays for all momentum configuration variables of a system.

concluded, "that the set of possible energies of microscopic systems is smaller than that for systems of our everyday experience."

Thus, the set of possible energies of a mechanical system is either continuous, or discrete.3

#### *3.1.3. Selection problem between CM and non-CM in terms of allowed configurations*

Einstein's alternative does not follow from purely mechanical arguing. For this, I continue the reasoning of subsection 3.1.1. The harmony between Newtonian and non-Newtonian CMs consists in that they both build an alternative to a non-CM mechanics, in which the set of allowed configurations comprises the *whole* configuration space, *Call* = *CNewton* ⋃ *Cnon-Newton* = {**r**}. Since the motions along paths in *CNewton* and *Cnon-Newton* are incompatible each to another, the motion of non-CM systems does not proceed along paths.

Thus, a mechanical system either moves along paths (CM), or it moves not along paths (non-CM). The configuration of a non-CM system can assume any element of *Call* even in the stationary states.

#### *3.1.4. Selection problem between mechanics and non-mechanics*

**3.1. The relationship between CM and non-CM as selection problem**

"We are faced here with the full force of the logical opposition between an

central to the approach exposed here.

either – or (point mechanics)

548 Advances in Quantum Mechanics

both – and (wave mechanics)

*PNewton*={**p**|*T*(**p**)*≤E*}.2

system moves along paths, **r**(*t*).

momentum configurations.

for all momentum configuration variables of a system.

and a

In his Nobel Award speech, Schrödinger ([44] p. 315) pointed to the logical aspect, which is

This would not matter much, if the old system were to be dropped entirely and to be *replaced* by the new."

Consider a linear undamped oscillator. For each stationary state of total energy *E*, Newton's equation of motion confines its position, *x*, to the interval between and including the classical turning points: *xmin≤x≤xmax*. Its momentum is confined as *pmin≤p≤pmax*. More generally speaking, a system obeying Newton's equation of motion moves within the sets *CNewton*={**r**|*V*(**r**)*≤E*} and

Alternatively, a classical, though non-Newtonian mechanics is conceivable, where *dp*/*dt*=-**F** and *E*=*V*-*T*. A linear oscillator would move beyond the turning points: *x≤xmin* or/and *x≥xmax*. In general, the set of possible configurations equals *Cnon-Newton*={**r**|*V*(**r**)*≥E*}. The momentum

Thus, a conservative CM system obeys either the laws of Newtonian CM, where *d***p**/*dt*=+**F** and *V*(**r**(*t*))≤*E*, or the laws of non-Newtonian CM, where *d***p**/*dt*=-**F** and *V*(**r**(*t*))≥*E*. In both cases, the

For both Newtonian and non-Newtonian classical systems, the set of possible energies (the energy spectrum) is continuous. Einstein [6] has observed that this leads to a temperatureindependent specific heat (Dulong-Petit's law) for an ensemble of classical oscillators. In contrast, the discrete set of possible energies of a Planck oscillator yields a specific heat that decreases with decreasing temperature, in agreement with then recent experiments. He

1 I will deviate from the exposition in [19][10] to make it shorter, though clearer and to correct few statements about the

2 V(r)≥0, since it equals the "disposable work storage of a system" [33]. 'r' stays for all configuration variables, 'p' stays

This "logical opposition" consists in a hierarchy of selection problems.1

*3.1.1. Selection problem between Newtonian and non-Newtionian CM*

configuration is no longer limited: *Pnon-Newton*=*Pall*={**p**}.

*3.1.2. Einsteinian selection problem between CM and non-CM*

For completeness I note that a system is either a mechanical, or a non-mechanical one.

Like Euler's axioms E1 and E2, these alternatives exclude each other and, at once, are "in harmony each with another". They dialectically determine each another in the sense of Hegel [29].

The question thus is how a linear oscillator *without turning points* is to be described?

#### **3.2. Non-classical representation of the potential and kinetic energies**

*V*(**r**) [*T*(**p**)] is no longer the contribution of the (momentum) configuration **r** (**p**) to the total energy, *E*, since it is unbounded in the domain *Call* (*Pall*). For this, I define 'limiting factors', *FE*(**r**) and *GE*(**p**), such, that

$$\begin{aligned} V\_E^{ncl}(\mathbf{r}) &= F\_E(\mathbf{r}) V(\mathbf{r}) \le E; & \mathbf{r} \in \mathbb{C}^{all} \\ T\_E^{ncl}(\mathbf{p}) &= G\_E(\mathbf{p}) T(\mathbf{p}) \le E; & \mathbf{p} \in P^{all} \end{aligned} \tag{6}$$

The contribution of the (momentum) configuration **r** (**p**) of a non-classical system to its total energy, *VE ncl*(**r**) [*TE ncl*(**p**)], depends on the energy, because the inequality is no longer realized through the restriction of the (momentum) configuration space.

*FE*(**r**) and *GE*(**p**) are non-negative. *FE*(**r**)<0 would mean, that *VE ncl*(**r**) is attractive (repulsive), while *V*(**r**) is repulsive (attractive). *GE*(**p**)<0 would mean, that *TE ncl*(**p**) becomes negative. For simplic‐ ity, I chose the one-dimensional representation of unity and set

<sup>3</sup> Because of the finite resolution of measurement apparatus, the set of rational numbers is physically equivalent to the set of real numbers.

$$F\_E(\mathbf{r}) = \left| f\_E(\mathbf{r}) \right|^2; \qquad G\_E(\mathbf{p}) = \left| \mathbb{g}\_E(\mathbf{p}) \right|^2 \tag{7}$$

transform. The operators become 2x2 matrices. It remains to explore whether their free

Lacking orbits, such a system does not assume a definite configuration, say, **r**1, and momentum

stationary state, *E*. The partial contribution of the single (momentum) configuration, **r** (**p**), is determined by the weight function according to eq. (6). The total energy thus becomes

() () ()()

**r r p p**

*F V dr G T dp*

*all all*

*C P*

= + òòò òòò

d

of *E* is given by the initial preparation of the system.

*all*

òòò

*C*

*all*

òòò

*P*

*g* ¯

equations are

*all*

òòò

*C*

*all*

òòò

*P*

ˆ () () ()

**rr r**

*f H f dr*

*E E*

*E E*

*E E*

*E E*

ˆ () () ()

**pp p**

*g H g dp*

*all all*

òòò òòò

*C P*

*E E*

( ) ( )

**r p**

*F dr G dp*

The denominators have been added for dimensional reasons. The classical representation is

The occurrence of *E* on the r.h.s. makes eq. (10) to be an implicit equation for *E*. This suggests *E* to be an *internal* system parameter being determined solely by system properties like the oscillation frequency of an undamped harmonic oscillator [19]. However, as in CM, the value

The Fourier transform (9) enables me to eliminate one of the weight amplitudes from eq. (10).

<sup>ˆ</sup> ; () () ( ) () ()

**r r r r r**

<sup>ˆ</sup> ; () ( ) () () ()

*H V ir p T g g dp*

*<sup>E</sup>* are linearly independent of *fE* and *gE*, respectively, necessary conditions for fulfilling these

**p p p**

(Other positions of the weight amplitudes lead to the same results.) Since, in general, *f*

3

3

*<sup>E</sup> H V T ip r f f dr*

¶ <sup>=</sup> º +-

¶ <sup>=</sup> º +

3

3

*E E*

3 3 ( ) ( ( )); ( ) ( ( )) *E ref E ref Fr t G p t* **r rr p pp** =- = -

*ncl*(**r**1)+*TE*

3 3

 d

*ref ref*

¶

*ref ref*

ˆ ˆ ( ) ( ) ( ); ( ) ( ) ( ) *EEE E Ef H f Eg H g* **r rr p pp** = = (13)

**p p**

¶

3 3

*ncl*(**p**1). Instead, they *all* contribute to the

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Quantization as Selection Rather Than Eigenvalue Problem

(11)

(10)

551

(12)

¯ *<sup>E</sup>*and

components can be exploited for the description of new effects.

configuration, **p**1, at a given time, *t*1, with *E*=*VE*

*E*

obtained through setting

"*ψψ*¯ [≡|*ψ*|<sup>2</sup> ] is a kind of *weight function* in the configuration space of the system. The *wavemechanical* configuration of the system is a *superposition* of many, strictly speaking, of *all* kinematically possible point-mechanical configurations. Thereby, each point-mechanical configuration contributes with a certain *weight* to the true wave-mechanical configuration, the weight of which is just given through *ψψ*¯. If one likes paradoxes, one can say, the system resides quasi in all kinematically thinkable positions at the same time, though not 'equally strongly'. " ([43] 4th Commun., p. 135)

Correspondingly, I call *FE* and *GE* weight functions, *fE* and *gE* - weight amplitudes. Since *FE*(**r**) and *GE*(**p**) are dimensionless, there are reference values, *rref* and *pref*, such, that actually, *FE*(**r**)=*FE*(**r**/*rref*) and *GE*(**p**)=*GE*(**p**/*pref*). In other words, each such system has got a characteristic length in configuration and in momentum configuration space.

$$\iiint\_{C^{\text{all}}} F\_E(\frac{\mathbf{r}}{r\_{ref}}) \frac{d^3 r}{r\_{ref}^3} = \iiint G\_E(\frac{\mathbf{p}}{p\_{ref}}) \frac{d^3 p}{p\_{ref}^3} = 1 \tag{8}$$

In order to simplify the notation, I will omit *rref* and *pref* wherever possible.

#### **3.3. The stationary Schrödinger equation**

Within CM, the balance between potential, *V*(**r**), and kinetic energies, *T*(**p**), to yield *V*(**r**) +*T*(**p**)=*E*=*const* is realized through the common path parameter time, *t*: **r**=**r**(*t*), **p**=**p**(*t*); *E*=*V*(**r**(*t*)) +*T*(**p**(*t*)). This common parameterization through *t* is absent for non-classical systems not moving along paths, **r**(*t*). Consequently, the balance between the potential, *VE ncl*(**r**), and kinetic energies, *TE ncl*(**p**), is not point-wise: **p**(*t*)↔**r**(*t*), but set-wise: {**p**}↔{**r**}. Set-wise relations are mediated through integral relations.4

$$g\_E(\mathbf{p}) = \frac{1}{\left(2\pi p\_{ref}r\_{ref}\right)^{\frac{3}{2}}} \iiint e^{i\frac{\mathbf{p}\cdot\mathbf{r}}{p\_{ref}r\_{ref}}} f\_E(\mathbf{r})d^3r; \qquad f\_E(\mathbf{r}) = \frac{1}{\left(2\pi p\_{ref}r\_{ref}\right)^{\frac{3}{2}}} \iiint e^{-i\frac{\mathbf{r}\cdot\mathbf{p}}{p\_{ref}r\_{ref}}} g\_E(\mathbf{p})d^3p \tag{9}$$

In view of the symmetric normalization (8) I have chosen symmetric normalization factors.

Alternatively, it is possible to avoid complex-valued weight amplitudes (wave functions) through using 2-component vectors for them and the Hartley transform in place of the Fourier

<sup>4</sup> The most general symmetric Fourier transform contains a free complex-valued parameter [49]. It appears to be merely a rescaling of rref and pref, respectively.

transform. The operators become 2x2 matrices. It remains to explore whether their free components can be exploited for the description of new effects.

2 2 () () ; ( ) ( ) *EE E E Ff G g* **rr pp** = = (7)

*r p r p* = = òòò òòò **r p** (8)

*ncl*(**r**), and kinetic

] is a kind of *weight function* in the configuration space of the system. The *wave-*

*mechanical* configuration of the system is a *superposition* of many, strictly speaking, of *all* kinematically possible point-mechanical configurations. Thereby, each point-mechanical configuration contributes with a certain *weight* to the true wave-mechanical configuration, the weight of which is just given through *ψψ*¯. If one likes paradoxes, one can say, the system resides quasi in all kinematically thinkable positions at the same time, though not 'equally strongly'.

Correspondingly, I call *FE* and *GE* weight functions, *fE* and *gE* - weight amplitudes. Since *FE*(**r**) and *GE*(**p**) are dimensionless, there are reference values, *rref* and *pref*, such, that actually, *FE*(**r**)=*FE*(**r**/*rref*) and *GE*(**p**)=*GE*(**p**/*pref*). In other words, each such system has got a characteristic

> 3 3 3 3 () ( ) 1

Within CM, the balance between potential, *V*(**r**), and kinetic energies, *T*(**p**), to yield *V*(**r**) +*T*(**p**)=*E*=*const* is realized through the common path parameter time, *t*: **r**=**r**(*t*), **p**=**p**(*t*); *E*=*V*(**r**(*t*)) +*T*(**p**(*t*)). This common parameterization through *t* is absent for non-classical systems not

*ncl*(**p**), is not point-wise: **p**(*t*)↔**r**(*t*), but set-wise: {**p**}↔{**r**}. Set-wise relations are

*ref ref ref ref*

× × -

**p r r p p r r p** (9)

> p

*i i p r p r*

3 3

length in configuration and in momentum configuration space.

*all all E E C P ref ref ref ref d r d p F G*

In order to simplify the notation, I will omit *rref* and *pref* wherever possible.

moving along paths, **r**(*t*). Consequently, the balance between the potential, *VE*

( ) ( )

*p r p r*

*ref ref ref ref*

= = òòò òòò

2 2

3 3 2 2 1 1 ( ) () ; () ( )

*g e f dr f e g dp*

*E E E E C P*

*all all*

In view of the symmetric normalization (8) I have chosen symmetric normalization factors.

Alternatively, it is possible to avoid complex-valued weight amplitudes (wave functions) through using 2-component vectors for them and the Hartley transform in place of the Fourier

4 The most general symmetric Fourier transform contains a free complex-valued parameter [49]. It appears to be merely

"*ψψ*¯ [≡|*ψ*|<sup>2</sup>

550 Advances in Quantum Mechanics

energies, *TE*

" ([43] 4th Commun., p. 135)

**3.3. The stationary Schrödinger equation**

mediated through integral relations.4

p

a rescaling of rref and pref, respectively.

Lacking orbits, such a system does not assume a definite configuration, say, **r**1, and momentum configuration, **p**1, at a given time, *t*1, with *E*=*VE ncl*(**r**1)+*TE ncl*(**p**1). Instead, they *all* contribute to the stationary state, *E*. The partial contribution of the single (momentum) configuration, **r** (**p**), is determined by the weight function according to eq. (6). The total energy thus becomes

$$E = \frac{\iiint F\_E(\mathbf{r}) V(\mathbf{r}) d^3 r}{\iiint F\_E(\mathbf{r}) d^3 r} + \frac{\iiint G\_E(\mathbf{p}) T(\mathbf{p}) d^3 p}{\iiint G\_E(\mathbf{p}) d^3 p} \tag{10}$$

The denominators have been added for dimensional reasons. The classical representation is obtained through setting

$$F\_E(\mathbf{r}) = r\_{ref}^3 \delta(\mathbf{r} - \mathbf{r}(t)); \qquad G\_E(\mathbf{p}) = p\_{ref}^3 \delta(\mathbf{p} - \mathbf{p}(t)) \tag{11}$$

The occurrence of *E* on the r.h.s. makes eq. (10) to be an implicit equation for *E*. This suggests *E* to be an *internal* system parameter being determined solely by system properties like the oscillation frequency of an undamped harmonic oscillator [19]. However, as in CM, the value of *E* is given by the initial preparation of the system.

The Fourier transform (9) enables me to eliminate one of the weight amplitudes from eq. (10).

$$\begin{aligned} E &= \frac{\iiint \overline{f}\_E(\mathbf{r}) \hat{H}(\mathbf{r}) f\_E(\mathbf{r}) d^3 r}{\iiint \overline{f}\_E(\mathbf{r}) f\_E(\mathbf{r}) d^3 r}; & \hat{H}(\mathbf{r}) \equiv V(\mathbf{r}) + T(-ip\_{ref} r\_{ref} \frac{\partial}{\partial \mathbf{r}}) \\\ &\stackrel{\text{cell}}{=} \iiint \overline{g}\_E(\mathbf{p}) \hat{H}(\mathbf{p}) g\_E(\mathbf{p}) d^3 p \\\ &= \frac{\iiint \overline{g}\_E(\mathbf{p}) g\_E(\mathbf{p}) d^3 p}{\iiint \overline{g}\_E(\mathbf{p}) g\_E(\mathbf{p}) d^3 p}; & \hat{H}(\mathbf{p}) \equiv V(i r\_{ref} p\_{ref} \frac{\partial}{\partial \mathbf{p}}) + T(\mathbf{p}) \end{aligned} \tag{12}$$

(Other positions of the weight amplitudes lead to the same results.) Since, in general, *f* ¯ *<sup>E</sup>*and *g* ¯ *<sup>E</sup>* are linearly independent of *fE* and *gE*, respectively, necessary conditions for fulfilling these equations are

$$E f\_E(\mathbf{r}) = \hat{H}(\mathbf{r}) f\_E(\mathbf{r}); \qquad E g\_E(\mathbf{p}) = \hat{H}(\mathbf{p}) g\_E(\mathbf{p}) \tag{13}$$

Moreover, these equations hold true for the minimum of the r.h.s. of eq. (12), *ie*, for the ground state. There is no indication for a difference between the stationary-state equation for the ground state and for the states of higher energy.

*E*~*ω*. Since *ω* does not occur as a self-standing parameter, the quantization is not affecting it;

1 1 2 2

x

 + æ öæ ö æ öæ ö ç ÷ç ÷ +- = -+ = ç ÷ç ÷ è øè ø è øè ø (17)

 x

 x  xx

Quantization as Selection Rather Than Eigenvalue Problem

http://dx.doi.org/10.5772/53847

553

/4) is not. This

x

1 1 1 1 ( ) 0; () 0 2 2 2 2

Therefore, the values *a*=±½ are mathematically distinguished against all other *a*-values. The corresponding solutions, *y*±½(*ξ*), are mathematically equivalent, but physically different. *y*-½

/4) is a limiting amplitude, while *y*+½(*ξ*)=*y*+½(0)×exp(+*ξ*<sup>2</sup>

If there would be no other distinguished *a*-values, there would be only *one* state (*a*=-½). However, a system having got just *one* state is not able to exchange energy with its environ‐ ment. In order to find further distinguished *a*-values, I examine two recurrence relations for

11 11 ( , ) ( 1, ) 0; ( , ) ( 1, ) 0 22 22

æ ö æ ö æ ö æ ö ç ÷ + ++ + = - -- - = ç ÷ ç ÷ ç ÷ è ø è ø è ø è ø (18)

Such recurrence relations can be obtained *without* solving the differential equation, *viz*, from Whittaker's representation of the solutions as contour integrals ([52] 16.61; [1] 19.5). This representation has been developed well before the advance of QM; it is thus independent of

**•** interrelate solution functions with a *finite* difference between their *a*-values, *viz*, Δ*a*=±1 (this

**•** reflect the *genuine* discrete structure wanted; in particular, this structure has nothing to do with boundary conditions, since *all* solutions exhibit this discrete structure, not only

**•** realize the "conviction", that "the true laws of quantum mechanics would consist not of specific prescriptions for the *single* orbit; but, in these true laws, the elements of the whole manifold of orbits of a system are connected by equations, so that there is apparently a

certain interaction between the various orbits." ([43] Second Commun., p. 508)

6 These factors are closely related to the variables that factorize the classical Hamiltonian.

 x x

 x

*d d Ua a Ua Va a Va*

x

Schrödinger's 3rd requirement is fulfilled.

x

 x

x

(*ξ*)=*y*-½(0)×exp(-*ξ*<sup>2</sup>

x

the needs of QM.

The recurrence relations

becomes Δ*E*=±*ħω* later on);

Schrödinger's eigensolutions;

x

 x

**4.2. The mathematically distinguished solutions**

 xx

distinguishes *physically* the value *a*=-½ over the value *a*=+½.

the standard solutions of eq. (16) ([1] 19.6.1, 19.6.5).

*d d*

**•** are not related to the usual, classical solution methods;

For and only for the values *a*=±½ the l.h.s. of eq. (16) factorizes.6

*d d d d y y d d d d*

A comparison with experiments reveals, that *rrefpref*=*ħ*, which I will use in what follows. Thus, with *f <sup>E</sup>*(*r*)=*rref* 3/2 *ψE*(*r*) and *gE*(*p*)= *pref* 3/2 *ϕE*(*p*), eqs. (13) are the stationary Schrödinger equations in configuration and momentum configuration spaces.

### **4. Quantization as selection problem — II. Non-classical solution to the stationary Schrödinger equation**

As observed by Schrödinger himself (!), the eigenvalue method used by himself does *not* properly account for the *quantum* nature of quantum systems, because it applies to (and had been developed for) classical systems like strings and pipes. In what follows, I will describe a solution method being free of that deficiency.

#### **4.1. The linear oscillator**

The stationary Schrödinger equation for a linear undamped harmonic oscillator reads

$$
\hat{H}(\mathbf{x}) f\_E(\frac{\mathbf{x}}{r\_{\rm ref}}) = E f\_E(\frac{\mathbf{x}}{r\_{\rm ref}}); \qquad \hat{H}(\mathbf{x}) = \frac{m}{2} \alpha^2 \mathbf{x}^2 - \frac{\hbar^2}{2m} \frac{\hat{\mathcal{C}}^2}{\hat{\mathcal{C}} \mathbf{x}^2} \tag{14}
$$

To see its essentials I introduce dimensionless variables as5

$$\xi = \frac{x}{r\_{ref}}; \qquad r\_{ref} = \sqrt{\frac{r\_{ref} p\_{ref}}{2m\alpha o}}; \qquad y\_a(\xi) = f\_E(\xi); \qquad a = -\frac{E}{r\_{ref} p\_{ref}\alpha} \tag{15}$$

to obtain

$$\frac{d^2 y\_a(\xi)}{d\xi^2} - \left(\frac{1}{4}\xi^2 + a\right) y\_a(\xi) = 0 \tag{16}$$

This is Weber's equation [48] being one of the equations of the parabolic cylinder [1]. Despite of the reference length, *rref*, the stationary states are determined solely through the energy parameter, -*a*. In contrast to the classical oscillator, where *E*~*ω*<sup>2</sup> , the quantum oscillator exhibits

<sup>5</sup> This yields pref=2mωrref; the classical maximum values are interrelated as pmax=mωrmax. I deviate from the exposition in [19][10] to make the following clearer.

*E*~*ω*. Since *ω* does not occur as a self-standing parameter, the quantization is not affecting it; Schrödinger's 3rd requirement is fulfilled.

#### **4.2. The mathematically distinguished solutions**

Moreover, these equations hold true for the minimum of the r.h.s. of eq. (12), *ie*, for the ground state. There is no indication for a difference between the stationary-state equation for the

A comparison with experiments reveals, that *rrefpref*=*ħ*, which I will use in what follows. Thus,

**4. Quantization as selection problem — II. Non-classical solution to the**

As observed by Schrödinger himself (!), the eigenvalue method used by himself does *not* properly account for the *quantum* nature of quantum systems, because it applies to (and had been developed for) classical systems like strings and pipes. In what follows, I will describe a

The stationary Schrödinger equation for a linear undamped harmonic oscillator reads

*r r m x*

¶ <sup>=</sup> = - ¶

ˆ ˆ ( ) ( ) ( ); ( ) 2 2 *E E*

*xx m H x f Ef H x x*

; ; ( ) ( ); <sup>2</sup> *ref ref*

w

*ref a E*

*ref ref ref x E r p r yf a r m r p*

2

æ ö -+ = ç ÷

x

*d y a y <sup>d</sup>*

( ) <sup>1</sup> () 0 <sup>4</sup>

This is Weber's equation [48] being one of the equations of the parabolic cylinder [1]. Despite of the reference length, *rref*, the stationary states are determined solely through the energy

5 This yields pref=2mωrref; the classical maximum values are interrelated as pmax=mωrmax. I deviate from the exposition in

*a*

 x

x

*ref ref*

To see its essentials I introduce dimensionless variables as5

2

parameter, -*a*. In contrast to the classical oscillator, where *E*~*ω*<sup>2</sup>

2

x

*a*

x

*ϕE*(*p*), eqs. (13) are the stationary Schrödinger equations

2 2

è ø (16)

h

2

 w

, the quantum oscillator exhibits

(14)

2 2

w

 x

<sup>=</sup> <sup>=</sup> <sup>=</sup> = - (15)

ground state and for the states of higher energy.

*ψE*(*r*) and *gE*(*p*)= *pref*

**stationary Schrödinger equation**

solution method being free of that deficiency.

in configuration and momentum configuration spaces.

3/2

with *f <sup>E</sup>*(*r*)=*rref*

552 Advances in Quantum Mechanics

3/2

**4.1. The linear oscillator**

x

[19][10] to make the following clearer.

to obtain

For and only for the values *a*=±½ the l.h.s. of eq. (16) factorizes.6

$$
\left(\frac{d}{d\xi} + \frac{1}{2}\xi\right)\left(\frac{d}{d\xi} - \frac{1}{2}\xi\right)y\_{+\frac{1}{2}\xi}(\xi) = 0; \qquad \left(\frac{d}{d\xi} - \frac{1}{2}\xi\right)\left(\frac{d}{d\xi} + \frac{1}{2}\xi\right)y\_{-\frac{1}{2}\xi}(\xi) = 0. \tag{17}
$$

Therefore, the values *a*=±½ are mathematically distinguished against all other *a*-values. The corresponding solutions, *y*±½(*ξ*), are mathematically equivalent, but physically different. *y*-½ (*ξ*)=*y*-½(0)×exp(-*ξ*<sup>2</sup> /4) is a limiting amplitude, while *y*+½(*ξ*)=*y*+½(0)×exp(+*ξ*<sup>2</sup> /4) is not. This distinguishes *physically* the value *a*=-½ over the value *a*=+½.

If there would be no other distinguished *a*-values, there would be only *one* state (*a*=-½). However, a system having got just *one* state is not able to exchange energy with its environ‐ ment. In order to find further distinguished *a*-values, I examine two recurrence relations for the standard solutions of eq. (16) ([1] 19.6.1, 19.6.5).

$$\left(\frac{d}{d\xi} + \frac{1}{2}\xi\right) \mathcal{U}(a,\xi) + \left(a + \frac{1}{2}\right) \mathcal{U}(a+1,\xi) = 0; \qquad \left(\frac{d}{d\xi} - \frac{1}{2}\xi\right) \mathcal{V}(a,\xi) - \left(a - \frac{1}{2}\right) \mathcal{V}(a-1,\xi) = 0 \tag{18}$$

Such recurrence relations can be obtained *without* solving the differential equation, *viz*, from Whittaker's representation of the solutions as contour integrals ([52] 16.61; [1] 19.5). This representation has been developed well before the advance of QM; it is thus independent of the needs of QM.

The recurrence relations


<sup>6</sup> These factors are closely related to the variables that factorize the classical Hamiltonian.

Moreover, the recurrence relations divide the set of *a*-values as follows.

Set (1) *a* =..., −5/2, −3/2, −1/2; the 2nd relation (18) breaks at *a*=−½ being one of the two mathe‐ matically distinguished values found above;

1 2

+¥ -

<sup>4</sup> <sup>1</sup> <sup>2</sup> 2

**4.4. The non-classical potential energy and the tunnel effect**

x


the normalized solutions [see eq. (8)] read

of the oscillator above equals

*ncl*

x

function decreases exponentially.

Hence, the inequalities (6) are fulfilled.

*Hen*(*ξ*)|<exp(*ξ*<sup>2</sup>

*ncl*

x

1 2  xx

*<sup>n</sup> <sup>n</sup> vy n*

x

<sup>2</sup> <sup>2</sup> () 2 ! *<sup>n</sup> e He d n*

1 2


x

2, *<sup>n</sup> <sup>n</sup> y e He*

!

actual contribution of a configuration, **r**, to the total energy is not *V*(**r**), but *VE*

2

2 2 1

2 <sup>1</sup> () () ; ; 0,1,2, <sup>2</sup>


 xx

*n*

p

<sup>1</sup> ( ) ( ); 0,1,

 x

The observation of quantum particles crossing spatial domains, where *V>E*, has led to the notion 'tunnel effect' [34][38]. Being a nice illustration, this wording masks the fact, that the

In terms of the dimensionless variables (15) the dimensionless non-classical potential energy

<sup>2</sup> <sup>1</sup>

Using the recurrence formula *ξHen*(*ξ*)=*Hen+1*(*ξ*)+*nHen-1*(*ξ*) ([1] 22.7.14) and the inequality |

The occurrence of the 'smaller than' sign means, that - in contrast to the classical oscillator there are no stationary states with, (i), vanishing potential energy (in particular, the ground state is not a state at rest) and, (ii) vanishing kinetic energy (there are no turning points).

The picture of the tunnel is partially correct, in that the classical turning points are points of inflection such, that beyond them, in the forbidden domains of Newtonian CM, the wave

Notice that these results follow solely from the most general principles of state description according to Leibniz [35], Euler, Helmholtz and Schrödinger, *without* solving any stationarystate equation or equation of motion and without assuming particular boundary conditions.

*n*

p

 x

x*n* - - = < + - ¥ < < +¥ = K (24)

22 2 2 1 1 () () ( ) <sup>2</sup> 2 2

/4)√(*n*!)*k*, *k*≈1.086435 (*ibid*., 22.14.17), one can prove, that

*n n <sup>n</sup> v y e He*

xx

 p

*n*


1 2

x

 x


<sup>=</sup> ò (21)

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*ncl*(**r**)<*E*, see eq. (6).

(23)

Set (2) *a* =..., 5/2, 3/2, 1/2; the 1st relation (18) breaks at *a*=+½ being the other mathematically distinguished value found above;

Set (3) *a* = {..., -2+*ς*, -1+*ς*, *ς*, *ς*+1, *ς*+2...|-½ < *ς* < +½}; there is no break in the recurrence relations (18) for this set.

The smallest interval representing *all* solutions is the closed interval *a*=[-½,+½], all other solutions being related to it through the recursion formulae. The values *a*=-½ (set (1)) and *a*= +½ (set (2)) are mathematically distinguished, again; this time as the boundary points of that interval. All inner interval points, -½<*a*<+½ (set (3)), are mathematically equivalent among each another and, consequently, not distinguished. The physically relevant set of *a*-values is a mathematically distinguished set.

#### **4.3. The physically distinguished solutions**

The mathematically distinguished set (1) contains the physically relevant value *a*=−½, while the mathematically distinguished set (2) contains the unphysical value *a*=+½. The recurrence relations (18) show, that all functions *U*(*a*,*ξ*) with *a*-values from set (1) are limiting amplitudes, while all functions *V*(*a*,*ξ*) with *a*-values from set (2) are not. For the *a*-values of set (3), neither *U*(*a*,*ξ*), nor *V*(*a*,*ξ*) is a limiting amplitude.

Moreover, set (1) exhibits a *finite minimum* of total energy, *E*=½*rrefprefω*, while sets (2) and (3) do not. A system of sets (2) or (3) can deliver an unlimited amount of energy to its environment, it is a *perpetuum mobile* of 1st kind. This makes set (1) to be physically distinguished against sets (2) and (3).

Hence, starting from the ground state, *y*-½(*ξ*)=*y*-½(0)×exp(-*ξ*<sup>2</sup> /4), and using the recursion formula (18) for *U*(*a*,*ξ*), the physically relevant solutions are obtained as

$$y\_{-n-\frac{1}{2}}(\xi) = e^{-\frac{1}{4}\xi^2} \operatorname{He}\_n(\xi); \qquad n = 0, 1, 2, \dots \tag{19}$$

where

$$H e\_n(\xi) = (-1)^n e^{\frac{1}{2}\xi^2} \frac{d^n}{d\xi^n} e^{-\frac{1}{2}\xi^2} \tag{20}$$

is the *n*th Hermite polynomial ([1] 19.13.1). Schrödinger's boundary condition (the wave function should vanish at infinity) is fulfilled automatically.

Since ([1] 22.2)

$$\int\_{-\infty}^{+\infty} e^{-\frac{1}{2}\xi^2} H e\_n(\xi)^2 d\xi = \sqrt{2\pi}n! \tag{21}$$

the normalized solutions [see eq. (8)] read

Moreover, the recurrence relations divide the set of *a*-values as follows.

matically distinguished values found above;

distinguished value found above;

mathematically distinguished set.

**4.3. The physically distinguished solutions**

*U*(*a*,*ξ*), nor *V*(*a*,*ξ*) is a limiting amplitude.

Hence, starting from the ground state, *y*-½(*ξ*)=*y*-½(0)×exp(-*ξ*<sup>2</sup>

(18) for *U*(*a*,*ξ*), the physically relevant solutions are obtained as

<sup>4</sup> <sup>1</sup> 2

> x

x

function should vanish at infinity) is fulfilled automatically.

1 2


( ) ( ); 0,1,2, *<sup>n</sup> <sup>n</sup> y e He n* x

 x

2 2 ( ) ( 1) *<sup>n</sup> <sup>n</sup>*

*n n <sup>d</sup> He e e*

1 1 2 2

 x

*d* x

is the *n*th Hermite polynomial ([1] 19.13.1). Schrödinger's boundary condition (the wave

x



(18) for this set.

554 Advances in Quantum Mechanics

(2) and (3).

where

Since ([1] 22.2)

Set (1) *a* =..., −5/2, −3/2, −1/2; the 2nd relation (18) breaks at *a*=−½ being one of the two mathe‐

Set (2) *a* =..., 5/2, 3/2, 1/2; the 1st relation (18) breaks at *a*=+½ being the other mathematically

Set (3) *a* = {..., -2+*ς*, -1+*ς*, *ς*, *ς*+1, *ς*+2...|-½ < *ς* < +½}; there is no break in the recurrence relations

The smallest interval representing *all* solutions is the closed interval *a*=[-½,+½], all other solutions being related to it through the recursion formulae. The values *a*=-½ (set (1)) and *a*= +½ (set (2)) are mathematically distinguished, again; this time as the boundary points of that interval. All inner interval points, -½<*a*<+½ (set (3)), are mathematically equivalent among each another and, consequently, not distinguished. The physically relevant set of *a*-values is a

The mathematically distinguished set (1) contains the physically relevant value *a*=−½, while the mathematically distinguished set (2) contains the unphysical value *a*=+½. The recurrence relations (18) show, that all functions *U*(*a*,*ξ*) with *a*-values from set (1) are limiting amplitudes, while all functions *V*(*a*,*ξ*) with *a*-values from set (2) are not. For the *a*-values of set (3), neither

Moreover, set (1) exhibits a *finite minimum* of total energy, *E*=½*rrefprefω*, while sets (2) and (3) do not. A system of sets (2) or (3) can deliver an unlimited amount of energy to its environment, it is a *perpetuum mobile* of 1st kind. This makes set (1) to be physically distinguished against sets

/4), and using the recursion formula

$$\mathcal{Y}\_{-n-\frac{1}{2}}(\xi) = \frac{1}{\sqrt{2\pi n!}} e^{-\frac{1}{4}\xi^2} \operatorname{He}\_n(\xi); \qquad n = 0, 1, 2, \ldots \tag{22}$$

#### **4.4. The non-classical potential energy and the tunnel effect**

The observation of quantum particles crossing spatial domains, where *V>E*, has led to the notion 'tunnel effect' [34][38]. Being a nice illustration, this wording masks the fact, that the actual contribution of a configuration, **r**, to the total energy is not *V*(**r**), but *VE ncl*(**r**)<*E*, see eq. (6).

In terms of the dimensionless variables (15) the dimensionless non-classical potential energy of the oscillator above equals

$$
\psi\_n^{\rm ncl}(\xi) = y^2 \, \_{-n-\frac{1}{2}}(\xi) \frac{1}{2} \xi^2 = \frac{1}{2\sqrt{2\pi n}} \xi^2 e^{-\frac{1}{2}\xi^2} \, \operatorname{He}\_n(\xi)^2 \tag{23}
$$

Using the recurrence formula *ξHen*(*ξ*)=*Hen+1*(*ξ*)+*nHen-1*(*ξ*) ([1] 22.7.14) and the inequality | *Hen*(*ξ*)|<exp(*ξ*<sup>2</sup> /4)√(*n*!)*k*, *k*≈1.086435 (*ibid*., 22.14.17), one can prove, that

$$w\_n^{\text{nd}}(\xi) = y\_{-n-\frac{1}{2}}(\xi)^2 \frac{1}{2} \xi^2 < n + \frac{1}{2}; \qquad -\infty < \xi < +\infty; \qquad n = 0, 1, 2, \dots \tag{24}$$

Hence, the inequalities (6) are fulfilled.

The occurrence of the 'smaller than' sign means, that - in contrast to the classical oscillator there are no stationary states with, (i), vanishing potential energy (in particular, the ground state is not a state at rest) and, (ii) vanishing kinetic energy (there are no turning points).

The picture of the tunnel is partially correct, in that the classical turning points are points of inflection such, that beyond them, in the forbidden domains of Newtonian CM, the wave function decreases exponentially.

Notice that these results follow solely from the most general principles of state description according to Leibniz [35], Euler, Helmholtz and Schrödinger, *without* solving any stationarystate equation or equation of motion and without assuming particular boundary conditions.

#### **5. The time dependent case**

While Heisenberg [30] and Schrödinger [43] started from a time-dependent equation7 , I have worked so far with the set of all possible (momentum) configurations of systems in their stationary states, where time plays no role. In order to incorporate time, I will proceed as Newton and Euler did in the classical case and will consider first the time-dependence of the stationary states, then, the change of these states, and, finally, I will arrive at the timedependent Schrödinger equation as the equation of motion.

#### **5.1. The time dependence of the stationary states**

According to their definitions (6), the stationary weight functions, *FE*(**r**) and *GE*(**p**), are time independent. Hence, if there is a time dependence of the stationary weight amplitudes, and correspondingly of the wave functions, it is of the form

$$
\rho\_E \left( \mathbf{r}\_\prime t \right) = e^{i\phi\_\mathbb{E}\left(t\right)} \boldsymbol{\nu}\_E \left( \mathbf{r} \right); \qquad \boldsymbol{\varphi}\_\mathbb{E} \left( \mathbf{p}\_\prime t \right) = e^{i\phi\_\mathbb{E}\left(t\right)} \boldsymbol{\varphi}\_\mathbb{E} \left( \mathbf{p} \right) \tag{25}
$$

ˆ ˆ ( ) ( ) ( , ) ( ); ( , ) ( ) *i i H t H t*

ˆ ˆ (,) (,) (,) ( ,) ( ,)( ,)

**r rr p pp**

*f tH t f tdr g tH tg tdp*

(,) (,) ( ,)( ,)

The analog to the classical principles of change of state, CS1...CS5, reads as follows. Up to first

QS1) the change of stationary-state quantities (*dZ*) depends solely on the external causes (*Zext*),

QS2) the changes of the stationary-state quantities (*dZ*) are independent of the stationary-state

QS3) the changes of state-of-motion quantities (*df*, *dg*) depend directly solely on stationary-

QS4) the changes of stationary-state (*dZ*) and of state-of-motion quantities (*df*, *dg*) are inde‐

QS5) as soon as the external causes (*Zext*) vanish, the system remains in the (not necessarily

ˆ ˆˆ (,) (,) (,) (,) (,) (,) (,) (,) (,) (,) (,)

*f tf t*

+ +

**r rr r rr r r r r r**

*ncl df t H t f t f t dH t f t f t H t df t*

<sup>ˆ</sup> (,) (,) (,) (,) (,) (,) (,) (,) (,) (,) (,)

*df t f t f t df t f tH t f t f tf t f tf t*

8 The modification against CS5 is a consequence of the discreteness of the energetic spectrum.

**r rr rr r r r r r r**

ˆ ˆ <sup>3</sup> *f tH t f tdr f t H t f t* (,) (,) (,) (,) (,) (,) <sup>=</sup> òòò **r rr r r r** (31)

**r r p p**

*f t f tdr g tg tdp* = =

 f

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(30)

557

(32)

**r rp p** h h (29)

3 3

3 3

*E EE E*

 *t e t e* - - = = **r p**

yf

The analog to the Hamiltonian as classical stationary-state function is the function

*all all*

òòò òòò

*ncl C P*

*all all*

òòò òòò

*C P*

y

**5.2. The equation-of-state-change**

*Z*

quantities (*Z*) themselves;

pendent each of another;

the equation of state change becomes

=

!

=

*dZ*

state quantities (*Z*);

stationary8

Hence, writing

but not on state-of-motion quantities (*f*, *g*);

) state assumed in this moment.

ˆ (,) (,) (,) (,) (,)

*f t dH t f t f tf t*

**r rr r r**

<sup>+</sup> -

order in *dt*,

The phase, *φE*(*t*), is the same for both functions, since the Fourier transform (9) is timeindependent.

For a free particle,

$$i\,\nu\_E(\mathbf{r},t) \sim \exp\left\{i\mathbf{k}\_E \cdot \mathbf{r} - i\nu\_E t\right\}; \qquad E = \frac{\hbar^2 k\_E^2}{2m} \tag{26}$$

The group velocity equals the time-independent particle velocity.

$$\mathbf{v}\_g = \frac{d o o\_E}{d \mathbf{k}\_E} = \frac{\hbar \mathbf{k}\_E}{m} \qquad \Longrightarrow \qquad o o\_E = \frac{\hbar k\_E^2}{2m} = \frac{E}{\hbar} \tag{27}$$

Therefore,

$$\boldsymbol{\nu}\_{E}(\mathbf{r},t) = e^{-i\frac{E}{\hbar}t} \boldsymbol{\nu}\_{E}(\mathbf{r}); \qquad \boldsymbol{\phi}\_{E}(\mathbf{p},t) = e^{-i\frac{E}{\hbar}t} \boldsymbol{\phi}\_{E}(\mathbf{p}) \tag{28}$$

For later use I remark that this can be written as

<sup>7</sup> In fn. 2, p. 489, of the 2nd Commun., Schrödinger has distanced himself from the time-independent approach of the 1st Commun.

$$\boldsymbol{\nu}\_{E}(\mathbf{r},t) = e^{-\frac{i}{\hbar}\hat{H}(\mathbf{r})t}\boldsymbol{\nu}\_{E}(\mathbf{r}); \qquad \boldsymbol{\phi}\_{E}(\mathbf{p},t) = e^{-\frac{i}{\hbar}\hat{H}(\mathbf{p})t}\boldsymbol{\phi}\_{E}(\mathbf{p})\tag{29}$$

#### **5.2. The equation-of-state-change**

**5. The time dependent case**

556 Advances in Quantum Mechanics

While Heisenberg [30] and Schrödinger [43] started from a time-dependent equation7

dependent Schrödinger equation as the equation of motion.

**5.1. The time dependence of the stationary states**

y

y

**v**

y

For later use I remark that this can be written as

independent.

Therefore,

Commun.

For a free particle,

correspondingly of the wave functions, it is of the form

f

*E EE*

The group velocity equals the time-independent particle velocity.

*E*

w

**k**

worked so far with the set of all possible (momentum) configurations of systems in their stationary states, where time plays no role. In order to incorporate time, I will proceed as Newton and Euler did in the classical case and will consider first the time-dependence of the stationary states, then, the change of these states, and, finally, I will arrive at the time-

According to their definitions (6), the stationary weight functions, *FE*(**r**) and *GE*(**p**), are time independent. Hence, if there is a time dependence of the stationary weight amplitudes, and

> ( ) ( ) ( , ) ( ); ( , ) ( ) *E E i t i t E EE E t e t e*

The phase, *φE*(*t*), is the same for both functions, since the Fourier transform (9) is time-

yj

{ }

( , ) exp ; <sup>2</sup>

**r kr**×- = <sup>h</sup>

*<sup>k</sup> t i it E*

 w

*E E E*

( , ) ( ); ( , ) ( ) *E E i t i t E EE E*

7 In fn. 2, p. 489, of the 2nd Commun., Schrödinger has distanced himself from the time-independent approach of the 1st

*t e t e* - -

yf

= = Þ ==

*d k E dm m*

h h

w

*g E*

**k**

 f

 j**r rp p** = = (25)

2 2

*m*

2 2

> f

**r rp p** = = h h (28)

: (26)

*E*

<sup>h</sup> (27)

, I have

The analog to the Hamiltonian as classical stationary-state function is the function

$$Z^{\rm trcl} = \frac{\iiint \overline{f}(\mathbf{r}, t) \hat{H}(\mathbf{r}, t) f(\mathbf{r}, t) d^3 r}{\iiint \overline{f}(\mathbf{r}, t) f(\mathbf{r}, t) d^3 r} = \frac{\iiint \overline{g}(\mathbf{p}, t) \hat{H}(\mathbf{p}, t) g(\mathbf{p}, t) d^3 p}{\iiint \overline{g}(\mathbf{p}, t) g(\mathbf{p}, t) d^3 p} \tag{30}$$

The analog to the classical principles of change of state, CS1...CS5, reads as follows. Up to first order in *dt*,

QS1) the change of stationary-state quantities (*dZ*) depends solely on the external causes (*Zext*), but not on state-of-motion quantities (*f*, *g*);

QS2) the changes of the stationary-state quantities (*dZ*) are independent of the stationary-state quantities (*Z*) themselves;

QS3) the changes of state-of-motion quantities (*df*, *dg*) depend directly solely on stationarystate quantities (*Z*);

QS4) the changes of stationary-state (*dZ*) and of state-of-motion quantities (*df*, *dg*) are inde‐ pendent each of another;

QS5) as soon as the external causes (*Zext*) vanish, the system remains in the (not necessarily stationary8 ) state assumed in this moment.

Hence, writing

$$\iiint \left| \overline{f}(\mathbf{r},t)\hat{H}(\mathbf{r},t)f(\mathbf{r},t)d^3r = \left\langle f(\mathbf{r},t) \middle| \hat{H}(\mathbf{r},t) \middle| f(\mathbf{r},t) \right\rangle \tag{31}$$

the equation of state change becomes

$$\begin{split}dZ^{md} &= \frac{\left\{df(\mathbf{r},t)\left|\hat{f}\hat{\mathbf{1}}(\mathbf{r},t)\right|f(\mathbf{r},t)\right\} + \left\{f(\mathbf{r},t)\left|d\hat{I}\hat{\mathbf{1}}(\mathbf{r},t)\right|f(\mathbf{r},t)\right\} + \left\{f(\mathbf{r},t)\left|\hat{I}\hat{I}(\mathbf{r},t)\right|d f(\mathbf{r},t)\right\}}{\left\{f(\mathbf{r},t)\right\}\left|f(\mathbf{r},t)\right\}} \\ &- \frac{\left\{df(\mathbf{r},t)\left|f(\mathbf{r},t)\right\rangle + \left\{f(\mathbf{r},t)\right|d f(\mathbf{r},t)\right\}}{\left\{f(\mathbf{r},t)\left|f(\mathbf{r},t)\right\}} \frac{\left\{f(\mathbf{r},t)\left|\hat{I}\hat{I}(\mathbf{r},t)\right|f(\mathbf{r},t)\right\}}{\left\{f(\mathbf{r},t)\right\}f(\mathbf{r},t)\right\}}\tag{32} \\ &= \frac{\left\{f(\mathbf{r},t)\left|d\hat{I}\hat{I}(\mathbf{r},t)\right|f(\mathbf{r},t)\right\}}{\left\{f(\mathbf{r},t)\right\}f(\mathbf{r},t)\right\}}\end{split}\tag{33}$$

<sup>8</sup> The modification against CS5 is a consequence of the discreteness of the energetic spectrum.

#### **5.3. Derivation of the time-dependent Schrödinger equation**

The requirement in eq. (32) implies two conditions.

$$\begin{aligned} \left\langle df(\mathbf{r},t) \Big| \hat{H}(\mathbf{r},t) \Big| f(\mathbf{r},t) \right\rangle + \left\langle f(\mathbf{r},t) \Big| \hat{H}(\mathbf{r},t) \Big| df(\mathbf{r},t) \right\rangle &= 0\\ \left\langle df(\mathbf{r},t) \Big| f(\mathbf{r},t) \right\rangle + \left\langle f(\mathbf{r},t) \Big| df(\mathbf{r},t) \right\rangle &= d \left\langle f(\mathbf{r},t) \Big| f(\mathbf{r},t) \right\rangle = 0 \end{aligned} \tag{33}$$

It is often assumed, that the difference between classical and quantum systems is caused by the existence of the quantum of action. I have shown that this assumption is not necessary. It is sufficient to make different assumptions about the set of (momentum) configurations a

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559

The "logical opposition" between CM and QM observed by Schrödinger [44] is actually a dialectic relationship, which resembles that between the Finite and the Infinite. Each determi‐ nation draws a Limit, where each Limit involves the existence of something beyond it (*cf* [29] Logik I p. 145). The notion of the Finite does not exist without the notion of the Infinite (*ibid*. pp. 139ff.). The Infinite is the Other of the Finite - in turn, the Finite is the Other of the Infinite. Now, the Finite and the Infinite are not simply opposites; a border between them would contradict the very meaning of infinity. The True Infinite includes the Finite, it is the unity of

The solution of the stationary Schrödinger equation without using boundary conditions shows, that it actually does "carry the quantum conditions in itself" (Schrödinger's 1st requirement, see

CM contains the necessary means for going beyond its own frame. This way, the relationship between CM and non-CM becomes well defined, and the physical content of non-CM is formulated on equal footing with the mathematical method (and *vice versa*). An example for this is the reformulation of Einstein's criterion (the number of stationary states) in terms of

Ad-hoc assumptions, which may be suggested by experimental results, but are not supported by the axiomatic of CM, can be avoided. The wave and particle aspects can be obtained from the time-dependent Schrödinger equation and its solutions [10]. The classical path in phase space is replaced with the wave functions in space and momentum representations. The wave functions take also the role of the initial conditions, which "are not free, but also have to obey

The dynamics in space and in momentum space are treated in parallel. As a consequence, the Schrödinger equation in momentum representation is obtained at once with the one in position representation. This, too, enables one to keep maximum contact to CM and to explain, why QM is a non-classical mechanics of conservative systems, where the classical potential and kinetic energy functions and, consequently, the classical Lagrange and Hamilton functions still apply. This includes a natural explanation of "the peculiar significance of the energy in

Modern representations of CM favor equations of motion as the foundation (the variational principles belong to this class). The state variables are position and velocity (Lagrange, Laplace), or position and momentum (Hamilton). Hence, there are 6 state variables for a single body. In contrast, there are only 3 quantum numbers for a spinless particle. And there are only 3 stationary-state variables for a single body within Newton's (the 3 components of the momentum vector) and Euler's (the 3 components of the velocity vector) representations of

the transition from CM to QM than Lagrange's and Hamilton's representations.

This is another indication for the fact, that the latter are more suitable for

Introduction). Hence, it has got "maximum strength" in the sense of Einstein [8].

mechanical system can assume in its stationary states.

the Finite and the Infinite (*cf ibid*. p. 158).

recurrence relations.

certain laws" [7].

quantum mechanics" [51].

CM, respectively.9

The second condition means that there is a unitary time development operator,

$$
\hat{\mathbf{U}}(\mathbf{r}, t\_2, t\_1) f(\mathbf{r}, t\_1) = f(\mathbf{r}, t\_2); \qquad \hat{\mathbf{U}}(\mathbf{r}, t\_2, t\_1)^{\dagger} = \hat{\mathbf{U}}(\mathbf{r}, t\_2, t\_1)^{-1} \tag{34}
$$

such, that

$$
\left\langle f(\mathbf{r}, t\_2) \middle| f(\mathbf{r}, t\_2) \right\rangle = \left\langle \hat{\mathcal{U}}(\mathbf{r}, t\_2, t\_1) f(\mathbf{r}, t\_1) \middle| \hat{\mathcal{U}}(\mathbf{r}, t\_2, t\_1) f(\mathbf{r}, t\_1) \right\rangle = \left\langle f(\mathbf{r}, t\_1) \middle| f(\mathbf{r}, t\_1) \right\rangle \tag{35}
$$

Now I insert eq. (34) into the first requirement (33).

$$
\left\langle d\hat{\mathcal{U}}(\mathbf{r},t,0)f(\mathbf{r},0) \middle| \hat{H}(\mathbf{r},t) \middle| \hat{\mathcal{U}}(\mathbf{r},t,0)f(\mathbf{r},0) \right\rangle \\
+ \left\langle \hat{\mathcal{U}}(\mathbf{r},t,0)f(\mathbf{r},0) \middle| \hat{H}(\mathbf{r},t) \middle| d\hat{\mathcal{U}}(\mathbf{r},t,0)f(\mathbf{r},0) \right\rangle \\
= 0 \tag{36}
$$

The unitary solution to this equation reads *dU* ^ (*r*, *<sup>t</sup>*, 0)=*iu*(*<sup>H</sup>* ^ (*r*, *<sup>t</sup>*))*dt*, where *u*(*<sup>H</sup>* ^ ) is a realvalued rational function of the self-adjoint Hamiltonian, *H* ^ . Compatibility with the stationary case (29) yields *u*(*H* ^ )= <sup>−</sup> <sup>1</sup> <sup>ℏ</sup> *<sup>H</sup>* ^ . Hence,

$$
\hat{\mathcal{U}}(\mathbf{r}, t, 0) = \hat{P}\left(\exp\left\{\frac{-i}{\hbar}\int\_0^t \hat{H}(\mathbf{r}, t')dt'\right\}\right) \tag{37}
$$

where *P* ^ denotes Dyson's time-ordering operator [4]. The time-dependent Schrödinger equation for *f*(r,*t*) follows immediately.

The momentum representation can be derived quite analogously.

Both representations of the time-dependent Schrödinger form two equivalent equations of motion. As in the classical case, the equation of motion is a dynamic equation for nonstationary-state entities.

#### **6. Summary and conclusions**

I have presented a relatively novel approach to quantization, *viz*, quantization as selection rather than eigenvalue problem. It starts from Euler's rather than Newton's axiomatic and exploits Helmholtz's [32][33] treatment of the energy conservation law. It fulfills all four of Schrödinger's methodical requirements quoted in the Introduction.

It is often assumed, that the difference between classical and quantum systems is caused by the existence of the quantum of action. I have shown that this assumption is not necessary. It is sufficient to make different assumptions about the set of (momentum) configurations a mechanical system can assume in its stationary states.

**5.3. Derivation of the time-dependent Schrödinger equation**

ˆ ˆ (,) (,) (,) (,) (,) (,) 0

*df t H t f t f t H t df t*

**r rr r r r**

The second condition means that there is a unitary time development operator,

*df t f t f t df t d f t f t*

**rr r r rr**

(,) (,) (,) (,) (,) (,) 0

+ =

21 1 2 21 21 <sup>ˆ</sup> ˆ ˆ *U ttf t f t U tt U tt* ( , , ) ( , ) ( , ); ( , , ) ( , , )- **r rr r r** = = (34)

^ (*r*, *<sup>t</sup>*, 0)=*iu*(*<sup>H</sup>*

2 2 21 1 21 1 1 1 ˆ ˆ *f t f t U ttf t U ttf t f t f t* (, ) (, ) (, , )(, ) (, , )(, ) (, ) (, ) **rr r r r r rr** = = (35)

ˆ ˆˆ ˆ ˆ ˆ *dU t f H t U t f U t f H t dU t f* ( , ,0) ( ,0) ( , ) ( , ,0) ( ,0) ( , ,0) ( ,0) ( , ) ( , ,0) ( ,0) 0 **r r rr r r r r r r** + = (36)

0

Both representations of the time-dependent Schrödinger form two equivalent equations of motion. As in the classical case, the equation of motion is a dynamic equation for non-

I have presented a relatively novel approach to quantization, *viz*, quantization as selection rather than eigenvalue problem. It starts from Euler's rather than Newton's axiomatic and exploits Helmholtz's [32][33] treatment of the energy conservation law. It fulfills all four of

denotes Dyson's time-ordering operator [4]. The time-dependent Schrödinger

æ ö ì ü ï ï - <sup>=</sup> ç ÷ í ý è ø ï ï î þ <sup>ò</sup> **r r**

ˆˆ ˆ ( , ,0) exp ( , ') ' *<sup>t</sup> <sup>i</sup> U t P H t dt*

† 1

^ (*r*, *<sup>t</sup>*))*dt*, where *u*(*<sup>H</sup>*

<sup>h</sup> (37)

^ . Compatibility with the stationary

(33)

^ ) is a real-

+= =

The requirement in eq. (32) implies two conditions.

Now I insert eq. (34) into the first requirement (33).

The unitary solution to this equation reads *dU*

^ )= <sup>−</sup> <sup>1</sup> <sup>ℏ</sup> *<sup>H</sup>*

equation for *f*(r,*t*) follows immediately.

**6. Summary and conclusions**

valued rational function of the self-adjoint Hamiltonian, *H*

^ . Hence,

The momentum representation can be derived quite analogously.

Schrödinger's methodical requirements quoted in the Introduction.

such, that

558 Advances in Quantum Mechanics

case (29) yields *u*(*H*

stationary-state entities.

where *P* ^ The "logical opposition" between CM and QM observed by Schrödinger [44] is actually a dialectic relationship, which resembles that between the Finite and the Infinite. Each determi‐ nation draws a Limit, where each Limit involves the existence of something beyond it (*cf* [29] Logik I p. 145). The notion of the Finite does not exist without the notion of the Infinite (*ibid*. pp. 139ff.). The Infinite is the Other of the Finite - in turn, the Finite is the Other of the Infinite. Now, the Finite and the Infinite are not simply opposites; a border between them would contradict the very meaning of infinity. The True Infinite includes the Finite, it is the unity of the Finite and the Infinite (*cf ibid*. p. 158).

The solution of the stationary Schrödinger equation without using boundary conditions shows, that it actually does "carry the quantum conditions in itself" (Schrödinger's 1st requirement, see Introduction). Hence, it has got "maximum strength" in the sense of Einstein [8].

CM contains the necessary means for going beyond its own frame. This way, the relationship between CM and non-CM becomes well defined, and the physical content of non-CM is formulated on equal footing with the mathematical method (and *vice versa*). An example for this is the reformulation of Einstein's criterion (the number of stationary states) in terms of recurrence relations.

Ad-hoc assumptions, which may be suggested by experimental results, but are not supported by the axiomatic of CM, can be avoided. The wave and particle aspects can be obtained from the time-dependent Schrödinger equation and its solutions [10]. The classical path in phase space is replaced with the wave functions in space and momentum representations. The wave functions take also the role of the initial conditions, which "are not free, but also have to obey certain laws" [7].

The dynamics in space and in momentum space are treated in parallel. As a consequence, the Schrödinger equation in momentum representation is obtained at once with the one in position representation. This, too, enables one to keep maximum contact to CM and to explain, why QM is a non-classical mechanics of conservative systems, where the classical potential and kinetic energy functions and, consequently, the classical Lagrange and Hamilton functions still apply. This includes a natural explanation of "the peculiar significance of the energy in quantum mechanics" [51].

Modern representations of CM favor equations of motion as the foundation (the variational principles belong to this class). The state variables are position and velocity (Lagrange, Laplace), or position and momentum (Hamilton). Hence, there are 6 state variables for a single body. In contrast, there are only 3 quantum numbers for a spinless particle. And there are only 3 stationary-state variables for a single body within Newton's (the 3 components of the momentum vector) and Euler's (the 3 components of the velocity vector) representations of CM, respectively.9 This is another indication for the fact, that the latter are more suitable for the transition from CM to QM than Lagrange's and Hamilton's representations.

For the quantization of fields, finally, this approach yields an explanation for the fact, that, within the method of normal-mode expansion, only the temporal, but not the spatial part of the field variables is concerned (*cf* [42]). Indeed, only those variables are subject to the quantization procedure, the possible values of which are restricted by the energy law. The spatial extension of the normal modes is fixed by the boundary conditions and thus not subject to quantization. The classical field energy (density) is determined by the normal-mode amplitudes and thus limits these. As a consequence, the time-dependent coefficients in the normal-mode expansion are quantized. When formulating this expansion such, that these expansion coefficients get the dimension of length, their quantization can be performed in complete analogy to that of the harmonic oscillator, without invoking additional assumptions or new constants [10]. Moreover, one could try to quantize a field in the space spanned by independent dynamical field variables. This could separate the quantization problem from the spatial and temporal field distributions and, thus, simplify the realization of Einstein's imagination of a "spatially granular" [5] structure of the electromagnetic field.

**References**

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[2] Birkhoff, G, & Von Neumann, J. *The logic of quantum mechanics*, Ann. of Math. [2]

[3] Bohr, N. (1913). On the Constitution of Atoms and Molecules", Phil. Mag. , 26, 1-13.

[4] Dyson, F. J. (1949). The radiation theories of Tomonaga, Schwinger, and Feynman",

[5] Einstein, A. (1905). Über einen die Erzeugung und Verwandlung des Lichtes betref‐

[6] Einstein, A. (1907). Die Plancksche Theorie der Strahlung und die Theorie der spezi‐

[7] Einstein, A. (1923). Bietet die Feldtheorie Möglichkeiten für die Lösung des Quanten‐ problems?", Sitzungsber. Preuss. Ak. Wiss. phys.-math. Kl., 13. Dez., XXXIII, , 359ff.

[8] Einstein, A. (1977). *Grundzüge der Relativitätstheorie* (Akademie-Verlag, Berlin), , 132.

[9] Enders, P. (2004). *Equality and Identity and (In)distinguishability in Classical and Quan‐ tum Mechanics from the Point of View of Newton's Notion of State*, 6th Int. Symp. Frontiers of Fundamental and Computational Physics, Udine; in: Sidharth, Honsell & De An‐

[10] Enders, P. (2006). *Von der klassischen Physik zur Quantenphysik. Eine historisch-kritische deduktive Ableitung mit Anwendungsbeispielen aus der Festkörperphysik*, Berlin Heidel‐

[11] Enders, P. *Is Classical Statistical Mechanics Self-Consistent?* (A paper of honour of C. F. von Weizsäcker, 1912-2007), Progr. Phys. (2007). http://www.allbusiness.com/

[12] Enders, P. *Equality and Identity and (In)distinguishability in Classical and Quantum Me‐ chanics from the Point of View of Newton's Notion of State*, Icfai Univ. J. Phys. I ((2008). http://www.iupindia.org/108/IJP\_Classical\_and\_Quantum\_Mechanics\_71.html [13] Enders, P. *Towards the Unity of Classical Physics*, Apeiron 16 ((2009). http://

[14] Enders, P. *Huygens principle as universal model of propagation*, Latin Am. J. Phys. Educ. (2009). http://dialnet.unirioja.es/servlet/articulo?codigo=3688899, 3(2009), 19-32. [15] Enders, P. *Gibbs' Paradox in the Light of Newton's Notion of State*, Entropy (2009). http://

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www.mdpi.com/1099-4300/11/3/454, 11(2009), 454-456.

#### **Acknowledgements**

I feel highly indebted to Dr. D. Suisky with whom the basic ideas of 'quantization as selection problem' have been elaborated [46][47][18][19]. Over the years I have benefited from numerous discussions with Dr. M. Altaisky, Prof. Y. Dabaghian, Dr. M. Daumer, Dr. K. Ellmer, Dr. D. B. Fairlie, Dr. A. Förster, Prof. L. Fritsche, Prof. W. Greiner, Dr. H. Hecht, H. Hille, Prof. J. Keller, Prof. J. R. Klauder, Prof. H. Kröger, Dr. Th. Krüger, Prof. H. Lübbig, Prof. G. Mann, Prof. Matone, Prof. S. N. Mayburow, Prof. P. Mittelstaedt, Dr. R. Müller, Prof. J. G. Muroz, Prof. G. Nimtz, Prof. H. Paul, Prof. Th. Pöschel, Prof. J. Rosen, A. Rothenberg, Prof. W. P. Schleich, Prof. J. Schröter, Prof. J. Schnakenberg, Prof. J. Schröter, Dr. W. Smilga, Dr. E. V. Stefanovich, Dr. L. Teufel, Dr. R. Tomulka, Prof. H. Tributsch, Dr. M. Vogt, Prof. R. F. Werner and many more. I am indebted to Profs. M. Müller-Preußker and W. Nolting for their continuous interest and support. Early stages of this work were supported by the Deutsche Akademie der Natur‐ forscher Leopoldina [18], Prof. Th. Elsässer and Prof. E. Siegmund.

#### **Author details**

Peter Enders\*

Address all correspondence to: enders@dekasges.de

University of Applied Sciences, Wildau, Königs Wusterhausen, Germany

<sup>9</sup> Some implications of this similarity between Newton's and Euler's notions of state on the classical and Schrödinger's and Pauli's [39][40] on the quantum sides have been investigated in [9][11][12][15][16].

#### **References**

For the quantization of fields, finally, this approach yields an explanation for the fact, that, within the method of normal-mode expansion, only the temporal, but not the spatial part of the field variables is concerned (*cf* [42]). Indeed, only those variables are subject to the quantization procedure, the possible values of which are restricted by the energy law. The spatial extension of the normal modes is fixed by the boundary conditions and thus not subject to quantization. The classical field energy (density) is determined by the normal-mode amplitudes and thus limits these. As a consequence, the time-dependent coefficients in the normal-mode expansion are quantized. When formulating this expansion such, that these expansion coefficients get the dimension of length, their quantization can be performed in complete analogy to that of the harmonic oscillator, without invoking additional assumptions or new constants [10]. Moreover, one could try to quantize a field in the space spanned by independent dynamical field variables. This could separate the quantization problem from the spatial and temporal field distributions and, thus, simplify the realization of Einstein's

imagination of a "spatially granular" [5] structure of the electromagnetic field.

forscher Leopoldina [18], Prof. Th. Elsässer and Prof. E. Siegmund.

University of Applied Sciences, Wildau, Königs Wusterhausen, Germany

and Pauli's [39][40] on the quantum sides have been investigated in [9][11][12][15][16].

9 Some implications of this similarity between Newton's and Euler's notions of state on the classical and Schrödinger's

Address all correspondence to: enders@dekasges.de

I feel highly indebted to Dr. D. Suisky with whom the basic ideas of 'quantization as selection problem' have been elaborated [46][47][18][19]. Over the years I have benefited from numerous discussions with Dr. M. Altaisky, Prof. Y. Dabaghian, Dr. M. Daumer, Dr. K. Ellmer, Dr. D. B. Fairlie, Dr. A. Förster, Prof. L. Fritsche, Prof. W. Greiner, Dr. H. Hecht, H. Hille, Prof. J. Keller, Prof. J. R. Klauder, Prof. H. Kröger, Dr. Th. Krüger, Prof. H. Lübbig, Prof. G. Mann, Prof. Matone, Prof. S. N. Mayburow, Prof. P. Mittelstaedt, Dr. R. Müller, Prof. J. G. Muroz, Prof. G. Nimtz, Prof. H. Paul, Prof. Th. Pöschel, Prof. J. Rosen, A. Rothenberg, Prof. W. P. Schleich, Prof. J. Schröter, Prof. J. Schnakenberg, Prof. J. Schröter, Dr. W. Smilga, Dr. E. V. Stefanovich, Dr. L. Teufel, Dr. R. Tomulka, Prof. H. Tributsch, Dr. M. Vogt, Prof. R. F. Werner and many more. I am indebted to Profs. M. Müller-Preußker and W. Nolting for their continuous interest and support. Early stages of this work were supported by the Deutsche Akademie der Natur‐

**Acknowledgements**

560 Advances in Quantum Mechanics

**Author details**

Peter Enders\*


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berlin.de/webprogramm.html;http://www.information-philosophie.de/philosophie/ leibniz2001.html, 1247.

**Chapter 24**

**Provisional chapter**

**Entanglement, Nonlocality, Superluminal Signaling**

**Entanglement, Nonlocality, Superluminal Signaling**

Entanglement has been considered by E. Schrödinger [1] as: *The most characteristic trait of Quantum Mechanics, the one which enforces its entire departure from classical lines of thought*. Actually, the just mentioned unavoidable departure from the classical worldview raises some serious problems when entanglement of far away quantum systems is considered in conjunction with the measurement process on one of the constituents. These worries have been, once more, expressed with great lucidity by Schrödinger himself [1]: *It is rather discomforting that the theory should allow a system to be steered or piloted into one or the other type*

All those who are familiar with quantum theory will have perfectly clear the formal and physical aspects to which the above sentences make clear reference: they consist in the fact that, when dealing with a composite quantum system whose constituents are entangled and far apart, the free will choice of an observer to perform a measurement at one wing of the apparatus and the quantum reduction postulate imply that the far away state "jumps" in a state which depends crucially from the free will choice of the observer performing the measurement and from the random outcome he has got. Just to present a quite elementary case, let us consider a quantum composite system *S* = *S*(1) + *S*(2), in an entangled state

**and Cloning**

GianCarlo Ghirardi

**1. Introduction**


**and Cloning**

GianCarlo Ghirardi

http://dx.doi.org/10.5772/56429

http://dx.doi.org/10.5772/CHAPTERDOI

Additional information is available at the end of the chapter

*of state at the experimenter's mercy, in spite of his having no access to it*.

<sup>|</sup>*ψ*(1, 2)� <sup>=</sup> ∑

*i*

*pi*|*φ*(1)

In this equation (the Schmidt biorthonormal decomposition) the sets {|*φ*(1)

*<sup>i</sup>* �⊗|*γ*(2)

are two orthonormal sets of the Hilbert spaces of system *S*(1) and *S*(2), respectively, and,

*<sup>i</sup>* �, *pi* <sup>≥</sup> 0, ∑

©2012 Ghirardi, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Ghirardi; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

© 2013 Ghirardi; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

distribution, and reproduction in any medium, provided the original work is properly cited.

*i*

*pi* = 1. (1)

*<sup>i</sup>* �} and {|*γ*(2)

*<sup>i</sup>* �}

Additional information is available at the end of the chapter


**Provisional chapter**

### **Entanglement, Nonlocality, Superluminal Signaling and Cloning and Cloning**

**Entanglement, Nonlocality, Superluminal Signaling**

GianCarlo Ghirardi Additional information is available at the end of the chapter

GianCarlo Ghirardi

berlin.de/webprogramm.html;http://www.information-philosophie.de/philosophie/

[47] Suisky, D, & Enders, P. (2003). On the derivation and solution of the Schrödinger equation. Quantization as selection problem", Proc. 5th Int. Symp. Frontiers of Funda‐

[48] Weber, H. *Über die Integration der partiellen Differentialgleichung…*, Math. Ann. I

[49] Weisstein, E. W. (2012). Fourier Transform", From MathWorld-A Wolfram Web Re‐ source; http://mathworld.wolfram.com/FourierTransform.htmlMarch 13, 2012)

[51] Weyl, H. (1950). *The Theory of Groups in Quantum Mechanics* (Dover, New York), § II.8 [52] Whittaker, E. T, & Watson, G. N. *A Course of Modern Analysis*, Cambridge: Cambridge

[50] Weizsäcker, C. F. v. (2002). *Aufbau der Physik* (dtv, München, 4th ed.), 235.

1927, new ed. (1996). (Cambr. Math. Libr. Ser.)

leibniz2001.html, 1247.

((1869).

564 Advances in Quantum Mechanics

Univ. Press 4

mental Physics, Hyderabad (India), Jan. , 8-11.

Additional information is available at the end of the chapter http://dx.doi.org/10.5772/CHAPTERDOI

http://dx.doi.org/10.5772/56429

### **1. Introduction**

Entanglement has been considered by E. Schrödinger [1] as: *The most characteristic trait of Quantum Mechanics, the one which enforces its entire departure from classical lines of thought*. Actually, the just mentioned unavoidable departure from the classical worldview raises some serious problems when entanglement of far away quantum systems is considered in conjunction with the measurement process on one of the constituents. These worries have been, once more, expressed with great lucidity by Schrödinger himself [1]: *It is rather discomforting that the theory should allow a system to be steered or piloted into one or the other type of state at the experimenter's mercy, in spite of his having no access to it*.

All those who are familiar with quantum theory will have perfectly clear the formal and physical aspects to which the above sentences make clear reference: they consist in the fact that, when dealing with a composite quantum system whose constituents are entangled and far apart, the free will choice of an observer to perform a measurement at one wing of the apparatus and the quantum reduction postulate imply that the far away state "jumps" in a state which depends crucially from the free will choice of the observer performing the measurement and from the random outcome he has got. Just to present a quite elementary case, let us consider a quantum composite system *S* = *S*(1) + *S*(2), in an entangled state |*ψ*(1, 2)�:

$$|\psi(1,2)\rangle = \sum\_{i} p\_i |\phi\_i^{(1)}\rangle \otimes |\gamma\_i^{(2)}\rangle, \ p\_i \ge 0, \ \sum\_{i} p\_i = 1. \tag{1}$$

In this equation (the Schmidt biorthonormal decomposition) the sets {|*φ*(1) *<sup>i</sup>* �} and {|*γ*(2) *<sup>i</sup>* �} are two orthonormal sets of the Hilbert spaces of system *S*(1) and *S*(2), respectively, and,

Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Ghirardi; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Ghirardi; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

©2012 Ghirardi, licensee InTech. This is an open access chapter distributed under the terms of the Creative

as such, they are eigenstates of appropriate observables Φ(1) and Γ(2) of such subsystems. Suppose now that subsystem *S*(2) is subjected to a measurement of the observable Γ(2) and suppose that in the measurement the outcome (one of its eigenvalues) Γ(2) = *gr* is obtained. Then, reduction of the wave packet leads instantly to the state <sup>|</sup>*φ*(1) *<sup>r</sup>* �⊗|*γ*(2) *<sup>r</sup>* � for which one can claim that if system *S*(1) is subjected to a measurement of the observable Φ(1) the outcome Φ(1) = *fr* will occur with certainty. Since this outcome, before the measurement on system *S*(2), has a nonepistemic probability *p*<sup>2</sup> *<sup>r</sup>* of occurrence, one can state that the observation of Γ(2) has caused the instantaneous emergence at-a-distance of a definite property (which, according to quantum mechanics, one cannot consider as possessed in advance) of subsystem *S*(1), i.e. the one associated to the eigenvalue *fr* of the observable Φ(1). An analogous argument can obviously be developed without making reference to the Schmidt decomposition but to an arbitrary measurement on subsystem *S*(2), and will lead, in general, to the emergence of a different property of subsystem *S*(1) (typically the outcome of another appropriate measurement on this system becomes certain).

http://dx.doi.org/10.5772/CHAPTERDOI

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567

Entanglement, Nonlocality, Superluminal Signaling and Cloning

modern science1. In spite of that, the cleverly devised prescription of wave packet reduction, which was elaborated without having in mind relativistic potential oddities, turned out to be such that, in spite of its nonlocal nature and instantaneity, it did not allow to violate the

A brief outline of the organization of the chapter follows. After recalling the relevant aspects of the way in which quantum mechanics accounts for natural processes, we will describe various proposals for achieving faster-than-light signaling which have been put forward, and point out the reasons for which they are basically incorrect. To conclude this part we will present a general theorem showing that quantum mechanics, in its standard version,

However, the most interesting part of the debate is not the one we have just mentioned. In 1982 an analogous but quite different proposal of faster-than-light signaling has been put forward by N. Herbert [4]. The idea consisted in taking, as usual, advantage of the entanglement of far away subsystems and of wave packet reduction, but a new device was called into play: an hypothetical machine which could perform the task of creating many copies of an arbitrary state of a quantum system (a sort of "quantum xeroxing machine"). The interesting point is that at that time no general argument had been developed proving this task impossible. So, the mistaken suggestion of Herbert triggered the derivation of a theorem, the so-called no-cloning theorem, which was not known and which represents a quite relevant achievement which stays at the very basis of many important recent developments and which, besides proving that Herbert's proposal was unviable, plays a fundamental role

As it is obvious, an hypothetical quantum device allowing faster-than-light communication would give rise to a direct and serious conflict with the special theory of relativity. As already stated, such a device is excluded by quantum mechanics. This, however, does not eliminate completely the potential tension of the nonlocal nature of quantum theory with the basic principles of relativity theory. The central issue is that the instantaneous collapse of the statevector of the far away system, even if it cannot be used to transfer energy or information at a superluminal speed, indicates that, in a way or another, an action performed in a given space-time region has some "effect" on systems at a space-like separation. Einstein has qualified this aspect of the theory as "a spooky action at-a-distance" which he could not accept. A. Shimony, by stressing the fact that the theory cannot be used to actually communicate superluminally has expressed his opinion that there is some sort of "peaceful coexistence of quantum mechanics and relativity" and has suggested to speak, in place of an "action" of a "passion" at-a-distance, to stress the peculiar nature of the perfect correlations of the outcomes, which, before any measurement process, individually have a fundamentally

Recently, the just mentioned problem has seen a revival due to the elaboration of the so-called "collapse models", i.e., modifications of quantum mechanics which, on the basis of a unique, universal dynamical principle account both for the quantum evolution of microscopic

<sup>1</sup> For this one should wait the celebrated EPR paper, which appeared, just as Schödinger's paper [1] in which the instantaneity of the reduction is seen as problematic, 2 years after von Neumann's precise formalization of the effect

basic relativistic request of no-faster-than-light signaling.

cannot in principle lead to superluminal communication.

for quantum cryptography and quantum computation.

random nature, i.e., only a certain probability of occurrence.

of a measurement.

The just described process makes perfectly clear the nonlocal character of quantum mechanics, a fact that subsequently has been precisely identified by the illuminating work of J.S. Bell [2].

The situation we have just described will allow the reader to understand how it has given rise to the so called problem of faster-than-light signaling. If my action on system *B*, which takes place and is completed at a space like separation from system *A*, affects this system making instantaneously actual one of its potentialities, I can hope to be able to take advantage of this quantum peculiarity to make the observer *A* aware of the fact that I am performing some precise action on subsystem *B* at a space-like separation from him.

And, actually, this is what happened. From the seventies up to now, an innumerable set of proposals of taking advantage of entanglement and the reduction of the wave packet to achieve superluminal communication between distant observers appeared in the literature, proposals aiming to exploit this exciting possibility and to put into evidence the incompatibility of quantum mechanics with special relativity. Fortunately, as I will show in this paper, all proposal advanced so far, and, in view of some general theorems I will discuss below, all conceivable proposals of this kind, can been proven to be basically flawed in a way or another.

This chapter is devoted to discuss this important and historically crucial aspect of modern physics. As such, it has more an historical than a research interest. However, I believe that the reconsideration of the debate about this issue will be useful for the reader since many not so well known and subtle aspects of quantum mechanics will enter into play.

Before coming to a sketchy outline of the organization of the whole paper I would like to call the attention of the reader to a quite peculiar fact. When the so-called quantum measurement problem arose and was formalized by J. von Neumann [3], the attention of the scientific community was not concentrated on the possible conflicts between quantum mechanics and relativity; quantum mechanics was considered as a fundamentally nonrelativistic description of natural processes. Obviously, everybody had clear that the problem of its relativistic generalization had to be faced, but the debate concerned the nonrelativistic aspects of the theory and nobody had raised the question of possible conflicts between the two pillars of modern science1. In spite of that, the cleverly devised prescription of wave packet reduction, which was elaborated without having in mind relativistic potential oddities, turned out to be such that, in spite of its nonlocal nature and instantaneity, it did not allow to violate the basic relativistic request of no-faster-than-light signaling.

2 Quantum Mechanics

J.S. Bell [2].

or another.

as such, they are eigenstates of appropriate observables Φ(1) and Γ(2) of such subsystems. Suppose now that subsystem *S*(2) is subjected to a measurement of the observable Γ(2) and suppose that in the measurement the outcome (one of its eigenvalues) Γ(2) = *gr* is obtained. Then, reduction of the wave packet leads instantly to the state <sup>|</sup>*φ*(1) *<sup>r</sup>* �⊗|*γ*(2) *<sup>r</sup>* � for which one can claim that if system *S*(1) is subjected to a measurement of the observable Φ(1) the outcome Φ(1) = *fr* will occur with certainty. Since this outcome, before the

state that the observation of Γ(2) has caused the instantaneous emergence at-a-distance of a definite property (which, according to quantum mechanics, one cannot consider as possessed in advance) of subsystem *S*(1), i.e. the one associated to the eigenvalue *fr* of the observable Φ(1). An analogous argument can obviously be developed without making reference to the Schmidt decomposition but to an arbitrary measurement on subsystem *S*(2), and will lead, in general, to the emergence of a different property of subsystem *S*(1) (typically the outcome

The just described process makes perfectly clear the nonlocal character of quantum mechanics, a fact that subsequently has been precisely identified by the illuminating work of

The situation we have just described will allow the reader to understand how it has given rise to the so called problem of faster-than-light signaling. If my action on system *B*, which takes place and is completed at a space like separation from system *A*, affects this system making instantaneously actual one of its potentialities, I can hope to be able to take advantage of this quantum peculiarity to make the observer *A* aware of the fact that I am performing some

And, actually, this is what happened. From the seventies up to now, an innumerable set of proposals of taking advantage of entanglement and the reduction of the wave packet to achieve superluminal communication between distant observers appeared in the literature, proposals aiming to exploit this exciting possibility and to put into evidence the incompatibility of quantum mechanics with special relativity. Fortunately, as I will show in this paper, all proposal advanced so far, and, in view of some general theorems I will discuss below, all conceivable proposals of this kind, can been proven to be basically flawed in a way

This chapter is devoted to discuss this important and historically crucial aspect of modern physics. As such, it has more an historical than a research interest. However, I believe that the reconsideration of the debate about this issue will be useful for the reader since many

Before coming to a sketchy outline of the organization of the whole paper I would like to call the attention of the reader to a quite peculiar fact. When the so-called quantum measurement problem arose and was formalized by J. von Neumann [3], the attention of the scientific community was not concentrated on the possible conflicts between quantum mechanics and relativity; quantum mechanics was considered as a fundamentally nonrelativistic description of natural processes. Obviously, everybody had clear that the problem of its relativistic generalization had to be faced, but the debate concerned the nonrelativistic aspects of the theory and nobody had raised the question of possible conflicts between the two pillars of

not so well known and subtle aspects of quantum mechanics will enter into play.

*<sup>r</sup>* of occurrence, one can

measurement on system *S*(2), has a nonepistemic probability *p*<sup>2</sup>

of another appropriate measurement on this system becomes certain).

precise action on subsystem *B* at a space-like separation from him.

A brief outline of the organization of the chapter follows. After recalling the relevant aspects of the way in which quantum mechanics accounts for natural processes, we will describe various proposals for achieving faster-than-light signaling which have been put forward, and point out the reasons for which they are basically incorrect. To conclude this part we will present a general theorem showing that quantum mechanics, in its standard version, cannot in principle lead to superluminal communication.

However, the most interesting part of the debate is not the one we have just mentioned. In 1982 an analogous but quite different proposal of faster-than-light signaling has been put forward by N. Herbert [4]. The idea consisted in taking, as usual, advantage of the entanglement of far away subsystems and of wave packet reduction, but a new device was called into play: an hypothetical machine which could perform the task of creating many copies of an arbitrary state of a quantum system (a sort of "quantum xeroxing machine"). The interesting point is that at that time no general argument had been developed proving this task impossible. So, the mistaken suggestion of Herbert triggered the derivation of a theorem, the so-called no-cloning theorem, which was not known and which represents a quite relevant achievement which stays at the very basis of many important recent developments and which, besides proving that Herbert's proposal was unviable, plays a fundamental role for quantum cryptography and quantum computation.

As it is obvious, an hypothetical quantum device allowing faster-than-light communication would give rise to a direct and serious conflict with the special theory of relativity. As already stated, such a device is excluded by quantum mechanics. This, however, does not eliminate completely the potential tension of the nonlocal nature of quantum theory with the basic principles of relativity theory. The central issue is that the instantaneous collapse of the statevector of the far away system, even if it cannot be used to transfer energy or information at a superluminal speed, indicates that, in a way or another, an action performed in a given space-time region has some "effect" on systems at a space-like separation. Einstein has qualified this aspect of the theory as "a spooky action at-a-distance" which he could not accept. A. Shimony, by stressing the fact that the theory cannot be used to actually communicate superluminally has expressed his opinion that there is some sort of "peaceful coexistence of quantum mechanics and relativity" and has suggested to speak, in place of an "action" of a "passion" at-a-distance, to stress the peculiar nature of the perfect correlations of the outcomes, which, before any measurement process, individually have a fundamentally random nature, i.e., only a certain probability of occurrence.

Recently, the just mentioned problem has seen a revival due to the elaboration of the so-called "collapse models", i.e., modifications of quantum mechanics which, on the basis of a unique, universal dynamical principle account both for the quantum evolution of microscopic

<sup>1</sup> For this one should wait the celebrated EPR paper, which appeared, just as Schödinger's paper [1] in which the instantaneity of the reduction is seen as problematic, 2 years after von Neumann's precise formalization of the effect of a measurement.

systems as well as for the reduction process when macroscopic systems enter into play. Such theories, the best known of which is the one presented in ref.[ 5] usually quoted as "The GRW Theory", have been worked out with the aim of solving the macro-objectification or measurement problem at the nonrelativistic level, and the fact that they get the desired result in a clean, mathematically rigorous and conceptually precise way has raised the interest of various scientists, among them the one of Bell [6-8]. After the complete formalization of such approaches, it has been natural to start investigating whether they admit relativistic generalizations. Since they, agreeing with the quantum predictions concerning microsystems, exhibit (essentially) the same nonlocal aspects as standard quantum mechanics, the question of wether they actually can be made compatible with relativity has attracted a lot of attention. The serious work of various physicists in recent years has made clear that the program can be pursued, which means that one can have a theory inducing instantaneous collapses at-a-distance which does not violate any relativistic request. We consider it interesting to devote the conclusive part of this chapter to outline the investigations along these lines and to discuss their compatibility with the principles of special relativity.

### **2. The relevant formal aspects of the theory**

#### **2.1. The general rules**

As is well known, quantum mechanics asserts that the most accurate specification of the state of a physical system is given by the statevector |Ψ�, an element of the Hilbert space H associated to the system itself. When one deals with a statistical ensemble of identical systems, an equivalent and practical mathematical object is the statistical operator *ρ* which is the weighted sum of the statistical operators |*ψi*��*ψi*| corresponding to the pure states |*ψi*� of the members of the ensemble: *<sup>ρ</sup>* <sup>=</sup> <sup>∑</sup>*<sup>i</sup> pi*|*ψi*��*ψi*|, with *pi* <sup>≥</sup> 0, <sup>∑</sup>*<sup>i</sup> pi* <sup>=</sup> 1. For an homogeneous ensemble or an individual system in a pure state |Ψ�, the statistical operator is a projection operator: *ρ* = |Ψ��Ψ|.

The observables of the theory are represented by self-adjoint operators of the Hilbert space H, which are characterized by their eigenvalues and eigenvectors. For the observable Ω one writes its eigenvalue equation as 2:

$$
\Omega|\omega\_{k,\mathfrak{a}}\rangle = \omega\_k|\omega\_{k,\mathfrak{a}}\rangle\_\prime \tag{2}
$$

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Entanglement, Nonlocality, Superluminal Signaling and Cloning

*Pkρbef orePk*, (3)

measure: <sup>|</sup>Ψ� <sup>=</sup> <sup>∑</sup>*k*,*<sup>α</sup> ck*,*α*|*ωk*,*α*�. Then the theory asserts that the probability *<sup>P</sup>*(<sup>Ω</sup> <sup>=</sup> *<sup>ω</sup>r*|Ψ) of getting the outcome *ω<sup>r</sup>* in the measurement of Ω when the system is in the pure state |Ψ�, is

projection of the state onto the relevant eigenmanifold . This rule becomes, in the statistical operator language, *P*(Ω = *ωr*|*ρ*) = *Tr*[*Prρ*], where the symbol *Tr* means that the sum of the diagonal elements of the quantity in square brackets in an arbitrary orthonormal complete basis of H must be taken (this sum is easily proved not to depend on the chosen basis). Note that, using the complete set of the eigenstates of an operator Ω to evaluate the Trace, one immediately sees that its quantum average can be simply expressed as �Ω� = *Tr*[Ω*ρ*]. It is an important mathematical fact that the Trace operation is linear and enjoys of the following formal feature: given two arbitrary (bounded) operators Λ and Γ of H, *Tr*[ΛΓ] = *Tr*[ΓΛ].

Before concluding this subsection we must also mention the effect on the statevector of performing a measurement process. Actually, two kinds of measurements can be carried out: the nonselective and the selective ones, i.e. those in which one measures an observable without isolating the cases in which a precise eigenvalue is obtained or, alternatively, those in which one is interested only in a definite outcome. They are represented, in the statistical

*k*

From now on we will be mainly interested in dealing with quantum systems *S* composed of two constituents, *S*(1) and *S*(2). Accordingly, their statevector |Ψ(1, 2)� is an element of the tensor product H(1) ⊗ H(2) of the Hibert spaces of the constituents. As is well known, in the considered case two radically different situations may occur: in the first one the statevector is simply the direct product of precise statevectors for the constituents |Ψ(1, 2)� = |*φ*(1)�⊗|*γ*(2)�, and in such a case both constituents possess precise physical properties; alternatively, the statevector is entangled, i.e., it cannot be written in this form but it involves the superposition of factorized states, typically |Ψ(1, 2)� = ∑*<sup>i</sup> ci*|*φi*(1)�⊗|*γi*(2)�. An extremely important point concerning composite systems is the following. Suppose one has a composite system and he is interested only in the outcomes of perspective measurement processes on one of the constituents. Then, one can easily convince himself that the simplest way of dealing with this problem is to consider the reduced statistical operator *ρ*˜(1), obtained by taking the partial trace of the full statistical operator on the Hilbert space of the subsystem *S*(2) one is not interested in. At this point, to evaluate the probability of the outcomes of measurements of observables of the system of interest *S*(1), one can use the reduced statistical

*ρbef ore* → *ρa f ter* = *Pkρbef orePk*/*Tr*[*Pkρbef ore*], (4)

*<sup>ρ</sup>bef ore* <sup>→</sup> *<sup>ρ</sup>a f ter* <sup>=</sup> ∑

operator language, by the two following formal expressions:

with obvious meaning of the symbols.

**2.2. Composite systems**

2, a quantity which coincides with the square of the norm ||*Pr*|Ψ�||<sup>2</sup> of the

given by <sup>∑</sup>*<sup>α</sup>* <sup>|</sup>*cr*,*α*<sup>|</sup>

where the index *α* is associated to the possible degeneracy of the eigenvalue *ωk*. A crucial feature implied by the assumption of self-adjointness of the operators representing physical observables is that their spectral family, i.e. the projection operators *Pr* <sup>=</sup> <sup>∑</sup>*<sup>α</sup>* <sup>|</sup>*ωr*,*α*��*ωr*,*α*<sup>|</sup> on their eigenmanifolds, correspond to a resolution of the identity: ∑*<sup>r</sup> Pr* = *I*, *I* being the identity operator on H.

For what concerns the physical predictions, it is stipulated that in the case of a system in a pure state |Ψ� one has to express it as a linear combination of the eigensates of the observable (let us call it Ω) corresponding to the microscopic physical quantity which one intends to

<sup>2</sup> For simplicity we will deal with observables with a purely discrete spectrum, the changes for the continuous case been obvious.

measure: <sup>|</sup>Ψ� <sup>=</sup> <sup>∑</sup>*k*,*<sup>α</sup> ck*,*α*|*ωk*,*α*�. Then the theory asserts that the probability *<sup>P</sup>*(<sup>Ω</sup> <sup>=</sup> *<sup>ω</sup>r*|Ψ) of getting the outcome *ω<sup>r</sup>* in the measurement of Ω when the system is in the pure state |Ψ�, is given by <sup>∑</sup>*<sup>α</sup>* <sup>|</sup>*cr*,*α*<sup>|</sup> 2, a quantity which coincides with the square of the norm ||*Pr*|Ψ�||<sup>2</sup> of the projection of the state onto the relevant eigenmanifold . This rule becomes, in the statistical operator language, *P*(Ω = *ωr*|*ρ*) = *Tr*[*Prρ*], where the symbol *Tr* means that the sum of the diagonal elements of the quantity in square brackets in an arbitrary orthonormal complete basis of H must be taken (this sum is easily proved not to depend on the chosen basis). Note that, using the complete set of the eigenstates of an operator Ω to evaluate the Trace, one immediately sees that its quantum average can be simply expressed as �Ω� = *Tr*[Ω*ρ*]. It is an important mathematical fact that the Trace operation is linear and enjoys of the following formal feature: given two arbitrary (bounded) operators Λ and Γ of H, *Tr*[ΛΓ] = *Tr*[ΓΛ].

Before concluding this subsection we must also mention the effect on the statevector of performing a measurement process. Actually, two kinds of measurements can be carried out: the nonselective and the selective ones, i.e. those in which one measures an observable without isolating the cases in which a precise eigenvalue is obtained or, alternatively, those in which one is interested only in a definite outcome. They are represented, in the statistical operator language, by the two following formal expressions:

$$
\rho\_{before} \to \rho\_{after} = \sum\_{k} P\_k \rho\_{before} P\_{k\prime} \tag{3}
$$

$$
\rho\_{\text{before}} \rightarrow \rho\_{\text{after}} = \text{P}\_{\text{k}} \rho\_{\text{before}} \text{P}\_{\text{k}} / \text{Tr} \left[ \text{P}\_{\text{k}} \rho\_{\text{before}} \right], \tag{4}
$$

with obvious meaning of the symbols.

#### **2.2. Composite systems**

4 Quantum Mechanics

systems as well as for the reduction process when macroscopic systems enter into play. Such theories, the best known of which is the one presented in ref.[ 5] usually quoted as "The GRW Theory", have been worked out with the aim of solving the macro-objectification or measurement problem at the nonrelativistic level, and the fact that they get the desired result in a clean, mathematically rigorous and conceptually precise way has raised the interest of various scientists, among them the one of Bell [6-8]. After the complete formalization of such approaches, it has been natural to start investigating whether they admit relativistic generalizations. Since they, agreeing with the quantum predictions concerning microsystems, exhibit (essentially) the same nonlocal aspects as standard quantum mechanics, the question of wether they actually can be made compatible with relativity has attracted a lot of attention. The serious work of various physicists in recent years has made clear that the program can be pursued, which means that one can have a theory inducing instantaneous collapses at-a-distance which does not violate any relativistic request. We consider it interesting to devote the conclusive part of this chapter to outline the investigations along these lines and

As is well known, quantum mechanics asserts that the most accurate specification of the state of a physical system is given by the statevector |Ψ�, an element of the Hilbert space H associated to the system itself. When one deals with a statistical ensemble of identical systems, an equivalent and practical mathematical object is the statistical operator *ρ* which is the weighted sum of the statistical operators |*ψi*��*ψi*| corresponding to the pure states |*ψi*� of the members of the ensemble: *<sup>ρ</sup>* <sup>=</sup> <sup>∑</sup>*<sup>i</sup> pi*|*ψi*��*ψi*|, with *pi* <sup>≥</sup> 0, <sup>∑</sup>*<sup>i</sup> pi* <sup>=</sup> 1. For an homogeneous ensemble or an individual system in a pure state |Ψ�, the statistical operator is a projection

The observables of the theory are represented by self-adjoint operators of the Hilbert space H, which are characterized by their eigenvalues and eigenvectors. For the observable Ω one

where the index *α* is associated to the possible degeneracy of the eigenvalue *ωk*. A crucial feature implied by the assumption of self-adjointness of the operators representing physical observables is that their spectral family, i.e. the projection operators *Pr* <sup>=</sup> <sup>∑</sup>*<sup>α</sup>* <sup>|</sup>*ωr*,*α*��*ωr*,*α*<sup>|</sup> on their eigenmanifolds, correspond to a resolution of the identity: ∑*<sup>r</sup> Pr* = *I*, *I* being the

For what concerns the physical predictions, it is stipulated that in the case of a system in a pure state |Ψ� one has to express it as a linear combination of the eigensates of the observable (let us call it Ω) corresponding to the microscopic physical quantity which one intends to

<sup>2</sup> For simplicity we will deal with observables with a purely discrete spectrum, the changes for the continuous case

<sup>Ω</sup>|*ωk*,*α*� <sup>=</sup> *<sup>ω</sup>k*|*ωk*,*α*�, (2)

to discuss their compatibility with the principles of special relativity.

**2. The relevant formal aspects of the theory**

**2.1. The general rules**

operator: *ρ* = |Ψ��Ψ|.

identity operator on H.

been obvious.

writes its eigenvalue equation as 2:

From now on we will be mainly interested in dealing with quantum systems *S* composed of two constituents, *S*(1) and *S*(2). Accordingly, their statevector |Ψ(1, 2)� is an element of the tensor product H(1) ⊗ H(2) of the Hibert spaces of the constituents. As is well known, in the considered case two radically different situations may occur: in the first one the statevector is simply the direct product of precise statevectors for the constituents |Ψ(1, 2)� = |*φ*(1)�⊗|*γ*(2)�, and in such a case both constituents possess precise physical properties; alternatively, the statevector is entangled, i.e., it cannot be written in this form but it involves the superposition of factorized states, typically |Ψ(1, 2)� = ∑*<sup>i</sup> ci*|*φi*(1)�⊗|*γi*(2)�.

An extremely important point concerning composite systems is the following. Suppose one has a composite system and he is interested only in the outcomes of perspective measurement processes on one of the constituents. Then, one can easily convince himself that the simplest way of dealing with this problem is to consider the reduced statistical operator *ρ*˜(1), obtained by taking the partial trace of the full statistical operator on the Hilbert space of the subsystem *S*(2) one is not interested in. At this point, to evaluate the probability of the outcomes of measurements of observables of the system of interest *S*(1), one can use the reduced statistical operator and the same prescriptions we have used for the general case <sup>3</sup> :

$$\begin{aligned} \tilde{\rho}(1) &= Tr^{(2)}[\rho(1,2)]; \; P(\Phi^{(1)} = f\_r | \rho(1,2)) = Tr^{(1+2)}[P\_r^{(1)}\rho(1,2)] \equiv Tr^{(1)}[P\_r^{(1)}\tilde{\rho}(1)];\\ \langle \Phi^{(1)} \rangle &= Tr^{(1)}[\Phi^{(1)}\tilde{\rho}(1)]. \end{aligned} \tag{5}$$

It goes without saying that the operator *<sup>P</sup>*(1) *<sup>r</sup>* in the previous equation is the projection operator onto the linear eigenmanifold associated to the eigenvalue *fr* of Φ(1).

#### **2.3. von Neumann's ideal measurement scheme and its limitations**

For the subsequent analysis it is important to briefly recall the so-called Ideal Measurement Scheme introduced by von Neumann in his celebrated book, ref.[3], and its limitations. The idea is quite simple: we are interested in "measuring" a microscopic observable, which is not directly accessible to our senses. Suppose then we have a microsystem *<sup>s</sup>* in a state |*ϕ*(*s*) *i* � which is in an eigenstate of a micro-observable Σ(*s*) pertaining to the eigenvalue *si*. How can one ascertain such a value, which, if a measurement is performed, according to the quantum rules will be obtained with certainity ? Von Neumann assumed that there exists a macroscopic object *<sup>M</sup>* which can be prepared in a ready state |*m*0� and can be put into interaction with the microsystem. The interaction leaves unaltered the microstate while it induces, in a quite short time interval, the following evolution of the microsystem+apparatus:

$$|\langle \mathfrak{p}\_{\mathrm{i}} \rangle \otimes |\mathfrak{m}\_{0} \rangle \to |\mathfrak{q}\_{\mathrm{i}} \rangle \otimes |\mathfrak{m}\_{\mathrm{i}} \rangle,\tag{6}$$

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2. We will not enter, here, in this deep

Entanglement, Nonlocality, Superluminal Signaling and Cloning

Eq.(7) has given rise to one of the most debated problems of quantum mechanics, the so-called measurement or macro-objectification problem. In fact its r.h.s. corresponds to an entangled state of the system and the apparatus and in no way whatsoever to a state corresponding to a precise outcome<sup>4</sup> . The orthodox way out from this puzzle consists in resorting to the postulate of wave packet reduction: when a superposition of different macrostates emerges, a sudden change of the statevector occurs, so that one has to replace the r.h.s. of the previous equation with one of its terms, let us say |*ϕj*�⊗|*mj*�. This

debate, we simply mention that it amounts to accept (as many scientists did) that the linear character of the theory is violated (the reduction process is nonlinear and stochastic while the quantum evolution is linear and deterministic) at an appropriate ( but not precisely specified) macroscopic level. Incidentally, von Neumann himself has proposed that the transition from the superposition to one of its terms occurs when a conscious observer becomes aware of the outcome (reduction by consciousness). Recently, various proposals of theories which, on the basis of a unique dynamical principle, account both for the linear nature of the evolution at the microscopic level as well as for the discontinuous changes (collapses) occurring when macrosystems are involved, have been put forward. We refer the reader to ref.[10] for an

There are limitations to the von Neumann ideal scheme that we must mention because they have played a role in the refutation of some proposals of faster-than-light communication. Such limitations have been identified by Wigner [11], Araki and Yanase [12,13] in a series of interesting papers and subsequently they have been generalized in refs.[14,15]. The analysis by these authors takes into account the existence of additive conserved quantities for the system+apparatus system to derive precise conditions on a process like the one of Eq.(6) which stays at the basis of the von Neumann treatment. Let us summarize the procedure in a sketchy way. The process described by Eq.(6) represents the unitary evolution of the system+apparatus during the measurement process of the observable Σ(*s*) with eigenstates |*ϕi*�. Let us therefore write it as: *<sup>U</sup>*|*ϕi*, *<sup>m</sup>*0� = |*ϕi*, *mi*�. Let us suppose that there exists an additive conserved quantity Γ = *γ*(*s*) ⊗ *I*(*A*) + *I*(*s*) ⊗ *γ*(*A*) of the whole system and let us evaluate the matrix element of <sup>Γ</sup>, �*ϕi*, *<sup>m</sup>*0|Γ|*ϕj*, *<sup>m</sup>*0� by taking into account that <sup>Γ</sup> commutes

)|*ϕjm*0� = �*ϕi*, *<sup>m</sup>*0|*U*†(*γ*(*s*) ⊗ *<sup>I</sup>*

<sup>4</sup> Note that in ref.[9] it has been proven that the occurrence of the embarrassing superpositions of macroscopically different states does not require that the measurement proceeds according to the ideal scheme of von Neumann. The same conclusion can be derived as a consequence of the necessary request that quantum mechanics governes the whole process and that one can perform a reasonably reliable measurement ascertaining the microproperty of the

(*A*) + *I*

(*s*) ⊗ *γ*(*A*)

)*U*|*ϕjm*0�. (8)

specific reduction occurs with probability |*cj*|

exhaustive analysis of such model theories.

with *U*, which implies :

(*A*) + *I*

(*s*) ⊗ *γ*(*A*)

�*ϕi*, *<sup>m</sup>*0|(*γ*(*s*) ⊗ *<sup>I</sup>*

measured system.

*2.3.1. Limits to the ideal scheme due to additive conservation laws*

where the states |*mi*� are assumed to be orthogonal (�*mi*|*mj*� = *<sup>δ</sup>i*,*j*), macroscopically and perceptively different (typically they are associated to different locations of the pointer of the macro-apparatus). Then, an observer, by looking at the measuring apparatus gets immediately the desired information concerning the value (*si*) of the microvariable.

The scheme is usually qualified as ideal because, in practice, the final apparatus states are not perfectly orthogonal and because very often the state of the microsystem is disturbed (or even the system is absorbed) in the measurement. The just mentioned scheme has an immediate important implication; the validity of Eq.(6) and the linear nature of Schrödinger's evolution equation imply that if one triggers the macroapparatus in its ready state with a superposition of the eigenstates of Σ(*s*), one has:

$$\sum\_{i} c\_{i} |\varphi\_{i}\rangle \otimes |m\_{0}\rangle \to \sum\_{i} c\_{i} |\varphi\_{i}\rangle \otimes |m\_{i}\rangle,\tag{7}$$

which is an entangled state of the microsystem and the macroapparatus.

<sup>3</sup> An elementary way to see this is to evaluate the probabilities of the joint outcomes of the measurement of a pair of observables, one for system *S*(1) and one for *S*(2), and then to sum on all possible outcomes of the measurement on the system we are not interested in.

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Eq.(7) has given rise to one of the most debated problems of quantum mechanics, the so-called measurement or macro-objectification problem. In fact its r.h.s. corresponds to an entangled state of the system and the apparatus and in no way whatsoever to a state corresponding to a precise outcome<sup>4</sup> . The orthodox way out from this puzzle consists in resorting to the postulate of wave packet reduction: when a superposition of different macrostates emerges, a sudden change of the statevector occurs, so that one has to replace the r.h.s. of the previous equation with one of its terms, let us say |*ϕj*�⊗|*mj*�. This specific reduction occurs with probability |*cj*| 2. We will not enter, here, in this deep debate, we simply mention that it amounts to accept (as many scientists did) that the linear character of the theory is violated (the reduction process is nonlinear and stochastic while the quantum evolution is linear and deterministic) at an appropriate ( but not precisely specified) macroscopic level. Incidentally, von Neumann himself has proposed that the transition from the superposition to one of its terms occurs when a conscious observer becomes aware of the outcome (reduction by consciousness). Recently, various proposals of theories which, on the basis of a unique dynamical principle, account both for the linear nature of the evolution at the microscopic level as well as for the discontinuous changes (collapses) occurring when macrosystems are involved, have been put forward. We refer the reader to ref.[10] for an exhaustive analysis of such model theories.

#### *2.3.1. Limits to the ideal scheme due to additive conservation laws*

6 Quantum Mechanics

�Φ(1)

*ρ*˜(1) = *Tr*(2)

� = *Tr*(1)

of the eigenstates of Σ(*s*), one has:

the system we are not interested in.

∑ *i*

[Φ(1)

operator and the same prescriptions we have used for the general case <sup>3</sup> :

[*ρ*(1, 2)]; *P*(Φ(1) = *fr*|*ρ*(1, 2)) = *Tr*(1+2)

operator onto the linear eigenmanifold associated to the eigenvalue *fr* of Φ(1).

**2.3. von Neumann's ideal measurement scheme and its limitations**

It goes without saying that the operator *<sup>P</sup>*(1) *<sup>r</sup>* in the previous equation is the projection

For the subsequent analysis it is important to briefly recall the so-called Ideal Measurement Scheme introduced by von Neumann in his celebrated book, ref.[3], and its limitations. The idea is quite simple: we are interested in "measuring" a microscopic observable, which is not directly accessible to our senses. Suppose then we have a microsystem *<sup>s</sup>* in a state |*ϕ*(*s*)

which is in an eigenstate of a micro-observable Σ(*s*) pertaining to the eigenvalue *si*. How can one ascertain such a value, which, if a measurement is performed, according to the quantum rules will be obtained with certainity ? Von Neumann assumed that there exists a macroscopic object *<sup>M</sup>* which can be prepared in a ready state |*m*0� and can be put into interaction with the microsystem. The interaction leaves unaltered the microstate while it induces, in a quite short time interval, the following evolution of the microsystem+apparatus:

where the states |*mi*� are assumed to be orthogonal (�*mi*|*mj*� = *<sup>δ</sup>i*,*j*), macroscopically and perceptively different (typically they are associated to different locations of the pointer of the macro-apparatus). Then, an observer, by looking at the measuring apparatus gets

The scheme is usually qualified as ideal because, in practice, the final apparatus states are not perfectly orthogonal and because very often the state of the microsystem is disturbed (or even the system is absorbed) in the measurement. The just mentioned scheme has an immediate important implication; the validity of Eq.(6) and the linear nature of Schrödinger's evolution equation imply that if one triggers the macroapparatus in its ready state with a superposition

*i*

<sup>3</sup> An elementary way to see this is to evaluate the probabilities of the joint outcomes of the measurement of a pair of observables, one for system *S*(1) and one for *S*(2), and then to sum on all possible outcomes of the measurement on

immediately the desired information concerning the value (*si*) of the microvariable.

*ci*|*ϕi*�⊗|*m*0� → ∑

which is an entangled state of the microsystem and the macroapparatus.

[*P*(1) *<sup>r</sup> <sup>ρ</sup>*(1, 2)] <sup>≡</sup> *Tr*(1)

*ρ*˜(1)]. (5)


*ci*|*ϕi*�⊗|*mi*�, (7)

[*P*(1) *<sup>r</sup> <sup>ρ</sup>*˜(1)];

*i* �

There are limitations to the von Neumann ideal scheme that we must mention because they have played a role in the refutation of some proposals of faster-than-light communication. Such limitations have been identified by Wigner [11], Araki and Yanase [12,13] in a series of interesting papers and subsequently they have been generalized in refs.[14,15]. The analysis by these authors takes into account the existence of additive conserved quantities for the system+apparatus system to derive precise conditions on a process like the one of Eq.(6) which stays at the basis of the von Neumann treatment. Let us summarize the procedure in a sketchy way. The process described by Eq.(6) represents the unitary evolution of the system+apparatus during the measurement process of the observable Σ(*s*) with eigenstates |*ϕi*�. Let us therefore write it as: *<sup>U</sup>*|*ϕi*, *<sup>m</sup>*0� = |*ϕi*, *mi*�. Let us suppose that there exists an additive conserved quantity Γ = *γ*(*s*) ⊗ *I*(*A*) + *I*(*s*) ⊗ *γ*(*A*) of the whole system and let us evaluate the matrix element of <sup>Γ</sup>, �*ϕi*, *<sup>m</sup>*0|Γ|*ϕj*, *<sup>m</sup>*0� by taking into account that <sup>Γ</sup> commutes with *U*, which implies :

$$\langle \varphi\_{\dot{\nu}} \cdot m\_0 | (\gamma^{(s)} \otimes I^{(A)} + I^{(s)} \otimes \gamma^{(A)}) | \varphi\_{\dot{f}} m\_0 \rangle = \langle \varphi\_{\dot{\nu}} \cdot m\_0 | \mathcal{U}^{\dagger} (\gamma^{(s)} \otimes I^{(A)} + I^{(s)} \otimes \gamma^{(A)}) \mathcal{U} | \varphi\_{\dot{f}} m\_0 \rangle. \tag{8}$$

<sup>4</sup> Note that in ref.[9] it has been proven that the occurrence of the embarrassing superpositions of macroscopically different states does not require that the measurement proceeds according to the ideal scheme of von Neumann. The same conclusion can be derived as a consequence of the necessary request that quantum mechanics governes the whole process and that one can perform a reasonably reliable measurement ascertaining the microproperty of the measured system.

We then have:

$$\begin{split} \langle \boldsymbol{\varrho}\_{i}, \boldsymbol{m}\_{0} | (\boldsymbol{\gamma}^{(s)} \otimes \boldsymbol{I}^{(A)} + \boldsymbol{I}^{(s)} \otimes \boldsymbol{\gamma}^{(A)}) | \boldsymbol{\varrho}\_{j} \boldsymbol{m}\_{0} \rangle &= \langle \boldsymbol{\varrho}\_{i} | \boldsymbol{\gamma}^{(s)} | \boldsymbol{\varrho}\_{j} \rangle + \delta\_{ij} \langle \boldsymbol{m}\_{0} | \boldsymbol{\gamma}^{(A)} | \boldsymbol{m}\_{0} \rangle = \\ \langle \boldsymbol{\varrho}\_{i}, \boldsymbol{m}\_{0} | \boldsymbol{\Lambda}^{\dagger} (\boldsymbol{\gamma}^{(s)} \otimes \boldsymbol{I}^{(A)} + \boldsymbol{I}^{(s)} \otimes \boldsymbol{\gamma}^{(A)}) \boldsymbol{\Lambda} | \boldsymbol{\varrho}\_{j} \boldsymbol{m}\_{0} \rangle &= \langle \boldsymbol{\varrho}\_{i}, \boldsymbol{m}\_{i} | (\boldsymbol{\gamma}^{(s)} \otimes \boldsymbol{I}^{(A)} + \boldsymbol{I}^{(s)} \otimes \boldsymbol{\gamma}^{(A)}) | \boldsymbol{\varrho}\_{j}, \boldsymbol{m}\_{j} \rangle = \\ \delta\_{ij} \langle \boldsymbol{\varrho}\_{i} | \boldsymbol{\gamma}^{(s)} | \boldsymbol{\varrho}\_{j} \rangle + \delta\_{ij} \langle \boldsymbol{m}\_{i} | \boldsymbol{\gamma}^{(A)} | \boldsymbol{m}\_{j} \rangle. \end{split} \tag{9}$$

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573

*<sup>i</sup>* = *I*. (10)

Entanglement, Nonlocality, Superluminal Signaling and Cloning

*<sup>ρ</sup>* <sup>→</sup> ∑ *i A*† *<sup>i</sup> <sup>ρ</sup>Ai*, ∑ *i*

*AiA*†

When considering measurement processes we will make reference to Eq.(3) or to the just written equation as expressing the effect of the measurement on the statistical operator.

As already anticipated, after the clear cut proof by J.S. Bell of the fundamentally nonlocal nature of physical processes involving far away constituents in an entangled state, many proposals have been put forward, either in private correspondence or in scientific papers, suggesting how to put into evidence superluminal effects. We will begin by reviewing a series of proposal whose rebuttal did require only to resort to the standard formalism or to well established facts, such as those put into evidence by the Wigner-Araki-Yanase theorems.

**3. Proposals of faster-than-light communication and their rebuttal**

**3.1. Proposals taking advantage of the conservation of angular momentum**

in this section have some strict links with the just mentioned context.

<sup>|</sup>Ψ� <sup>=</sup> <sup>1</sup> √2

state is:

In the year 1979 various papers appeared asserting the possibility of superluminal communication by taking advantage of the change in the angular momentum of a far away constituent due to a measurement performed on its partner. The scientific and social context of these first investigations aiming to take advantage of quantum nonlocality have been described in the interesting and funny book [17] by D. Kaiser *How the hippies saved physics*, which intends to point out how the actions of a peculiar community of scientists and non scientists trying to justify various sort of paranormal effects on the basis of quantum nonlocality have drawn, in the US, the attention of the scientific community to Bell's fundamental theorem and its implications. The three papers [18-20] that I intend to consider

Let us start with refs.[18,19]. Their argument is quite straightforward: one considers two far away spin 1/2 particles in the singlet state which interact with 2 apparatuses aimed to measure the spin *<sup>z</sup>*-component and are in their "ready" states |*A*0� and |*B*0�, so that the initial

Here the indices + and - denote the values (in the usual units) of the *z*-component of the spin of the particles. Suppose now that the interaction of particle 2 with the apparatus *B* takes place before the other particle reaches *A* (*A* and *B* being at rest in a given inertial frame). Wave packet reduction occurs, and we are left, with the same probability, with one of the two states |1+, 2−, *<sup>A</sup>*0, *<sup>B</sup>*−� and |1−, 2+, *<sup>A</sup>*0, *<sup>B</sup>*+�, where |*B*±� are the states of the apparatus *B* after the measurement. We can now evaluate the mean value of the square of the spin angular momentum when the state is the one of Eq.(11) and when it is one of the states of the mixture. In the first case we get: �*S*2�*singlet* = 0, while in the second case we get the value *h*¯ 2. Now one takes advantage of the conservation of the angular momentum **L** = **M** + **S**

[|1+, 2−�−|1−, 2+�] ⊗ |*A*0, *<sup>B</sup>*0�. (11)

Comparison of the final expression with the one after the equality sign in the first line shows that, in the considered case, one must have, for *<sup>i</sup>* �= *<sup>j</sup>*, �*ϕi*|*γ*(*s*)|*ϕj*� = 0 which amounts to the condition that the observable Σ(*s*) which we want to measure on the microsystem must commute with the microsystem part *γ*(*s*) of the conserved additive quantity. If this is not the case (as it happens when Σ(*s*) is a component of the angular momentum of the system which does not commute with the other components), a process like the one of Eq.(6) turns out to be impossible; terms must be added to the r.h.s. involving other states of the microsystem besides |*ϕi*� and also other states of the apparatus. In refs.[13-15] it has been shown that in order to go as near as possible to the ideal case one must make more and more large the square of the norm of the state *<sup>γ</sup>*(*A*)|*m*0�. In the case of an angular momentum measurement this means to make the mean value of the square of the angular momentum component extremely large. Actually, in the case of the measurement of the spin component of a spin 1/2 particle, the "distorsion" of the state by the measurement, a quantity which can be estimated by the squared norm *ǫ*<sup>2</sup> of the state which has to be added to the r.h.s. of Eq.(6), must satisfy: *<sup>ǫ</sup>*<sup>2</sup> ≥ *<sup>h</sup>*2/32*π*2�*m*0|*L*2|*m*0�, where *<sup>L</sup>*<sup>2</sup> is the square of the angular momentum operator of the apparatus: to make the error extremely small one has to make extremely large �*m*0|*L*2|*m*0� .

#### **2.4. More realistic formalizations of the measurement process**

Up to this point, when accounting for the occurrence of measurement processes, we have always made reference to the projection operators on the eigenmanifolds of the operators associated to the measurement. However, in practice, it is quite difficult to have apparatuses whose effect on the statevector can be accounted precisely by a projection operator. A simple example is the one of a detector of the position of a particle in a given interval ∆ which has different efficiency in different portions of the interval ∆ so that it detects for sure a particle impinging on its central region but only with a certain probability a particle which is detected near its extreme points. Another example is given by a measurement process which corresponds to two different successive measurements of two noncommuting observables, the outcome being represented by the pair of results which have been obtained. Also in this case the probability of "an outcome" cannot be expressed in terms of a single projection operator. The appropriate consideration of situations like those just mentioned has led to the consideration of more general processes affecting the statistical operator than the one of Eq.(3). One can then take advantage of a fundamental theorem by Kraus [16] asserting that the most general map of trace class and trace one semipositive definite operators onto themselves which respects also the condition of complete positivity (which has strong physical reasons to be imposed5) has the form:

<sup>5</sup> For a definition and a discussion of completely positive maps we refer the reader to ref.[16]

$$
\rho \to \sum\_{i} A\_{i}^{\dagger} \rho A\_{i\prime} \sum\_{i} A\_{i} A\_{i}^{\dagger} = I. \tag{10}
$$

When considering measurement processes we will make reference to Eq.(3) or to the just written equation as expressing the effect of the measurement on the statistical operator.

### **3. Proposals of faster-than-light communication and their rebuttal**

8 Quantum Mechanics

We then have:

*<sup>δ</sup>ij*�*ϕi*|*γ*(*s*)

�*ϕi*, *<sup>m</sup>*0|(*γ*(*s*) ⊗ *<sup>I</sup>*

large �*m*0|*L*2|*m*0� .

�*ϕi*, *<sup>m</sup>*0|*U*†(*γ*(*s*) ⊗ *<sup>I</sup>*

(*A*) + *I*


(*A*) + *I*

(*s*) ⊗ *γ*(*A*)

(*s*) ⊗ *γ*(*A*)

**2.4. More realistic formalizations of the measurement process**

has strong physical reasons to be imposed5) has the form:

<sup>5</sup> For a definition and a discussion of completely positive maps we refer the reader to ref.[16]

)|*ϕjm*0� = �*ϕi*|*γ*(*s*)

Comparison of the final expression with the one after the equality sign in the first line shows that, in the considered case, one must have, for *<sup>i</sup>* �= *<sup>j</sup>*, �*ϕi*|*γ*(*s*)|*ϕj*� = 0 which amounts to the condition that the observable Σ(*s*) which we want to measure on the microsystem must commute with the microsystem part *γ*(*s*) of the conserved additive quantity. If this is not the case (as it happens when Σ(*s*) is a component of the angular momentum of the system which does not commute with the other components), a process like the one of Eq.(6) turns out to be impossible; terms must be added to the r.h.s. involving other states of the microsystem besides |*ϕi*� and also other states of the apparatus. In refs.[13-15] it has been shown that in order to go as near as possible to the ideal case one must make more and more large the square of the norm of the state *<sup>γ</sup>*(*A*)|*m*0�. In the case of an angular momentum measurement this means to make the mean value of the square of the angular momentum component extremely large. Actually, in the case of the measurement of the spin component of a spin 1/2 particle, the "distorsion" of the state by the measurement, a quantity which can be estimated by the squared norm *ǫ*<sup>2</sup> of the state which has to be added to the r.h.s. of Eq.(6), must satisfy: *<sup>ǫ</sup>*<sup>2</sup> ≥ *<sup>h</sup>*2/32*π*2�*m*0|*L*2|*m*0�, where *<sup>L</sup>*<sup>2</sup> is the square of the angular momentum operator of the apparatus: to make the error extremely small one has to make extremely

Up to this point, when accounting for the occurrence of measurement processes, we have always made reference to the projection operators on the eigenmanifolds of the operators associated to the measurement. However, in practice, it is quite difficult to have apparatuses whose effect on the statevector can be accounted precisely by a projection operator. A simple example is the one of a detector of the position of a particle in a given interval ∆ which has different efficiency in different portions of the interval ∆ so that it detects for sure a particle impinging on its central region but only with a certain probability a particle which is detected near its extreme points. Another example is given by a measurement process which corresponds to two different successive measurements of two noncommuting observables, the outcome being represented by the pair of results which have been obtained. Also in this case the probability of "an outcome" cannot be expressed in terms of a single projection operator. The appropriate consideration of situations like those just mentioned has led to the consideration of more general processes affecting the statistical operator than the one of Eq.(3). One can then take advantage of a fundamental theorem by Kraus [16] asserting that the most general map of trace class and trace one semipositive definite operators onto themselves which respects also the condition of complete positivity (which

)*U*|*ϕjm*0� = �*ϕi*, *mi*|(*γ*(*s*) ⊗ *<sup>I</sup>*




(*s*) ⊗ *γ*(*A*)

)|*ϕj*, *mj*� =

(*A*) + *I*

As already anticipated, after the clear cut proof by J.S. Bell of the fundamentally nonlocal nature of physical processes involving far away constituents in an entangled state, many proposals have been put forward, either in private correspondence or in scientific papers, suggesting how to put into evidence superluminal effects. We will begin by reviewing a series of proposal whose rebuttal did require only to resort to the standard formalism or to well established facts, such as those put into evidence by the Wigner-Araki-Yanase theorems.

#### **3.1. Proposals taking advantage of the conservation of angular momentum**

In the year 1979 various papers appeared asserting the possibility of superluminal communication by taking advantage of the change in the angular momentum of a far away constituent due to a measurement performed on its partner. The scientific and social context of these first investigations aiming to take advantage of quantum nonlocality have been described in the interesting and funny book [17] by D. Kaiser *How the hippies saved physics*, which intends to point out how the actions of a peculiar community of scientists and non scientists trying to justify various sort of paranormal effects on the basis of quantum nonlocality have drawn, in the US, the attention of the scientific community to Bell's fundamental theorem and its implications. The three papers [18-20] that I intend to consider in this section have some strict links with the just mentioned context.

Let us start with refs.[18,19]. Their argument is quite straightforward: one considers two far away spin 1/2 particles in the singlet state which interact with 2 apparatuses aimed to measure the spin *<sup>z</sup>*-component and are in their "ready" states |*A*0� and |*B*0�, so that the initial state is:

$$|\Psi\rangle = \frac{1}{\sqrt{2}} [|1\_+, 2\_-\rangle - |1\_-, 2\_+\rangle] \otimes |A\_{0'}B\_0\rangle. \tag{11}$$

Here the indices + and - denote the values (in the usual units) of the *z*-component of the spin of the particles. Suppose now that the interaction of particle 2 with the apparatus *B* takes place before the other particle reaches *A* (*A* and *B* being at rest in a given inertial frame). Wave packet reduction occurs, and we are left, with the same probability, with one of the two states |1+, 2−, *<sup>A</sup>*0, *<sup>B</sup>*−� and |1−, 2+, *<sup>A</sup>*0, *<sup>B</sup>*+�, where |*B*±� are the states of the apparatus *B* after the measurement. We can now evaluate the mean value of the square of the spin angular momentum when the state is the one of Eq.(11) and when it is one of the states of the mixture. In the first case we get: �*S*2�*singlet* = 0, while in the second case we get the value *h*¯ 2. Now one takes advantage of the conservation of the angular momentum **L** = **M** + **S**

where **M** is the angular momentum of the apparatus6. Since �**M** · **S**� = 0 in all above states, one concludes that the measurement induces a change of ¯*h*<sup>2</sup> in the apparatus which performs the measurement. This is not the whole story. If one, subsequently, leaves the second particle to interact with the apparatus measuring the spin state of the particle, the expectation value of **L**<sup>2</sup> does not change any more. So, actually, the angular momentum of the apparatus which is the first to perform the measurement changes of the indicated amount, while the one of the other remains unchanged. Now if Alice and Bob, sitting near *A* and *B*, have at their disposal a source of entangled particles in the singlet state, Bob, who interacts first with his particle, can choose to perform or not the measurement; correspondingly he can choose whether to leave unchanged or to change the angular momentum of the apparatus at *A*. If Alice can detect this change she can get information about the choice made, in each single instance, by Bob7. Superluminal communication becomes possible.

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575

Entanglement, Nonlocality, Superluminal Signaling and Cloning

a measurement of either plane (H,V) or circular (L,R) polarization. As a consequence of his measurement the far away photon is projected either onto a state of plane or of circular polarization. Subsequently, this photon impinges on a half-wave plate (near Alice). Since the photon when plane polarized crosses the plate without transmitting any angular momentum to it, while, when circularly polarized, it imparts a change of ±2¯*h* to the angular momentum of the plate, if Alice is able to check whether his plate has not changed or has changed its angular momentum she can know what kind of measurement (H,V) or (L,R) Bob has chosen to perform in any single case. Once more, entanglement and reduction of the wave packet

To prove why also this suggestion is inviable one has to analyze the functioning of the half-wave plate. The nice fact is that, as proved in ref.[22], one can develope precisely an argument analogous to the one of Wigner-Araki-Yanase, to prove that a half-wave plate can work as indicated only if a violation of the angular momentum conservation occurs. But such a conservation is necessary for the argument, so, once more, the proposal is contradictory.

Herbert and the Fundamental Fysiks Group made all they could do to spread out Herbert's conclusions. The debate involved scientists like H. Stapp and P. Eberhard. In june 1979, Stapp challenged the idea, building on Eberhard's argument that statistical averages would wash out any non local effect. But Herbert had worked out his reasoning for individual photons, and the above objection turned out to be not relevant for setting the issue. In the same year we (T. Weber and myself), became acquainted with Herbert's, as well as with Selleri's and others proposals. Accordingly, we wrote the paper [22] which presents the conclusions I have just described concerning refs.[18-20]. Beth's important experiment worked just because the experimenter sent an enormous number of photons at the half-wave plate. But, at the single-photon level, to get the same result, the half-wave plate would have to be infinitely massive, and, as such, it could not be put into rotation by the passage of an individual photon. This conclusion can be made rigorous with a little of mathematics, as we did in

Mention should also be made of the position of K. Popper concerning the problem of faster-than-light communication. In some previous writings, but specifically in his famous book [23] *Quantum Theory and the Schism in Physics* he raised the question of the conflict between quantum theory and special relativity theory, due to his alleged claim that "if quantum mechanical predictions are correct", then one would be able to send superluminal signals putting into evidence a conflict between the two pillars of our conception of the world. Unfortunately he was (mistankingly) convinced that the quantum rules would imply an effect that they actually exclude (a fact which he missed completely to understand), and, consequently, in his opinion they would allow superluminal signaling in an appropriate

The idea is quite simple (see Fig1, a,b): we have two perfectly correlated (in position) particles propagating towards two arrays of detectors placed at left (*L*) and right (*R*) of the emitting source at almost equal distances from it. Two slits, orthogonal to the direction *x* in the figure, are placed at both sides, along the *y*-axis, before the array of the counters, and, initially, only the counters lying behind the opening of the slits get activated. Subsequently, the slit

No superluminal communication is possible by resorting to the QUICK device.

allow superluminal transmission of information.

ref.[22].

**3.2. Popper enters the game**

experimental situation.

According to the above analysis and the remark in the footnote, the key ingredient which allows to draw the conclusion is the occurrence of an ideal nondistorting measurement of the spin component. This implies that the argument of refs.[18,19] is based on contradictory assumptions, since, as discussed in the previous section, the Wigner-Araki-Yanase theorem asserts that the occurrence of an ideal nondistorting measurement of a spin component of a subsystem contradicts the conservation of total angular momentum. Actually, to have an ideal measurement process one needs apparatuses with a divergent mean value of the square of the angular momentum, but then no change of this quantity can be detected. Alternatively, one should consider nonideal measurements which are compatible with angular momentum conservation, but then the previous argument does not work, just because Eq.(6) has to be modified.

Precisely in the same year in which the above described arguments were presented, N. Herbert circulated a paper [20] which made resort to the functioning of a half wave plate to get the same result. His proposal was stimulated by his reading of a paper [21] written in 1936 by R. Beth and included by the American Association of Physics Teachers in a collection of papers published as *Quantum and Statistical Aspects of Light*. Beth managed to measure the angular momentum of circularly polarized light due to the fact that when right-circularly polarized light is shone on the half-wave plate it sets the plate spinning in one direction, while left-circularly polarized light spun the half-wave plate in the opposite direction. Moreover, the plate flips the light polarization from left to right and viceversa. Beth had measured the effect for circularly polarized light waves, i.e., by using a huge collection of photons all acting together. Herbert, inspired by this work, suggested, in the paper he called QUICK, to play a similar game with the angular momentum of individual photons to get superluminal effects.

Once more the idea is quite simple: one imagines a source emitting pairs of entangled photons in two opposite directions, their state being the analogous of the singlet state, i.e. the rotationally invariant state: <sup>|</sup>Ψ(1, 2)� = [1/√2][|*H*1, *<sup>H</sup>*2� <sup>+</sup> <sup>|</sup>*V*1, *<sup>V</sup>*2�] <sup>≡</sup> [1/√2][|*R*1, *<sup>L</sup>*2� <sup>+</sup> |*L*1, *R*2�]. Here, the symbols *H*, *V*, *R* and *L* make reference to the states of horizontal, vertical, right and left circular polarizations, respectively. Bob can freely choose whether to perform

<sup>6</sup> Here, we disregard the orbital angular momentum of the particles, but the argument holds true also without this limitation.

<sup>7</sup> It is interesting to remark that the same argument can be developed if one does not take into account the reduction process, i.e., if one assumes that the interactions simply take place in accordance with the von Neumann scheme.

a measurement of either plane (H,V) or circular (L,R) polarization. As a consequence of his measurement the far away photon is projected either onto a state of plane or of circular polarization. Subsequently, this photon impinges on a half-wave plate (near Alice). Since the photon when plane polarized crosses the plate without transmitting any angular momentum to it, while, when circularly polarized, it imparts a change of ±2¯*h* to the angular momentum of the plate, if Alice is able to check whether his plate has not changed or has changed its angular momentum she can know what kind of measurement (H,V) or (L,R) Bob has chosen to perform in any single case. Once more, entanglement and reduction of the wave packet allow superluminal transmission of information.

To prove why also this suggestion is inviable one has to analyze the functioning of the half-wave plate. The nice fact is that, as proved in ref.[22], one can develope precisely an argument analogous to the one of Wigner-Araki-Yanase, to prove that a half-wave plate can work as indicated only if a violation of the angular momentum conservation occurs. But such a conservation is necessary for the argument, so, once more, the proposal is contradictory. No superluminal communication is possible by resorting to the QUICK device.

Herbert and the Fundamental Fysiks Group made all they could do to spread out Herbert's conclusions. The debate involved scientists like H. Stapp and P. Eberhard. In june 1979, Stapp challenged the idea, building on Eberhard's argument that statistical averages would wash out any non local effect. But Herbert had worked out his reasoning for individual photons, and the above objection turned out to be not relevant for setting the issue. In the same year we (T. Weber and myself), became acquainted with Herbert's, as well as with Selleri's and others proposals. Accordingly, we wrote the paper [22] which presents the conclusions I have just described concerning refs.[18-20]. Beth's important experiment worked just because the experimenter sent an enormous number of photons at the half-wave plate. But, at the single-photon level, to get the same result, the half-wave plate would have to be infinitely massive, and, as such, it could not be put into rotation by the passage of an individual photon. This conclusion can be made rigorous with a little of mathematics, as we did in ref.[22].

#### **3.2. Popper enters the game**

10 Quantum Mechanics

modified.

limitation.

where **M** is the angular momentum of the apparatus6. Since �**M** · **S**� = 0 in all above states, one concludes that the measurement induces a change of ¯*h*<sup>2</sup> in the apparatus which performs the measurement. This is not the whole story. If one, subsequently, leaves the second particle to interact with the apparatus measuring the spin state of the particle, the expectation value of **L**<sup>2</sup> does not change any more. So, actually, the angular momentum of the apparatus which is the first to perform the measurement changes of the indicated amount, while the one of the other remains unchanged. Now if Alice and Bob, sitting near *A* and *B*, have at their disposal a source of entangled particles in the singlet state, Bob, who interacts first with his particle, can choose to perform or not the measurement; correspondingly he can choose whether to leave unchanged or to change the angular momentum of the apparatus at *A*. If Alice can detect this change she can get information about the choice made, in each single instance, by

According to the above analysis and the remark in the footnote, the key ingredient which allows to draw the conclusion is the occurrence of an ideal nondistorting measurement of the spin component. This implies that the argument of refs.[18,19] is based on contradictory assumptions, since, as discussed in the previous section, the Wigner-Araki-Yanase theorem asserts that the occurrence of an ideal nondistorting measurement of a spin component of a subsystem contradicts the conservation of total angular momentum. Actually, to have an ideal measurement process one needs apparatuses with a divergent mean value of the square of the angular momentum, but then no change of this quantity can be detected. Alternatively, one should consider nonideal measurements which are compatible with angular momentum conservation, but then the previous argument does not work, just because Eq.(6) has to be

Precisely in the same year in which the above described arguments were presented, N. Herbert circulated a paper [20] which made resort to the functioning of a half wave plate to get the same result. His proposal was stimulated by his reading of a paper [21] written in 1936 by R. Beth and included by the American Association of Physics Teachers in a collection of papers published as *Quantum and Statistical Aspects of Light*. Beth managed to measure the angular momentum of circularly polarized light due to the fact that when right-circularly polarized light is shone on the half-wave plate it sets the plate spinning in one direction, while left-circularly polarized light spun the half-wave plate in the opposite direction. Moreover, the plate flips the light polarization from left to right and viceversa. Beth had measured the effect for circularly polarized light waves, i.e., by using a huge collection of photons all acting together. Herbert, inspired by this work, suggested, in the paper he called QUICK, to play a similar game with the angular momentum of individual photons to get superluminal effects. Once more the idea is quite simple: one imagines a source emitting pairs of entangled photons in two opposite directions, their state being the analogous of the singlet state, i.e. the rotationally invariant state: <sup>|</sup>Ψ(1, 2)� = [1/√2][|*H*1, *<sup>H</sup>*2� <sup>+</sup> <sup>|</sup>*V*1, *<sup>V</sup>*2�] <sup>≡</sup> [1/√2][|*R*1, *<sup>L</sup>*2� <sup>+</sup> |*L*1, *R*2�]. Here, the symbols *H*, *V*, *R* and *L* make reference to the states of horizontal, vertical, right and left circular polarizations, respectively. Bob can freely choose whether to perform

<sup>6</sup> Here, we disregard the orbital angular momentum of the particles, but the argument holds true also without this

<sup>7</sup> It is interesting to remark that the same argument can be developed if one does not take into account the reduction process, i.e., if one assumes that the interactions simply take place in accordance with the von Neumann scheme.

Bob7. Superluminal communication becomes possible.

Mention should also be made of the position of K. Popper concerning the problem of faster-than-light communication. In some previous writings, but specifically in his famous book [23] *Quantum Theory and the Schism in Physics* he raised the question of the conflict between quantum theory and special relativity theory, due to his alleged claim that "if quantum mechanical predictions are correct", then one would be able to send superluminal signals putting into evidence a conflict between the two pillars of our conception of the world. Unfortunately he was (mistankingly) convinced that the quantum rules would imply an effect that they actually exclude (a fact which he missed completely to understand), and, consequently, in his opinion they would allow superluminal signaling in an appropriate experimental situation.

The idea is quite simple (see Fig1, a,b): we have two perfectly correlated (in position) particles propagating towards two arrays of detectors placed at left (*L*) and right (*R*) of the emitting source at almost equal distances from it. Two slits, orthogonal to the direction *x* in the figure, are placed at both sides, along the *y*-axis, before the array of the counters, and, initially, only the counters lying behind the opening of the slits get activated. Subsequently, the slit at *R* is narrowed so as to produce an uncertainty principle scatter of the momentum *py*, which appreciably increases the set of counters which are activated with a non-negligible probability (see Fig.1b). Popper then argues: If quantum mechanics is correct, any increase in the knowledge of the position *y* at *R* like the one we get by making more precise the location along the *y*-axis of the particle which is there, implies an analogous increase of the knowledge of the position of the particle at *L*. As a consequence also the scatter at *L* should increase even though the width of the slit at this side has not been narrowed. This prediction is testable, since new counters would be activated with an appreciable probability, giving rise to a superluminal influence: Alice can know (with an appreciable probability) whether Bob has chosen to narrow or leave unchanged his slit. The conclusion of Popper is quite emblematic: in his opinion the increase of the spread at *L* would not occur and this *would show that quantum theory is wrong*. He also contemplates the other alternative: if the scatter at left would increase, then superluminal communication would be possible and relativity theory would be proven false; in both cases, a quite astonishing conclusion.

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Entanglement, Nonlocality, Superluminal Signaling and Cloning

• A unitary transformation describing the free evolution of the system at *R* under consideration and/or possibly of its interactions with other systems lying in a space-time

• A transformation corresponding to a non selective projective measurement (with wave

• A transformation like the one summarized in Eq.(10) corresponding, essentially, to the

• A transformation like the one of Eq.(4), corresponding to a selective measurement. To be strict, one should also consider the analogous of this transformation in the case of a non-ideal measurement, but, for the reasons which we will make clear below, this case

Now we can proceed to outline our proof, which represents a generalization and a more accurate formulation of some ideas put forward by P. Eberhard [28] about one year before

To start with we recall that all probabilistic predictions concerning a subsystem of a composite system can be obtained by considering the reduced statistical operator of the subsystem of interest. We suppose now to have a composite quantum system *S* = *S*(1) + *S*(2) associated to the statistical operator *ρ*(1, 2) and to be interested in predictions concerning prospective measurements on subsystem *S*(1). As already remarked, the physics of this subsystem is fully described by the reduced statistical operator *ρ*˜(1) = *Tr*(2)*ρ*(1, 2). We can

*ρ*(1, 2),

*U*(2)†*ρ*(1, 2)] = *Tr*(2)

<sup>2</sup>*ρ*(1, 2)] = *Tr*(2)

*<sup>i</sup> ρ*(1, 2)] = *Tr*(2)

*ρ*(1, 2) = *ρ*˜

*ρ*(1, 2) = *ρ*˜

*<sup>i</sup>* and of the fact that the sum of the operators

(1) ,

(1)

(1)

*<sup>i</sup> <sup>A</sup>*(2)†

. (12)

*<sup>i</sup>* of the

*ρ*(1, 2) = *ρ*˜

[*U*(2)

[∑ *i* [*P*(2) *<sup>i</sup>* ]

[∑ *i*

<sup>2</sup> = *P*(2)

*<sup>i</sup>* in the case of a projective measurement, as well as of the operators *<sup>A</sup>*(2)

*A*(2) *<sup>i</sup> <sup>A</sup>*(2)†

In these equations we have made use of the cyclic property of the trace over the Hilbert space H(2) when operators of the same Hilbert space are involved, of the unitarity relation

Kraus theorem [16], must equal the identity operator of the same space. We call the attention of the reader to the fact that in all considered cases, i.e., i). no action on *S*(2), ii). a unitary evolution of *S*(2), or iii). the fact that it is subjected to an operation corresponding to a (ideal or non-ideal) nonselective measurement process, the reduced statistical operator *ρ*˜(1) of subsystem *S*(1) does not change in any way whatsoever, and, accordingly, Alice, performing measurements on such a subsystem cannot get any information about the fact that Bob is

we wrote our paper [26] (see also A. Shimony [29] for an enlightening discussion).

packet reduction as described by Eq. (3)) of the considered subsystem.

does not have a physical relevance for faster-than-light signaling.

region which is space like with respect to the one at *L*.

occurrence of a non ideal measurement.

now consider the following set of equations:

*A*(2)†

*<sup>U</sup>*(2)*U*(2)† = *<sup>I</sup>*(2), of the propery [*P*(2)

making some specific action on subsystem *S*(2).

[*U*(2)†*ρ*(1, 2)*U*(2)

*<sup>i</sup> <sup>ρ</sup>*(1, 2)*P*(2)

*<sup>i</sup> <sup>ρ</sup>*(1, 2)*A*(2)

*Tr*(2)

*Tr*(2))[∑ *i P*(2)

*Tr*(2))[∑ *i*

*P*(2)

*ρ*˜

(1) = *Tr*(2)

*<sup>i</sup>* ] = *Tr*(2)

*<sup>i</sup>* ] = *Tr*(2)

*<sup>i</sup>* ]

] = *Tr*(2)

**Figure 1.** The set up and functioning of Popper's ideal experiment.

Unfortunately, in this passage of his important work, Popper shows his lack of understanding the quantum principles governing the unfolding of the considered experiment. In fact, it can be easily proved that quantum mechanics predicts precisely that no scatter at left will be induced by the narrowing of the slit at right. We do not consider it useful to enter in all technical details of the argument. The reader can look at ref.[24] or to Chapter 11 of ref.[25] for a detailed and punctual discussion. Here, we simply outline the argument: if the positions of the particles are really 100% correlated (and therefore associated to a Dirac delta like unnormalized state), then they are in a state which implies that, even when the two slits are fully opened, all counters are activated with large probabilities, while, if their correlations are only approximate (even with an extremely high degree of accuracy) the action at *R* by Bob does not change in any way whatsoever the outcomes at *L*. So, the argument is basically wrong.

### **4. The general proof of the impossibility of faster-than-light communication**

To present a completely general proof [26,27] of the fact that instantaneous wave-packet reduction does not allow superluminal signaling we must start by reconsidering all possible actions [3,16] which are permitted, by standard quantum mechanics, on a constituent of a composite system. Quite in general, quantum mechanics allows the possibility of:

• A unitary transformation describing the free evolution of the system at *R* under consideration and/or possibly of its interactions with other systems lying in a space-time region which is space like with respect to the one at *L*.

12 Quantum Mechanics

wrong.

**communication**

at *R* is narrowed so as to produce an uncertainty principle scatter of the momentum *py*, which appreciably increases the set of counters which are activated with a non-negligible probability (see Fig.1b). Popper then argues: If quantum mechanics is correct, any increase in the knowledge of the position *y* at *R* like the one we get by making more precise the location along the *y*-axis of the particle which is there, implies an analogous increase of the knowledge of the position of the particle at *L*. As a consequence also the scatter at *L* should increase even though the width of the slit at this side has not been narrowed. This prediction is testable, since new counters would be activated with an appreciable probability, giving rise to a superluminal influence: Alice can know (with an appreciable probability) whether Bob has chosen to narrow or leave unchanged his slit. The conclusion of Popper is quite emblematic: in his opinion the increase of the spread at *L* would not occur and this *would show that quantum theory is wrong*. He also contemplates the other alternative: if the scatter at left would increase, then superluminal communication would be possible and relativity

Unfortunately, in this passage of his important work, Popper shows his lack of understanding the quantum principles governing the unfolding of the considered experiment. In fact, it can be easily proved that quantum mechanics predicts precisely that no scatter at left will be induced by the narrowing of the slit at right. We do not consider it useful to enter in all technical details of the argument. The reader can look at ref.[24] or to Chapter 11 of ref.[25] for a detailed and punctual discussion. Here, we simply outline the argument: if the positions of the particles are really 100% correlated (and therefore associated to a Dirac delta like unnormalized state), then they are in a state which implies that, even when the two slits are fully opened, all counters are activated with large probabilities, while, if their correlations are only approximate (even with an extremely high degree of accuracy) the action at *R* by Bob does not change in any way whatsoever the outcomes at *L*. So, the argument is basically

To present a completely general proof [26,27] of the fact that instantaneous wave-packet reduction does not allow superluminal signaling we must start by reconsidering all possible actions [3,16] which are permitted, by standard quantum mechanics, on a constituent of a

theory would be proven false; in both cases, a quite astonishing conclusion.

**4. The general proof of the impossibility of faster-than-light**

composite system. Quite in general, quantum mechanics allows the possibility of:

**Figure 1.** The set up and functioning of Popper's ideal experiment.


Now we can proceed to outline our proof, which represents a generalization and a more accurate formulation of some ideas put forward by P. Eberhard [28] about one year before we wrote our paper [26] (see also A. Shimony [29] for an enlightening discussion).

To start with we recall that all probabilistic predictions concerning a subsystem of a composite system can be obtained by considering the reduced statistical operator of the subsystem of interest. We suppose now to have a composite quantum system *S* = *S*(1) + *S*(2) associated to the statistical operator *ρ*(1, 2) and to be interested in predictions concerning prospective measurements on subsystem *S*(1). As already remarked, the physics of this subsystem is fully described by the reduced statistical operator *ρ*˜(1) = *Tr*(2)*ρ*(1, 2). We can now consider the following set of equations:

$$\begin{aligned} \tilde{\rho}^{(1)} &= \text{Tr}^{(2)} \rho(1, 2), \\ \text{Tr}^{(2)}[\mathcal{U}^{(2)\dagger} \rho(1, 2) \mathcal{U}^{(2)}] &= \text{Tr}^{(2)}[\mathcal{U}^{(2)} \mathcal{U}^{(2)\dagger} \rho(1, 2)] = \text{Tr}^{(2)} \rho(1, 2) = \tilde{\rho}^{(1)}, \\ \text{Tr}^{(2)}[\sum\_{i} P\_{i}^{(2)} \rho(1, 2) P\_{i}^{(2)}] &= \text{Tr}^{(2)}[\sum\_{i} [P\_{i}^{(2)}]^2 \rho(1, 2)] = \text{Tr}^{(2)} \rho(1, 2) = \tilde{\rho}^{(1)} \\ \text{Tr}^{(2)}[\sum\_{i} A\_{i}^{(2)\dagger} \rho(1, 2) A\_{i}^{(2)}] &= \text{Tr}^{(2)}[\sum\_{i} A\_{i}^{(2)} A\_{i}^{(2)\dagger} \rho(1, 2)] = \text{Tr}^{(2)} \rho(1, 2) = \tilde{\rho}^{(1)}. \end{aligned} \tag{12}$$

In these equations we have made use of the cyclic property of the trace over the Hilbert space H(2) when operators of the same Hilbert space are involved, of the unitarity relation *<sup>U</sup>*(2)*U*(2)† = *<sup>I</sup>*(2), of the propery [*P*(2) *<sup>i</sup>* ] <sup>2</sup> = *P*(2) *<sup>i</sup>* and of the fact that the sum of the operators *P*(2) *<sup>i</sup>* in the case of a projective measurement, as well as of the operators *<sup>A</sup>*(2) *<sup>i</sup> <sup>A</sup>*(2)† *<sup>i</sup>* of the Kraus theorem [16], must equal the identity operator of the same space. We call the attention of the reader to the fact that in all considered cases, i.e., i). no action on *S*(2), ii). a unitary evolution of *S*(2), or iii). the fact that it is subjected to an operation corresponding to a (ideal or non-ideal) nonselective measurement process, the reduced statistical operator *ρ*˜(1) of subsystem *S*(1) does not change in any way whatsoever, and, accordingly, Alice, performing measurements on such a subsystem cannot get any information about the fact that Bob is making some specific action on subsystem *S*(2).

Up to now, we have not considered explicitly the case of selective ideal or non-ideal measurement processes, accounted for by Eq. (4) or by its analogue referring to processes like those governed by Eq.(10). If one considers the modifications to the general statistical operator in these cases and one uses the reduced statistical operator to evaluate the probabilities of the measurement outcomes on subsystem *S*(1), one would easily discover that the physics of such a system is actually changed by the action on its far-away partner. But the probabilistic changes depend crucially on the outcome that Bob has obtained in his measurement, so that Alice might take advantage of this fact only if she would be informed of the outcome obtained by Bob. This implies that Bob must inform Alice concerning his outcome and this can be done only by resorting to standard communication procedures which require a subluminal communication between the two. Accordingly, these cases can safely be disregarded within our context.

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Entanglement, Nonlocality, Superluminal Signaling and Cloning

measurement of either plane (H,V) or circular (L,R) polarization. As a consequence of his measurement the far away photon is projected either onto a state of plane or of circular polarization. At this point the far away photon is injected in the Laser gain tube which emits 4*N* photons with the same polarization, which, in turn, depends on the measurement performed by Bob and the outcome he has got. The 4*N* photons are then separated into 4 beams of N photons each, directed towards 4 detectors of *V*,*H*,*L* and *R* (mind the order) polarization, respectively. To see the game coming at an end we have now simply to recall that a detector registers for sure a photon with the polarization it is devised to measure, it does not detect a photon into an orthogonal state and detects with probability 1/2 a photon in a state of polarization which is the equal superposition of the state that it is devised to

We analyze the situation in detail specifying the measurements which Bob chooses to

• Suppose Bob chooses to perform a polarization measurement aimed to ascertain whether the photon (2) reaching him has vertical or horizontal polarization and that he finds the

• a). Initial state: [1/√2][|*R*1, *<sup>L</sup>*2� <sup>+</sup> <sup>|</sup>*L*1, *<sup>R</sup>*2�]; b). Measurement with outcome Left; c). Reduction of the state: |*R*1; *L*2�; d). Amplification: |*R*, 4*N*; *L*2�; e). Number of photons

Now, one has simply to remark that in the cases listed under the two first items (i.e. when Bob chooses to measure linear polarization) the detector which does not register any photon is either the first or the second, while, in the alternative case in which Bob chooses to measure circular polarization, it is either the third or the fourth detector which does not register any photon. Accordingly, Alice can become aware, instantaneously, of the choice made by Bob:

The FLASH paper was sent for refereeing to A. Peres and to me. Peres' answer [30] was rather peculiar: *I recommended to the editor that this paper should be published. I wrote that it was obviously wrong, but I expected that it would elicit considerable interest and that finding the error would lead to significant progress in our understanding of physics*. I also was rather worried for various reasons. I was not an expert on Lasers and I was informed that A. Gozzini and R. Peierls were trying to disprove Herbert's conclusion by invoking the unavoidable

a). Initial state: <sup>|</sup>Ψ(1, 2)� = [1/√2][|*H*1, *<sup>H</sup>*2� <sup>+</sup> <sup>|</sup>*V*1, *<sup>V</sup>*2�]; b). Measurement with outcome Vertical; c). Reduction of the state: |*V*1, *V*2�; d). Amplification: |*V*, 4*N*; *V*2�; e). Number of photons detected by the far away detectors (near Alice) for the 4 beams: N,0,N/2,N/2. • a). Initial state: <sup>|</sup>Ψ(1, 2)� = [1/√2][|*H*1, *<sup>H</sup>*2� <sup>+</sup> <sup>|</sup>*V*1, *<sup>V</sup>*2�]; b). Measurement with outcome Horizontal; c). Reduction of the state: |*H*1, *H*2�; d). Amplification: |*H*, 4*N*; *H*2�; e). Number of photons detected by the far away detectors on the 4 beams: 0,N,N/2,N/2. • a). Initial state: [1/√2][|*R*1, *<sup>L</sup>*2� <sup>+</sup> <sup>|</sup>*L*1, *<sup>R</sup>*2�]; b). Measurement with outcome Right; c). Reduction of the state: |*L*1, *R*2�; d). Amplification: |*L*, 4*N*; *R*2�; e). Number of photons

perform, the outcomes he gets and the records by the counters near Alice.

photon vertically polarized. In this case, the process goes as follows:

detected by the far away detectors on the 4 beams: N/2,N/2,N,0.

detected by the far away detectors on the 4 beams: N/2,N/2,0,N.

detect and of the state orthogonal to it.

they can communicate superluminally.

**5.2. The no-cloning theorem**

It should be clear that the general validity of our theorem implies that all previously discussed attempts to get faster-than-light signaling taking advantage of the instantaneous reduction at-a-distance of the statevector in the case of entangled states of far-away systems, were doomed to fail. We have discussed them in some detail to present an historically complete perspective of the debate on this fundamental issue, i.e. the one of the compatibility of quantum mechanics with relativistic requirements concerning the communication between far-away observers.

### **5. A radical change of perspective**

#### **5.1. Herbert's new proposal**

In 1981 N. Herbert submitted for publication to Foundations of Physics a paper [4] by the title: *FLASH–A superluminal communicator based on a new kind of quantum measurement* in which he added a new specific device to his previous proposal we have discussed in sect.3.1. The stimulus to do so came probably from our paper, ref.[26], as remarked by D. Kaiser in his book [17]: *From Ghirardi's intervention, Herbert came to appreciate the importance of amplifying the tiny distinction between various quantum states, to evade fundamental limits on signaling.* The crucial device which, in his opinion, could do the game, was a Laser gain tube exhibiting the following characteristics: if the laser was stimulated by a single photon in any state of polarization (the states which mattered were actually those of plane (V and H) and of circular (R and L) polarization) it would emit a relevant number, let us say 4N with N large, of identical copies (in particular with the same polarization) of the impinging photon. If we summarize the process by means of an arrow leading from the initial photon state to the bunch of final photons, Hebert's Laser gain tube has to work in the following way:

$$|V,1\rangle \to |V,4N\rangle, \ |H,1\rangle \to |H,4N\rangle, \ |R,1\rangle \to |R,4N\rangle, \ |L,1\rangle \to |L,4N\rangle,\tag{13}$$

with obvious meaning of the symbols. Here 1 denotes the photon propagating towards Alice.

By resorting to this machine Herbert's game becomes quite simple. One starts, as in his first proposal, with a source emitting pairs of entangled photons in two opposite directions, their state being the rotationally invariant state: <sup>|</sup>Ψ(1, 2)� = [1/√2][|*H*1, *<sup>H</sup>*2� <sup>+</sup> <sup>|</sup>*V*1, *<sup>V</sup>*2�] <sup>≡</sup> [1/√2][|*R*1, *<sup>L</sup>*2� <sup>+</sup> <sup>|</sup>*L*1, *<sup>R</sup>*2�]. Obviously, Bob can freely choose whether to perform a

http://dx.doi.org/10.5772/CHAPTERDOI

measurement of either plane (H,V) or circular (L,R) polarization. As a consequence of his measurement the far away photon is projected either onto a state of plane or of circular polarization. At this point the far away photon is injected in the Laser gain tube which emits 4*N* photons with the same polarization, which, in turn, depends on the measurement performed by Bob and the outcome he has got. The 4*N* photons are then separated into 4 beams of N photons each, directed towards 4 detectors of *V*,*H*,*L* and *R* (mind the order) polarization, respectively. To see the game coming at an end we have now simply to recall that a detector registers for sure a photon with the polarization it is devised to measure, it does not detect a photon into an orthogonal state and detects with probability 1/2 a photon in a state of polarization which is the equal superposition of the state that it is devised to detect and of the state orthogonal to it.

We analyze the situation in detail specifying the measurements which Bob chooses to perform, the outcomes he gets and the records by the counters near Alice.

• Suppose Bob chooses to perform a polarization measurement aimed to ascertain whether the photon (2) reaching him has vertical or horizontal polarization and that he finds the photon vertically polarized. In this case, the process goes as follows:

a). Initial state: <sup>|</sup>Ψ(1, 2)� = [1/√2][|*H*1, *<sup>H</sup>*2� <sup>+</sup> <sup>|</sup>*V*1, *<sup>V</sup>*2�]; b). Measurement with outcome Vertical; c). Reduction of the state: |*V*1, *V*2�; d). Amplification: |*V*, 4*N*; *V*2�; e). Number of photons detected by the far away detectors (near Alice) for the 4 beams: N,0,N/2,N/2.


Now, one has simply to remark that in the cases listed under the two first items (i.e. when Bob chooses to measure linear polarization) the detector which does not register any photon is either the first or the second, while, in the alternative case in which Bob chooses to measure circular polarization, it is either the third or the fourth detector which does not register any photon. Accordingly, Alice can become aware, instantaneously, of the choice made by Bob: they can communicate superluminally.

#### **5.2. The no-cloning theorem**

14 Quantum Mechanics

safely be disregarded within our context.

**5. A radical change of perspective**

**5.1. Herbert's new proposal**

far-away observers.

Up to now, we have not considered explicitly the case of selective ideal or non-ideal measurement processes, accounted for by Eq. (4) or by its analogue referring to processes like those governed by Eq.(10). If one considers the modifications to the general statistical operator in these cases and one uses the reduced statistical operator to evaluate the probabilities of the measurement outcomes on subsystem *S*(1), one would easily discover that the physics of such a system is actually changed by the action on its far-away partner. But the probabilistic changes depend crucially on the outcome that Bob has obtained in his measurement, so that Alice might take advantage of this fact only if she would be informed of the outcome obtained by Bob. This implies that Bob must inform Alice concerning his outcome and this can be done only by resorting to standard communication procedures which require a subluminal communication between the two. Accordingly, these cases can

It should be clear that the general validity of our theorem implies that all previously discussed attempts to get faster-than-light signaling taking advantage of the instantaneous reduction at-a-distance of the statevector in the case of entangled states of far-away systems, were doomed to fail. We have discussed them in some detail to present an historically complete perspective of the debate on this fundamental issue, i.e. the one of the compatibility of quantum mechanics with relativistic requirements concerning the communication between

In 1981 N. Herbert submitted for publication to Foundations of Physics a paper [4] by the title: *FLASH–A superluminal communicator based on a new kind of quantum measurement* in which he added a new specific device to his previous proposal we have discussed in sect.3.1. The stimulus to do so came probably from our paper, ref.[26], as remarked by D. Kaiser in his book [17]: *From Ghirardi's intervention, Herbert came to appreciate the importance of amplifying the tiny distinction between various quantum states, to evade fundamental limits on signaling.* The crucial device which, in his opinion, could do the game, was a Laser gain tube exhibiting the following characteristics: if the laser was stimulated by a single photon in any state of polarization (the states which mattered were actually those of plane (V and H) and of circular (R and L) polarization) it would emit a relevant number, let us say 4N with N large, of identical copies (in particular with the same polarization) of the impinging photon. If we summarize the process by means of an arrow leading from the initial photon state to the

bunch of final photons, Hebert's Laser gain tube has to work in the following way:


with obvious meaning of the symbols. Here 1 denotes the photon propagating towards Alice. By resorting to this machine Herbert's game becomes quite simple. One starts, as in his first proposal, with a source emitting pairs of entangled photons in two opposite directions, their state being the rotationally invariant state: <sup>|</sup>Ψ(1, 2)� = [1/√2][|*H*1, *<sup>H</sup>*2� <sup>+</sup> <sup>|</sup>*V*1, *<sup>V</sup>*2�] <sup>≡</sup> [1/√2][|*R*1, *<sup>L</sup>*2� <sup>+</sup> <sup>|</sup>*L*1, *<sup>R</sup>*2�]. Obviously, Bob can freely choose whether to perform a The FLASH paper was sent for refereeing to A. Peres and to me. Peres' answer [30] was rather peculiar: *I recommended to the editor that this paper should be published. I wrote that it was obviously wrong, but I expected that it would elicit considerable interest and that finding the error would lead to significant progress in our understanding of physics*. I also was rather worried for various reasons. I was not an expert on Lasers and I was informed that A. Gozzini and R. Peierls were trying to disprove Herbert's conclusion by invoking the unavoidable noise affecting the Laser which would inhibit its desired functioning. On the other hand, I was convinced that quantum theory in its general formulation and not due to limitations of practical nature would make unviable Herbert's proposal. After worrying for some days about this problem I got the general answer: while it is possible to devise an ideal apparatus which clones two orthogonal states with 100% efficiency, the same apparatus, if the linear quantum theory governs its functioning, cannot clone states which are linear combinations of the previous ones. Here is my argument, on the basis of which I recommended rejection of Herbert's paper. The assumption that the cloning machine acts as follows:

$$|V\rangle \to |V, \mathbf{4N}\rangle \text{ and } |H\rangle \to |H, \mathbf{4N}\rangle,\tag{14}$$

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Entanglement, Nonlocality, Superluminal Signaling and Cloning

Ψ = *S*Φ. (16)

nutrient, is: Φ = *ν* × *w*. When replication takes place the statevector will have the form: Ψ = *ν* × *ν* × *r*, i.e. two organisms each in the statevector *ν* will be present, while the vector *r* describes both the rest of the system, the rejected part of the nutrient and also the other coordinates (positions, etc.) of the two organisms. One assumes that the system lives in an *<sup>N</sup>*-dimensional Hilbert space H(<sup>N</sup> ), the part *<sup>r</sup>* in an *<sup>R</sup>*-dimensional Hilbert space H(R), while the "food" state *<sup>w</sup>* belongs to a *<sup>N</sup>* · *<sup>R</sup>* dimensional Hilbert space H(N R), so that <sup>Φ</sup> and <sup>Ψ</sup> live, as they must, in the same space. Suppose we do not know the state of the living system; however, since it belongs to H(<sup>N</sup> ) his knowledge requires to know *<sup>N</sup>* complex numbers. Analogously we do not know the state *r*, and the state *w*, which require the specification of *R* and *NR* complex numbers. We now assume, with Wigner, that the collision matrix which gives the final state resulting from the interaction, which will be denoted as *S*, of the organism and the nutrient is a random matrix, which, however, even though unknown to us explicitly, is completely determined by the laws of quantum mechanics. Obviously *S* must

Choosing a basis for the whole Hilbert space and projecting Eq. (16) on such a basis one gets *N*2*R* equations. And now the conclusion follows: our unknown quantities are the components of the states *ν*,*r* and *w* on their respective bases and are therefore *N* + *R* + *NR* in number. Thus, according to Wigner, the question is: given the matrix corresponding to *S*, it is possible to find vectors *ν*,*r* and *w* such that their components satisfy the above mentioned *N*2*R* equations? Since *N*2*R* ≫ *N* + *R* + *NR*, for extremely large *N* and *R*, according to him: *it would be a miracle if such equations could be satisfied*. In other words, a self-replicating quantum unit does not exist. One might state that Wigner has "derived" (with the proviso he is making - see below) the no-cloning theorem for a quantum system whose Hilbert space has an extremely high dimensionality N, while we have shown that it holds also for N=2. Wigner was perfectly aware that the argument is not fully rigorous and cannot be taken too seriously because of the many assumptions on which it is based. However, he seems inclined to attach a certain value to it. This is not surprising because at the time in which he wrote his paper he was adhering to von Neumann's idea that consciousness is responsible for the reduction of the wave packet, so that, in a certain sense, the fact that quantum mechanics is not able to account for the basic property of living organism (the self-reproduction) supported his view that such a theory cannot be used to describe the conscious perceptions of such organisms. In 1971 Eigen [36] responded to Wigner claiming that his choice of resorting to a typical unitary map to account for the process did not take into account the instructive

Strictly connected with Wigner argument, even though derived through a much more rigorous and general procedure is the proof of the no-cloning theorem presented in a beautiful paper by R. Alicki [35]. He considers a dynamical transformation T from an initial state *ϕ* ⊗ *ω*, where *ϕ* is the state of the organism and *ω* the fixed initial state of

As before, *σ* represents the state of the "food" after the replication. Alicki assumes that any dynamical process of a closed system (typically the one given by *T*) cannot reduce the

*ϕ* ⊗ *ω* → *T*(*ϕ* ⊗ *ω*) = *ϕ* ⊗ *ϕ* ⊗ *σ*. (17)

satisfy:

functions of informational macromolecules.

the environment designed as "food":

when the linear nature of the theory is taken into account, implies:

$$R \equiv \frac{1}{\sqrt{2}} [i|V\rangle + |H\rangle] \to \frac{1}{\sqrt{2}} [i|V, 4N\rangle + |H, 4N\rangle]\_{\prime} \tag{15}$$

and analogously for the left polarization. Now, the state at the r.h.s. of the last equation is by no means the state |*R*, 4*N*� which Herbert had assumed to occur in the case in which the Laser gain tube is triggered by a right polarized photon. But this is not the whole story: how it has been shown in ref.[31] the very linear nature of the theory implies that no difference in the detections of Alice occurs in dependence of the free will choice of Bob.

This is an account of how Herbert's ingenious, but mistaken, proposal has led me to be the first to derive the no-cloning theorem 8. About one and half year later Wootters and Zurek [32] and Dieks [33] derived independently the same result and published it9. The theorem is of remarkable importance in quantum theory, it has become known as "The no cloning theorem" and it has been quoted an innumerable number of times. Only subsequently I realized that it had been a mistake on my part not to publish my result. I discussed my precise argument with Gozzini and Peierls, by sending them a draft which was a sort of repetition of my referee report and I subsequently published it [31] in collaboration with my collaborator, T. Weber.

#### *5.2.1. More on quantum cloning*

In a paper like the present one, we believe it useful to mention that E.P. Wigner [34], in an essay of 1961 had already argued that the phenomena of self-replication of biological systems contradict the principles of quantum mechanics. His argument is quite straightforward. Following his notation let us suppose we have a living system in a state *ν* and an environment (assimilated to "food") in a state *w*, so that, the initial statevector of the system, organism +

<sup>8</sup> I have chosen to attach at the end of the paper, a document - a letter by A. van der Merwe - which officially attests this fact, since it is known only to a restricted community of physicists.

<sup>9</sup> I must confess that I have never understood why A. Peres, in mentioning my derivation, has stated that it was a special case of the theorems in refs. [32] and [33]. Comparison even only of the short page by A. van der Merwe with the just mentioned papers makes clear that the argument is precisely the same and has the same generality.

nutrient, is: Φ = *ν* × *w*. When replication takes place the statevector will have the form: Ψ = *ν* × *ν* × *r*, i.e. two organisms each in the statevector *ν* will be present, while the vector *r* describes both the rest of the system, the rejected part of the nutrient and also the other coordinates (positions, etc.) of the two organisms. One assumes that the system lives in an *<sup>N</sup>*-dimensional Hilbert space H(<sup>N</sup> ), the part *<sup>r</sup>* in an *<sup>R</sup>*-dimensional Hilbert space H(R), while the "food" state *<sup>w</sup>* belongs to a *<sup>N</sup>* · *<sup>R</sup>* dimensional Hilbert space H(N R), so that <sup>Φ</sup> and <sup>Ψ</sup> live, as they must, in the same space. Suppose we do not know the state of the living system; however, since it belongs to H(<sup>N</sup> ) his knowledge requires to know *<sup>N</sup>* complex numbers. Analogously we do not know the state *r*, and the state *w*, which require the specification of *R* and *NR* complex numbers. We now assume, with Wigner, that the collision matrix which gives the final state resulting from the interaction, which will be denoted as *S*, of the organism and the nutrient is a random matrix, which, however, even though unknown to us explicitly, is completely determined by the laws of quantum mechanics. Obviously *S* must satisfy:

16 Quantum Mechanics

collaborator, T. Weber.

*5.2.1. More on quantum cloning*

noise affecting the Laser which would inhibit its desired functioning. On the other hand, I was convinced that quantum theory in its general formulation and not due to limitations of practical nature would make unviable Herbert's proposal. After worrying for some days about this problem I got the general answer: while it is possible to devise an ideal apparatus which clones two orthogonal states with 100% efficiency, the same apparatus, if the linear quantum theory governs its functioning, cannot clone states which are linear combinations of the previous ones. Here is my argument, on the basis of which I recommended rejection

> 1 √2

and analogously for the left polarization. Now, the state at the r.h.s. of the last equation is by no means the state |*R*, 4*N*� which Herbert had assumed to occur in the case in which the Laser gain tube is triggered by a right polarized photon. But this is not the whole story: how it has been shown in ref.[31] the very linear nature of the theory implies that no difference in

This is an account of how Herbert's ingenious, but mistaken, proposal has led me to be the first to derive the no-cloning theorem 8. About one and half year later Wootters and Zurek [32] and Dieks [33] derived independently the same result and published it9. The theorem is of remarkable importance in quantum theory, it has become known as "The no cloning theorem" and it has been quoted an innumerable number of times. Only subsequently I realized that it had been a mistake on my part not to publish my result. I discussed my precise argument with Gozzini and Peierls, by sending them a draft which was a sort of repetition of my referee report and I subsequently published it [31] in collaboration with my

In a paper like the present one, we believe it useful to mention that E.P. Wigner [34], in an essay of 1961 had already argued that the phenomena of self-replication of biological systems contradict the principles of quantum mechanics. His argument is quite straightforward. Following his notation let us suppose we have a living system in a state *ν* and an environment (assimilated to "food") in a state *w*, so that, the initial statevector of the system, organism +

<sup>8</sup> I have chosen to attach at the end of the paper, a document - a letter by A. van der Merwe - which officially attests

<sup>9</sup> I must confess that I have never understood why A. Peres, in mentioning my derivation, has stated that it was a special case of the theorems in refs. [32] and [33]. Comparison even only of the short page by A. van der Merwe with the just mentioned papers makes clear that the argument is precisely the same and has the same generality.

this fact, since it is known only to a restricted community of physicists.


[*i*|*V*, 4*N*� + |*H*, 4*N*�], (15)

of Herbert's paper. The assumption that the cloning machine acts as follows:

[*i*|*V*� + |*H*�] →

the detections of Alice occurs in dependence of the free will choice of Bob.

when the linear nature of the theory is taken into account, implies:

*<sup>R</sup>* <sup>≡</sup> <sup>1</sup> √2

$$
\Psi = S\Phi.\tag{16}
$$

Choosing a basis for the whole Hilbert space and projecting Eq. (16) on such a basis one gets *N*2*R* equations. And now the conclusion follows: our unknown quantities are the components of the states *ν*,*r* and *w* on their respective bases and are therefore *N* + *R* + *NR* in number. Thus, according to Wigner, the question is: given the matrix corresponding to *S*, it is possible to find vectors *ν*,*r* and *w* such that their components satisfy the above mentioned *N*2*R* equations? Since *N*2*R* ≫ *N* + *R* + *NR*, for extremely large *N* and *R*, according to him: *it would be a miracle if such equations could be satisfied*. In other words, a self-replicating quantum unit does not exist. One might state that Wigner has "derived" (with the proviso he is making - see below) the no-cloning theorem for a quantum system whose Hilbert space has an extremely high dimensionality N, while we have shown that it holds also for N=2.

Wigner was perfectly aware that the argument is not fully rigorous and cannot be taken too seriously because of the many assumptions on which it is based. However, he seems inclined to attach a certain value to it. This is not surprising because at the time in which he wrote his paper he was adhering to von Neumann's idea that consciousness is responsible for the reduction of the wave packet, so that, in a certain sense, the fact that quantum mechanics is not able to account for the basic property of living organism (the self-reproduction) supported his view that such a theory cannot be used to describe the conscious perceptions of such organisms. In 1971 Eigen [36] responded to Wigner claiming that his choice of resorting to a typical unitary map to account for the process did not take into account the instructive functions of informational macromolecules.

Strictly connected with Wigner argument, even though derived through a much more rigorous and general procedure is the proof of the no-cloning theorem presented in a beautiful paper by R. Alicki [35]. He considers a dynamical transformation T from an initial state *ϕ* ⊗ *ω*, where *ϕ* is the state of the organism and *ω* the fixed initial state of the environment designed as "food":

$$
\varphi \otimes \omega \to T(\varphi \otimes \omega) = \varphi \otimes \varphi \otimes \sigma. \tag{17}
$$

As before, *σ* represents the state of the "food" after the replication. Alicki assumes that any dynamical process of a closed system (typically the one given by *T*) cannot reduce the indistinguishability of two states *ϕ* and *ψ*, which can be quantified by the "overlap" (*ϕ*|*ψ*) of the two states10, i.e., (*T*(*ϕ*)|*T*(*ψ*)) ≥ (*ϕ*|*ψ*) (which in Alicki's spirit can be considered as a form of the second law of thermodynamics: indistinguishability cannot decrease with the evolution). Then one has:

$$\begin{split} (\varphi|\boldsymbol{\varrho}') &= (\boldsymbol{\varrho}\otimes\boldsymbol{\omega}|\boldsymbol{\varrho}'\otimes\boldsymbol{\omega}) \leq (T(\boldsymbol{\varrho}\otimes\boldsymbol{\omega})|T(\boldsymbol{\varrho}'\otimes\boldsymbol{\omega})) \\ &= (\boldsymbol{\varrho}\otimes\boldsymbol{\varrho}\otimes\boldsymbol{\sigma}|\boldsymbol{\varrho}'\otimes\boldsymbol{\varrho}'\otimes\boldsymbol{\sigma}') = (\boldsymbol{\varrho}|\boldsymbol{\varrho}')^2(\boldsymbol{\sigma}|\boldsymbol{\sigma}') \leq (\boldsymbol{\varrho}|\boldsymbol{\varrho}')^2,\end{split} \tag{18}$$

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583

upon transmission. On the path of the photon emitted along *b*, after it goes through the first beam splitter, there is a phase-shifter *A* that shifts the phase of any photon passing through

A/B

i). *The phase shifter can be prepared not only in the states* |*A*� *and* |*B*�*, corresponding to its being inserted or removed from the path of the photon, but also in their orthogonal linear*

[|*A*� <sup>+</sup> <sup>|</sup>*B*�], <sup>|</sup>*v*�<sup>3</sup> <sup>=</sup> <sup>1</sup>

According to the author of [37], one can also switch on the Hamiltonian *H* for this macroscopic object, whose eigenstates are |*u*�<sup>3</sup> and |*v*�3, corresponding to slightly different energies, implying the development in time of relative phases with respect to each other. We will not go through the subsequent elementary calculations of the paper; we limit ourselves to mention that the above assumptions lead to the conclusion that, as the photons are nearing their final detectors represented in the figure by the 4 black squares, they will be

d

e

h

Entanglement, Nonlocality, Superluminal Signaling and Cloning

g

[|*A*�−|*B*�]. (20)

<sup>−</sup>*iα*|*h*�1|*d*′

*<sup>i</sup>γ*|*u*�3. (22)

�2)*<sup>e</sup>*

<sup>−</sup>*iβ*|*v*�<sup>3</sup> , (21)

f

√2

*<sup>i</sup>α*|*g*�1|*c*′

�<sup>2</sup> − *<sup>e</sup>*

c

it by *π*, and that can be inserted or removed from the beam at will.

a'

c'

d'

*combinations:*

<sup>|</sup>*ψ*�1,2,3 <sup>=</sup> <sup>1</sup>

2 (−*e*

**Figure 2.** Illustration of Greenberger's proposal as depicted in his paper.

<sup>|</sup>*u*�<sup>3</sup> <sup>=</sup> <sup>1</sup> √2

in the following entangled photon-phase shifter state:

�<sup>2</sup> + *<sup>e</sup>*

<sup>−</sup>*iα*|*g*�1|*c*′

�2)*<sup>e</sup>*

the phase factors *<sup>e</sup>*±*i<sup>α</sup>* and *<sup>e</sup>*±*i<sup>β</sup>* being due to the evolution of the states <sup>|</sup>*u*�<sup>3</sup> and <sup>|</sup>*v*�3. At this point Greeneberger puts forward his really crucial assumption. In his words:

*into the state eiγ*|*u*�*, where γ is the accumulated phase difference during this process.*

*<sup>T</sup>*|*u*�<sup>3</sup> = |*u*�3, *<sup>T</sup>*|*v*�<sup>3</sup> = *<sup>e</sup>*

ii). *In accordance with our assumption that one can manipulate these Cat states, one can turn off H for the state* |*v*�*, while leaving it in place for the state* |*u*�*. This will rotate the state* |*v*�

As it is obvious this amounts to accept that a nonunitary transformation *T* can be performed:

*<sup>i</sup>β*|*u*�<sup>3</sup> + (*<sup>e</sup>*

*<sup>i</sup>α*|*h*�1|*d*′

b'

S

At this point the first crucial assumption of the paper enters into play:

a

b

implying (*ϕ*|*ϕ*′ ) = 1 or (*ϕ*|*ϕ*′ ) = 0.

It is interesting to note that if, taking a strictly quantum perspective (which means to replace the round brackets in the above equation by Dirac's bras and kets), one identifies (as it is quite reasonable) the general concept of overlap≡indistinguishability with the scalar product of the Hilbert space and one assumes that the unfolding of the process is governed by a unitary transformation (which as such does not change the overlap), the above proof (slightly modified to take into account the complex nature of the scalar product) corresponds to a modern version of the no-cloning theorem which, in place of using the linearity of the evolution as we and the authors of refs.[32,33] did in deriving the theorem, makes resort to unitarity.

### **6. Further recent proposals which require new impossibility proofs**

#### **6.1. A proposal by D. Greenberger**

In spite of the lively debate and the many precise results which should have made fully clear why quantum mechanics does not allow superluminal communication, new papers claiming to have found a new way to achieve this result continue to appear. The first we want to mention is a proposal [37] of D. Greenberger which has been considered as inspiring even quite recently. Actually, in ref.[38] it is claimed that the proposal of Greenberger *has not yet been refused and calls into question the universality of the no-signaling theorem*, and, accordingly, it represents a stimulus to pursue the investigations on this line.

Greenberger proposal involves the simoultaneous emission of two photons by a source along two different opposite directions (*a*, *<sup>a</sup>*′ ) and (*b*, *<sup>b</sup>*′ ), so that the initial state is the entangled state:

$$|\psi\rangle\_{1,2} = \frac{1}{\sqrt{2}} [|a\rangle\_1 \otimes |a'\rangle\_2 + |b\rangle\_1 \otimes |b'\rangle\_2] \tag{19}$$

Subsequently, the two photons impinge on a series of beam splitters, as shown in figure 2. The horizontal gray boxes represent the beam splitters which are assumed to both reflect and transmit half the incident light, and produce a phase shift of *π*/2 upon reflection and none

<sup>10</sup> Here , the expressions (*α*|*β*) must not be identified with the Hilbert scalar product, since Alicki is taking a much more general perspective, which, however, requires to quantify the idea of distinguishability and its fundamental properties. He does this by introducing his symbol for the overlap which he takes, for simplicity, to be a real number between 0 and 1.

upon transmission. On the path of the photon emitted along *b*, after it goes through the first beam splitter, there is a phase-shifter *A* that shifts the phase of any photon passing through it by *π*, and that can be inserted or removed from the beam at will.

**Figure 2.** Illustration of Greenberger's proposal as depicted in his paper.

18 Quantum Mechanics

implying (*ϕ*|*ϕ*′

unitarity.

state:

between 0 and 1.

evolution). Then one has:

(*ϕ*|*ϕ*′

) = 1 or (*ϕ*|*ϕ*′

**6.1. A proposal by D. Greenberger**

two different opposite directions (*a*, *<sup>a</sup>*′

indistinguishability of two states *ϕ* and *ψ*, which can be quantified by the "overlap" (*ϕ*|*ψ*) of the two states10, i.e., (*T*(*ϕ*)|*T*(*ψ*)) ≥ (*ϕ*|*ψ*) (which in Alicki's spirit can be considered as a form of the second law of thermodynamics: indistinguishability cannot decrease with the

)=(*<sup>ϕ</sup>* <sup>⊗</sup> *<sup>ω</sup>*|*ϕ*′ <sup>⊗</sup> *<sup>ω</sup>*) <sup>≤</sup> (*T*(*<sup>ϕ</sup>* <sup>⊗</sup> *<sup>ω</sup>*)|*T*(*ϕ*′ <sup>⊗</sup> *<sup>ω</sup>*))

It is interesting to note that if, taking a strictly quantum perspective (which means to replace the round brackets in the above equation by Dirac's bras and kets), one identifies (as it is quite reasonable) the general concept of overlap≡indistinguishability with the scalar product of the Hilbert space and one assumes that the unfolding of the process is governed by a unitary transformation (which as such does not change the overlap), the above proof (slightly modified to take into account the complex nature of the scalar product) corresponds to a modern version of the no-cloning theorem which, in place of using the linearity of the evolution as we and the authors of refs.[32,33] did in deriving the theorem, makes resort to

**6. Further recent proposals which require new impossibility proofs**

In spite of the lively debate and the many precise results which should have made fully clear why quantum mechanics does not allow superluminal communication, new papers claiming to have found a new way to achieve this result continue to appear. The first we want to mention is a proposal [37] of D. Greenberger which has been considered as inspiring even quite recently. Actually, in ref.[38] it is claimed that the proposal of Greenberger *has not yet been refused and calls into question the universality of the no-signaling theorem*, and, accordingly,

Greenberger proposal involves the simoultaneous emission of two photons by a source along

) and (*b*, *<sup>b</sup>*′

[|*a*�<sup>1</sup> ⊗ |*a*′

Subsequently, the two photons impinge on a series of beam splitters, as shown in figure 2. The horizontal gray boxes represent the beam splitters which are assumed to both reflect and transmit half the incident light, and produce a phase shift of *π*/2 upon reflection and none

<sup>10</sup> Here , the expressions (*α*|*β*) must not be identified with the Hilbert scalar product, since Alicki is taking a much more general perspective, which, however, requires to quantify the idea of distinguishability and its fundamental properties. He does this by introducing his symbol for the overlap which he takes, for simplicity, to be a real number

�<sup>2</sup> <sup>+</sup> <sup>|</sup>*b*�<sup>1</sup> ⊗ |*b*′

)=(*ϕ*|*ϕ*′

)2(*σ*|*σ*′

) <sup>≤</sup> (*ϕ*|*ϕ*′

), so that the initial state is the entangled

�2] (19)

)2, (18)

= (*<sup>ϕ</sup>* <sup>⊗</sup> *<sup>ϕ</sup>* <sup>⊗</sup> *<sup>σ</sup>*|*ϕ*′ <sup>⊗</sup> *<sup>ϕ</sup>*′ <sup>⊗</sup> *<sup>σ</sup>*′

) = 0.

it represents a stimulus to pursue the investigations on this line.

<sup>|</sup>*ψ*�1,2 <sup>=</sup> <sup>1</sup> √2 At this point the first crucial assumption of the paper enters into play:

i). *The phase shifter can be prepared not only in the states* |*A*� *and* |*B*�*, corresponding to its being inserted or removed from the path of the photon, but also in their orthogonal linear combinations:*

$$|\mu\rangle\_3 = \frac{1}{\sqrt{2}} [|A\rangle + |B\rangle]\_\prime \qquad |v\rangle\_3 = \frac{1}{\sqrt{2}} [|A\rangle - |B\rangle]. \tag{20}$$

According to the author of [37], one can also switch on the Hamiltonian *H* for this macroscopic object, whose eigenstates are |*u*�<sup>3</sup> and |*v*�3, corresponding to slightly different energies, implying the development in time of relative phases with respect to each other.

We will not go through the subsequent elementary calculations of the paper; we limit ourselves to mention that the above assumptions lead to the conclusion that, as the photons are nearing their final detectors represented in the figure by the 4 black squares, they will be in the following entangled photon-phase shifter state:

$$|\psi\rangle\_{1,2,3} = \frac{1}{2} \left[ (-e^{i\mathbf{a}}|h\rangle\_1 |d'\rangle\_2 + e^{-i\mathbf{a}}|g\rangle\_1 |c'\rangle\_2)e^{i\beta}|u\rangle\_3 + (e^{i\mathbf{a}}|g\rangle\_1 |c'\rangle\_2 - e^{-i\mathbf{a}}|h\rangle\_1 |d'\rangle\_2)e^{-i\beta}|v\rangle\_3 \right]. \tag{21}$$

the phase factors *<sup>e</sup>*±*i<sup>α</sup>* and *<sup>e</sup>*±*i<sup>β</sup>* being due to the evolution of the states <sup>|</sup>*u*�<sup>3</sup> and <sup>|</sup>*v*�3.

At this point Greeneberger puts forward his really crucial assumption. In his words:

ii). *In accordance with our assumption that one can manipulate these Cat states, one can turn off H for the state* |*v*�*, while leaving it in place for the state* |*u*�*. This will rotate the state* |*v*� *into the state eiγ*|*u*�*, where γ is the accumulated phase difference during this process.*

As it is obvious this amounts to accept that a nonunitary transformation *T* can be performed:

$$T|\mu\rangle\_{\mathfrak{J}} = |\mu\rangle\_{\mathfrak{J}} \qquad T|\upsilon\rangle\_{\mathfrak{J}} = e^{i\gamma}|\mu\rangle\_{\mathfrak{J}}.\tag{22}$$

The conclusion follows. After this transformation the state becomes:

$$|\psi\_{f\text{final}}\rangle\_{1,2,3} = e^{i\gamma/2} \left[ -\cos(\mathfrak{a} + \mathfrak{z} - \gamma/2) |h\rangle\_1 |d'\rangle\_2 + \cos(\mathfrak{z} - \mathfrak{z} - \gamma/2) |g\rangle\_1 |c'\rangle\_2 \right] |\mathfrak{u}\rangle\_3. \tag{23}$$

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585

Entanglement, Nonlocality, Superluminal Signaling and Cloning

[|*h*+� + |*h*−�]. (27)

mirror

beam splitter

O |h-,t> |h+,t>

mirror

phase shifter

**6.2. A proposal involving a single system**

cannot be used to send superluminal signals.

O |h-,t> |h+,t>

mirror

a) b)

beam splitter

*x*-axis:

respectively.

Another proposal that has to be mentioned is the one [40] by Shiekh. His suggestion is different from all those which have appeared in the literature since the author does not make resort to an entangled state of two systems but he works with a single particle in a superposition of two states corresponding to its being in two far-away regions, and the measurement process involves only one of the two far-away parts of the wave function. So, in a sense, the argument of ref.[40] does not fall under the no-go theorems considered here and requires a separate comment. The author is inspired by the fact that when a single particle is associated to a wavefunction as the one just mentioned, any attempt to test whether it is "here" (at right), or "there" (at left) changes instantaneously the wavefunction on the whole real axis by making it equal to zero or enhancing it "there" according to whether we detect or we do not find the particle "here". The process seems to exhibit some nonlocal aspects due to the instantaneous change at-a-distance. Obviously, that this might lead to superluminal signaling is something that nobody can believe, but it is instructive to show that also in this case, to achieve the desired result, one has to resort to a nonunitary evolution. The elementary analysis which follows will lead once more to the conclusion that the process

We briefly review the argument by Shiekh. He considers a particle which is prepared, at time *t* = 0, in an equal weights superposition of two normalized states, |*h*+� and |*h*−�, propagating in two opposite directions, respectively, starting from the common origin of the

Subsequently the state |*h*+� is injected in an appropriate device behaving in a way rather similar, apart from the final stage, to a Mach-Zender interferometer. One also assumes that an observer, located near to it, can choose, at his free will, to insert or not a phase-shifter along one of the two paths of the interferometer. The two wave functions are then recombined by appropriate deflectors so that, by deciding whether or not to insert the phase-shifter, one can produce a constructive (no phase-shifter in place) or a destructive (the phase-shifter is present) interference of the two terms in which the impinging state |*h*+� has been split. Finally, a detector is placed along the direction of propagation of the final state and it induces wave packet reduction, since it either detects or fails to detect the particle. We

<sup>|</sup>*ψ*, 0� <sup>=</sup> <sup>1</sup> √2

have summarized the situation for the two considered cases in Figs. 3a,b.

mirror

**Figure 3.** The experimental arrangement of ref.[40]. The two cases refer to no Phase-shifter inserted or phase shifter inserted,

The author then concludes: *If the sender* (the guy who can choose to insert or not the phase shifter), *arranges for constructive interference, then some of the particles will be "taken up" by the*

And now the game is over: by appropriately choosing the angles *α*, *β* and *γ*, one can, at his free will, suppress one of the two terms of the superposition of the photon states, i.e. one can make certain either the firing of the detector in *<sup>d</sup>*′ or the one in *<sup>c</sup>*′ (and correspondingly the one in *h* or the one in *g*) allowing in this way a superluminal transfer of information from the phase shifter, which acts as the signaler, to the photon detectors.

The paper, since nobody had discussed it in spite of its revolutionary character, deserved some attention; we have reconsidered it in ref.[39]. Its weak points are:


We will not go on analyzing all the details of ref.[37] and of the punctual criticisms of ref.[39]. We believe that to show where it fails the simplest way is to resort to an example that we have devised in our paper. We consider an elementary EPR-Bohm like setup for two far away spin 1/2 particles in the singlet state:

$$|\psi\_{-}\rangle = \frac{1}{\sqrt{2}} [|\uparrow\_{1}\rangle|\downarrow\_{2}\rangle - |\downarrow\_{1}\rangle|\uparrow\_{2}\rangle].\tag{24}$$

In strict analogy with what has been assumed by Greenberger, suppose now we can rotate only one of the two spin states of particle 2 making it to coincide, apart from a controllable phase, with the other one:

$$T|\downarrow\downarrow\rangle = |\downarrow\downarrow\rangle, \qquad T|\uparrow\downarrow\rangle = e^{i\gamma}|\downarrow\downarrow\rangle. \tag{25}$$

Under this transformation the state [25] becomes a factorized state of the two particles:

$$|\psi\_T\rangle = \frac{1}{\sqrt{2}} [|\uparrow\_1\rangle - e^{i\gamma}|\downarrow\_1\rangle] |\downarrow\_2\rangle \tag{26}$$

In (26), the state referring to particle 1 is an eigenstate of *σ* · **d** for the direction **d** = (cos *γ*, sin *γ*, 0) pertaining to the eigenvalue −1. This means that a measurement of this observable by Alice (where particle 1 is) will give with certainty the outcome −1 if Bob has performed the transformation *T* on his particle, while, if Bob does nothing, the probability of getting such an outcome equals 1/2. Having such a device, one can easily implement superluminal transfer of information. Concluding: if assumption ii) were correct, one would not need all the complex apparatus involved in Greenberger's proposal which, at any rate, cannot work as indicated due to the nonlinear nature of *T*.

### **6.2. A proposal involving a single system**

20 Quantum Mechanics


*<sup>i</sup>γ*/2

is impossible to get in practice.

spin 1/2 particles in the singlet state:

phase, with the other one:

The conclusion follows. After this transformation the state becomes:

the phase shifter, which acts as the signaler, to the photon detectors.

some attention; we have reconsidered it in ref.[39]. Its weak points are:

the possibility of implementing a nonunitary transformation.

<sup>|</sup>*ψ*−� <sup>=</sup> <sup>1</sup> √2

> <sup>|</sup>*ψT*� <sup>=</sup> <sup>1</sup> √2

cannot work as indicated due to the nonlinear nature of *T*.

<sup>−</sup> cos(*<sup>α</sup>* <sup>+</sup> *<sup>β</sup>* <sup>−</sup> *<sup>γ</sup>*/2)|*h*�1|*d*′

And now the game is over: by appropriately choosing the angles *α*, *β* and *γ*, one can, at his free will, suppress one of the two terms of the superposition of the photon states, i.e. one can make certain either the firing of the detector in *<sup>d</sup>*′ or the one in *<sup>c</sup>*′ (and correspondingly the one in *h* or the one in *g*) allowing in this way a superluminal transfer of information from

The paper, since nobody had discussed it in spite of its revolutionary character, deserved

• The assumption that one can prepare the linear superposition of two macroscopically different states, corresponding to different locations of the macroscopic phase-shifter. This

• However, even ignoring the above critical feature of the hypothetical experiment, the really crucial and unacceptable fact is the one embodied in its second assumption, i.e.,

We will not go on analyzing all the details of ref.[37] and of the punctual criticisms of ref.[39]. We believe that to show where it fails the simplest way is to resort to an example that we have devised in our paper. We consider an elementary EPR-Bohm like setup for two far away

In strict analogy with what has been assumed by Greenberger, suppose now we can rotate only one of the two spin states of particle 2 making it to coincide, apart from a controllable

*<sup>T</sup>*| ↓2� = | ↓2�, *<sup>T</sup>*| ↑2� = *<sup>e</sup>*

Under this transformation the state [25] becomes a factorized state of the two particles:

[| ↑1� − *<sup>e</sup>*

In (26), the state referring to particle 1 is an eigenstate of *σ* · **d** for the direction **d** = (cos *γ*, sin *γ*, 0) pertaining to the eigenvalue −1. This means that a measurement of this observable by Alice (where particle 1 is) will give with certainty the outcome −1 if Bob has performed the transformation *T* on his particle, while, if Bob does nothing, the probability of getting such an outcome equals 1/2. Having such a device, one can easily implement superluminal transfer of information. Concluding: if assumption ii) were correct, one would not need all the complex apparatus involved in Greenberger's proposal which, at any rate,

�<sup>2</sup> <sup>+</sup> cos(*<sup>β</sup>* <sup>−</sup> *<sup>α</sup>* <sup>−</sup> *<sup>γ</sup>*/2)|*g*�1|*c*′

[| ↑1�| ↓2�−|↓1�| ↑2�]. (24)

*<sup>i</sup>γ*| ↓2�. (25)

*<sup>i</sup>γ*| ↓1�]| ↓2� (26)

�2 


Another proposal that has to be mentioned is the one [40] by Shiekh. His suggestion is different from all those which have appeared in the literature since the author does not make resort to an entangled state of two systems but he works with a single particle in a superposition of two states corresponding to its being in two far-away regions, and the measurement process involves only one of the two far-away parts of the wave function. So, in a sense, the argument of ref.[40] does not fall under the no-go theorems considered here and requires a separate comment. The author is inspired by the fact that when a single particle is associated to a wavefunction as the one just mentioned, any attempt to test whether it is "here" (at right), or "there" (at left) changes instantaneously the wavefunction on the whole real axis by making it equal to zero or enhancing it "there" according to whether we detect or we do not find the particle "here". The process seems to exhibit some nonlocal aspects due to the instantaneous change at-a-distance. Obviously, that this might lead to superluminal signaling is something that nobody can believe, but it is instructive to show that also in this case, to achieve the desired result, one has to resort to a nonunitary evolution. The elementary analysis which follows will lead once more to the conclusion that the process cannot be used to send superluminal signals.

We briefly review the argument by Shiekh. He considers a particle which is prepared, at time *t* = 0, in an equal weights superposition of two normalized states, |*h*+� and |*h*−�, propagating in two opposite directions, respectively, starting from the common origin of the *x*-axis:

$$<\langle \psi, 0 \rangle = \frac{1}{\sqrt{2}} [|h+\rangle + |h-\rangle]. \tag{27}$$

Subsequently the state |*h*+� is injected in an appropriate device behaving in a way rather similar, apart from the final stage, to a Mach-Zender interferometer. One also assumes that an observer, located near to it, can choose, at his free will, to insert or not a phase-shifter along one of the two paths of the interferometer. The two wave functions are then recombined by appropriate deflectors so that, by deciding whether or not to insert the phase-shifter, one can produce a constructive (no phase-shifter in place) or a destructive (the phase-shifter is present) interference of the two terms in which the impinging state |*h*+� has been split. Finally, a detector is placed along the direction of propagation of the final state and it induces wave packet reduction, since it either detects or fails to detect the particle. We have summarized the situation for the two considered cases in Figs. 3a,b.

**Figure 3.** The experimental arrangement of ref.[40]. The two cases refer to no Phase-shifter inserted or phase shifter inserted, respectively.

The author then concludes: *If the sender* (the guy who can choose to insert or not the phase shifter), *arranges for constructive interference, then some of the particles will be "taken up" by the* *sender, but none if destructive interference is arranged; in this way the sender can control the intensity of the beam detected by the receiver* (the observer located far away where the evolved of |*h*−� is concentrated). *So, a faster than light transmitter of information (but not energy or matter) might be possible*.

http://dx.doi.org/10.5772/CHAPTERDOI

http://dx.doi.org/10.5772/56429

Entanglement, Nonlocality, Superluminal Signaling and Cloning

2)

587

so-called hidden variables). The best known and rigorous example of this line is represented by Bohmian mechanics [42], a deterministic theory such that the assignement of the wavefunction and of the hidden variables (i.e. the initial positions of all particles which are chosen to be distributed according to the quantum probability |*ψ*(**r**1,**r**2, ...,**r***N*, *<sup>t</sup>*0)|

determines uniquely their positions at any subsequent time. The predictions of the theory concerning the probability distribution of the particles coincide with those of standard quantum mechanics and the theory overcomes the measurement problem in a clean and

I will spend only few words on the locality issue within Bohmian mechanics. Since this theory agrees with quantum mechanics in general and typically in an EPR-like situation, it must exhibit a specific sort of nonlocality. It has been proved [43] that any deterministic hidden variable theory equivalent to quantum mechanics admits only relativistic generalizations which must resort to a specific foliation of space-time. In other words, such theories are characterized by a preferred reference frame which, however, remains unaccessible. Accordingly, as stressed by Bell [8], they require a change of attitude concerning Lorentz invariance: the situation resembles the one of the theory of relativity in the Fitzgerald, Larmor, Lorentz and Poincaré formulation in which there is an absolute aether, but the contraction of space and the dilation of time fooled the moving observers by allowing them to consider themselves at rest. In spite of this remark, explicit and interesting relativistic generalizations of Bohmian mechanics have been presented. In particular bohmian-like relativistic models have been worked out both in first quantized

The second alternative corresponds to assuming that Schrödinger's equation has to be changed in such a way not to alter the well established predictions of quantum mechanics for all microscopic systems while leading to the collapse of the statevector with the desired features and probabilities when macroscopic systems enter into play. The first explicit example of this kind is the so called GRW theory [5] which we summarize in a very sketchy

The central idea is to modify the linear and deterministic evolution equation of standard quantum mechanics by adding nonlinear and stochastic (i.e. sharing the features of the reduction process) terms to it. As it is obvious, and as it has been stressed by many scientists, since the situations characterizing macro-objects correspond to perceptually different locations of their macroscopic parts (e.g. the pointer) the change in the dynamics must strive to make definite the positions of macroscopic bodies. The model is based on the

• A Hilbert space H is associated to any physical system and the state of the system is

• The evolution of the system is governed by Schrödinger's equation. Moreover, at random times, with mean frequency11 *λ*, each particle of any system is subjected to random

<sup>11</sup> Actually this frequency must be made proportional to the mass of the particles entering into play. The value we will

versions [44] as well as in the framework of quantum field theories [45].

logically consistent way.

**7.2. Collapse theories**

following assumptions:

choose below refers to nucleons.

represented by a normalized vector |*ψt*� of H,

way.

We believe that all readers will have clear the trivial mistake of the paper. In fact, what one can govern by deciding whether to insert or not the phase-shifter, is the interference at the central region of the final detector. Let us concentrate our considerations only on the normalized state |*h*+�. If triggered by such a state when it exhibits constructive interference, the counter will register (practically) for sure the particle described by such a state (the wavefunction has a peak just there), while, if there is destructive interference, the counter will not register the particle. But this does not mean that the wavefunction associated to |*h*+� disappears, as claimed by the author, it simply means that its support lies outside the interval covered by the counter. Actually, no one will doubt that if one places an array of counters covering all the line orthogonal to the final direction of propagation, one of them, different from the one of the experiment, will fire for sure. If one combines these considerations with the fact that actually the whole state of the particle is the superposition of |*h*−� and |*h*+� one realizes that the statements we have just made concerning what is going on at right have only probability 1/2 of occurrence, since the particle can be not detected in the right region. Accordingly, the probability that the particle is found at left remains equal to 1/2, as if no specific action would be made at right.

It seems rather peculiar that the author introduces an hypothetical process which can make zero a wavefunction (i.e. the normalized state |*h*+�), and as such it does not preserve unitarity, and, at the same time, he resorts to the overall conservation of probability (i.e. to unitarity) to claim that the action at right can change the norm of the state at left.

This concludes our analysis of the many proposals which have been presented to send superluminal signals.

### **7. Nonlocality and relativistic requirements**

As already stated, quantum mechanics suffers of an internal inconsistency, the one between the linear and deterministic evolution induced by Schrödinger's dynamics and the nonlinear and stochastic collapse of the state in a measurement. Many scientists, among them Einstein, Schrödinger, Bell and many others have been disturbed not only by the formal inconsistency between the two dynamical principles, but especially by the fact that the borderline between what is classical and what is quantum, what is reversible and what is irreversible, what is micro and what is macro is to a large extent ambiguous. Accordingly, many serious attempts to overcome this difficulty have been presented, inspired by the conviction that Bell has expressed [6] so lucidly:

*Either the wavefunction, as given by the Schrödinger equation is not everything or it is not right.*

### **7.1. Bohmian mechanics**

The first alternative corresponds to the idea that the specification of the state of a physical system given by the statevector has to be enriched or replaced by new variables (the so-called hidden variables). The best known and rigorous example of this line is represented by Bohmian mechanics [42], a deterministic theory such that the assignement of the wavefunction and of the hidden variables (i.e. the initial positions of all particles which are chosen to be distributed according to the quantum probability |*ψ*(**r**1,**r**2, ...,**r***N*, *<sup>t</sup>*0)| 2) determines uniquely their positions at any subsequent time. The predictions of the theory concerning the probability distribution of the particles coincide with those of standard quantum mechanics and the theory overcomes the measurement problem in a clean and logically consistent way.

I will spend only few words on the locality issue within Bohmian mechanics. Since this theory agrees with quantum mechanics in general and typically in an EPR-like situation, it must exhibit a specific sort of nonlocality. It has been proved [43] that any deterministic hidden variable theory equivalent to quantum mechanics admits only relativistic generalizations which must resort to a specific foliation of space-time. In other words, such theories are characterized by a preferred reference frame which, however, remains unaccessible. Accordingly, as stressed by Bell [8], they require a change of attitude concerning Lorentz invariance: the situation resembles the one of the theory of relativity in the Fitzgerald, Larmor, Lorentz and Poincaré formulation in which there is an absolute aether, but the contraction of space and the dilation of time fooled the moving observers by allowing them to consider themselves at rest. In spite of this remark, explicit and interesting relativistic generalizations of Bohmian mechanics have been presented. In particular bohmian-like relativistic models have been worked out both in first quantized versions [44] as well as in the framework of quantum field theories [45].

### **7.2. Collapse theories**

22 Quantum Mechanics

*possible*.

specific action would be made at right.

**7. Nonlocality and relativistic requirements**

superluminal signals.

expressed [6] so lucidly:

**7.1. Bohmian mechanics**

*right.*

*sender, but none if destructive interference is arranged; in this way the sender can control the intensity of the beam detected by the receiver* (the observer located far away where the evolved of |*h*−� is concentrated). *So, a faster than light transmitter of information (but not energy or matter) might be*

We believe that all readers will have clear the trivial mistake of the paper. In fact, what one can govern by deciding whether to insert or not the phase-shifter, is the interference at the central region of the final detector. Let us concentrate our considerations only on the normalized state |*h*+�. If triggered by such a state when it exhibits constructive interference, the counter will register (practically) for sure the particle described by such a state (the wavefunction has a peak just there), while, if there is destructive interference, the counter will not register the particle. But this does not mean that the wavefunction associated to |*h*+� disappears, as claimed by the author, it simply means that its support lies outside the interval covered by the counter. Actually, no one will doubt that if one places an array of counters covering all the line orthogonal to the final direction of propagation, one of them, different from the one of the experiment, will fire for sure. If one combines these considerations with the fact that actually the whole state of the particle is the superposition of |*h*−� and |*h*+� one realizes that the statements we have just made concerning what is going on at right have only probability 1/2 of occurrence, since the particle can be not detected in the right region. Accordingly, the probability that the particle is found at left remains equal to 1/2, as if no

It seems rather peculiar that the author introduces an hypothetical process which can make zero a wavefunction (i.e. the normalized state |*h*+�), and as such it does not preserve unitarity, and, at the same time, he resorts to the overall conservation of probability (i.e.

This concludes our analysis of the many proposals which have been presented to send

As already stated, quantum mechanics suffers of an internal inconsistency, the one between the linear and deterministic evolution induced by Schrödinger's dynamics and the nonlinear and stochastic collapse of the state in a measurement. Many scientists, among them Einstein, Schrödinger, Bell and many others have been disturbed not only by the formal inconsistency between the two dynamical principles, but especially by the fact that the borderline between what is classical and what is quantum, what is reversible and what is irreversible, what is micro and what is macro is to a large extent ambiguous. Accordingly, many serious attempts to overcome this difficulty have been presented, inspired by the conviction that Bell has

*Either the wavefunction, as given by the Schrödinger equation is not everything or it is not*

The first alternative corresponds to the idea that the specification of the state of a physical system given by the statevector has to be enriched or replaced by new variables (the

to unitarity) to claim that the action at right can change the norm of the state at left.

The second alternative corresponds to assuming that Schrödinger's equation has to be changed in such a way not to alter the well established predictions of quantum mechanics for all microscopic systems while leading to the collapse of the statevector with the desired features and probabilities when macroscopic systems enter into play. The first explicit example of this kind is the so called GRW theory [5] which we summarize in a very sketchy way.

The central idea is to modify the linear and deterministic evolution equation of standard quantum mechanics by adding nonlinear and stochastic (i.e. sharing the features of the reduction process) terms to it. As it is obvious, and as it has been stressed by many scientists, since the situations characterizing macro-objects correspond to perceptually different locations of their macroscopic parts (e.g. the pointer) the change in the dynamics must strive to make definite the positions of macroscopic bodies. The model is based on the following assumptions:


<sup>11</sup> Actually this frequency must be made proportional to the mass of the particles entering into play. The value we will choose below refers to nucleons.

spontaneous localization processes as follows. If particle *i* suffers a localization then the statevector changes according to:

$$|\psi\_l\rangle \rightarrow \frac{L\_i(\mathbf{x})|\psi\_l\rangle}{||\,L\_i(\mathbf{x})|\psi\_l\rangle}; \; L\_i(\mathbf{x}) = (\frac{\mathfrak{a}}{\pi})^{3/4} e^{[-\frac{\mathfrak{a}}{2}(\mathfrak{k}\_l - \mathbf{x})^2]}\,,\tag{28}$$

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589

Entanglement, Nonlocality, Superluminal Signaling and Cloning

The conclusion should be obvious. A universal dynamics has been worked out which leaves (practically) unaltered all quantum predictions for microsystems but it accounts for wave packet reduction with probabilities in agreement with the quantum ones and for the classical behaviour of macroscopic systems, as well as for our definite perceptions concerning them. Few remarks: i). The model has been generalized and formulated in a mathematically much more satisfactory but physically equivalent way [46,47] by resorting to stochastic dynamical equations of the Ito or Stratonowich type, ii) The model is manifestly phenomenological but it gives some clear indications concerning the fact that the macro-objectification or measurement problem admits a consistent solution, iii). The model, even though it almost completely agrees with quantum mechanics at the microscopic level qualifies itself as a rival theory of quantum mechanics, one which can be tested against it. Accordingly, it suggests where to look for putting into evidence possible violations of linearity. In recent years a lot

Obviously, also Collapse theories exhibit nonlocal features. However for them there is no theorem forbidding to get a generalization which does not resort to a preferred reference frame. Lot of work has been done along these lines; I will limit myself to mention some relevant steps. Before doing this, I consider it interesting to stress that the problem of having a theory inducing instantaneous reduction at-a-distance is a quite old one which has seen a lively debate and important contributions by Landau and Peierls [48], Bohr and Rosenfeld

Soon after the GRW theory has been formulated, P. Pearle [53] has presented a field theoretic relativistic generalization of it which has subsequently [54] been shown to be fully Lorentz invariant. Unfortunately, the model had some limitations arising from the occurrence of divergences which were not easily amendable. In 2000 the author of the present chapter has presented [55] a genuinely relativistic toy model of a theory inducing reductions. The model satisfies all the strict conditions identified in refs.[51,52]. F. Dowker and collaborators [56] have presented, in 2004, a relativistic collapse model on a discrete space that does not require

The really important steps, however, occurred starting from 2007. R. Tumulka has presented [57] a fully satisfactory and genuinely relativistic invariant dynamical reduction model for a system of noninteracting fermions. Another important contribution [58] came from D. Bedingham. Finally, few month ago, a convincing proof of the viability of the Collapse theories in the relativistic domain has been presented [59]. The nice fact is that the conceptual attitude which underlies this attempt is that what the theory assumes to be true of the world around us is the mass density of the whole universe. In this way one recovers a unified,

I believe that the best way to conclude this Chapter, which has dealt in detail with the compatibility of quantum effects and relativistic requirements, is to quote a clarifying sentence from R. Tumulka [57], who has studied in great detail both the Bohmian as well

*A somewhat surprising feature of the present situation is that we seem to arrive at the following alternative: Bohmian mechanics shows that one can explain quantum mechanics, exactly and completely, if one is willing to pay with using a preferred reference slicing of space-time; our model suggests that one should be able to avoid a preferred slicing if one is willing to pay with*

general picture of a quantum universe both at the micro and macro levels.

as the Collapse approaches to this fundamental problem:

*a certain deviation from quantum mechanics.*

of work in this direction is going on.

a preferred slicing of space-time.

[49], Hellwig and Kraus [50] and Aharonov and Albert [51,52].

where **ˆx***<sup>i</sup>* is the position operator of particle *i*,

• The probability density for a collapse at **<sup>x</sup>** is *<sup>p</sup>*(**x**) =� *Li*(**x**)|*ψt*� �2, so that localizations occur more frequently where the particle has a larger probability of being found in a standard position measurement.

The most relevant fact of the process is its "trigger mechanism", i.e. the fact that, as one can show by passing to the centre-of-mass and relative coordinates, the localization frequency of the c.o.m. of a composite system is amplified with the number of particles, while the internal motion, with the choice for *α* we will make, remains practically unaffected. We have summarized the situation for a micro (at left) and macroscopic (at right – a pointer) system in Fig.4.

**Figure 4.** The localization of a single particle and of a macro-object according to GRW.

With these premises we can now make the choice for the parameters *α* (note that <sup>√</sup> 1 *<sup>α</sup>* gives the localization accuracy) and *λ* of the theory. In ref.[5] we have chosen:

$$
\alpha \simeq 10^{10} cm^{-2} , \lambda \simeq 10^{-16} \text{sec}^{-1} . \tag{29}
$$

Note that with these choices a microscopic system suffers a localization about every 10<sup>7</sup> years, and this is why the theory agrees with quantum mechanics for such systems, a macroscopic body every 10−8*sec* (due to the trigger mechanism in the case of an Avogadro number of particles the frequency becomes 10<sup>24</sup> · <sup>10</sup>−<sup>16</sup> = 108*sec*<sup>−</sup>1). As commented [6] by J. Bell: *The cat is not both dead and alive for more than a split second.*

http://dx.doi.org/10.5772/CHAPTERDOI

The conclusion should be obvious. A universal dynamics has been worked out which leaves (practically) unaltered all quantum predictions for microsystems but it accounts for wave packet reduction with probabilities in agreement with the quantum ones and for the classical behaviour of macroscopic systems, as well as for our definite perceptions concerning them.

24 Quantum Mechanics

in Fig.4.

the statevector changes according to:

standard position measurement.

where **ˆx***<sup>i</sup>* is the position operator of particle *i*,

Here There

+

The hit occurs Here

Localization function

Ensuing situation

*cat is not both dead and alive for more than a split second.*

**Figure 4.** The localization of a single particle and of a macro-object according to GRW.

the localization accuracy) and *λ* of the theory. In ref.[5] we have chosen:

With these premises we can now make the choice for the parameters *α* (note that <sup>√</sup>

Note that with these choices a microscopic system suffers a localization about every 10<sup>7</sup> years, and this is why the theory agrees with quantum mechanics for such systems, a macroscopic body every 10−8*sec* (due to the trigger mechanism in the case of an Avogadro number of particles the frequency becomes 10<sup>24</sup> · <sup>10</sup>−<sup>16</sup> = 108*sec*<sup>−</sup>1). As commented [6] by J. Bell: *The*

Here There +

<sup>|</sup>*ψt*� → *Li*(**x**)|*ψt*�

spontaneous localization processes as follows. If particle *i* suffers a localization then

*<sup>π</sup>* )3/4*<sup>e</sup>*

[− *<sup>α</sup>* <sup>2</sup> (**ˆx***i*−**x**)<sup>2</sup>]

Here There +

+

*<sup>α</sup>* <sup>≃</sup> <sup>10</sup>10*cm*<sup>−</sup>2, *<sup>λ</sup>* <sup>≃</sup> <sup>10</sup>−16*sec*<sup>−</sup>1. (29)

Localize this individual (black) particle Here

, (28)

1 *<sup>α</sup>* gives

� *Li*(**x**)|*ψt*� � ; *Li*(**x**)=( *<sup>α</sup>*

• The probability density for a collapse at **<sup>x</sup>** is *<sup>p</sup>*(**x**) =� *Li*(**x**)|*ψt*� �2, so that localizations occur more frequently where the particle has a larger probability of being found in a

The most relevant fact of the process is its "trigger mechanism", i.e. the fact that, as one can show by passing to the centre-of-mass and relative coordinates, the localization frequency of the c.o.m. of a composite system is amplified with the number of particles, while the internal motion, with the choice for *α* we will make, remains practically unaffected. We have summarized the situation for a micro (at left) and macroscopic (at right – a pointer) system Few remarks: i). The model has been generalized and formulated in a mathematically much more satisfactory but physically equivalent way [46,47] by resorting to stochastic dynamical equations of the Ito or Stratonowich type, ii) The model is manifestly phenomenological but it gives some clear indications concerning the fact that the macro-objectification or measurement problem admits a consistent solution, iii). The model, even though it almost completely agrees with quantum mechanics at the microscopic level qualifies itself as a rival theory of quantum mechanics, one which can be tested against it. Accordingly, it suggests where to look for putting into evidence possible violations of linearity. In recent years a lot of work in this direction is going on.

Obviously, also Collapse theories exhibit nonlocal features. However for them there is no theorem forbidding to get a generalization which does not resort to a preferred reference frame. Lot of work has been done along these lines; I will limit myself to mention some relevant steps. Before doing this, I consider it interesting to stress that the problem of having a theory inducing instantaneous reduction at-a-distance is a quite old one which has seen a lively debate and important contributions by Landau and Peierls [48], Bohr and Rosenfeld [49], Hellwig and Kraus [50] and Aharonov and Albert [51,52].

Soon after the GRW theory has been formulated, P. Pearle [53] has presented a field theoretic relativistic generalization of it which has subsequently [54] been shown to be fully Lorentz invariant. Unfortunately, the model had some limitations arising from the occurrence of divergences which were not easily amendable. In 2000 the author of the present chapter has presented [55] a genuinely relativistic toy model of a theory inducing reductions. The model satisfies all the strict conditions identified in refs.[51,52]. F. Dowker and collaborators [56] have presented, in 2004, a relativistic collapse model on a discrete space that does not require a preferred slicing of space-time.

The really important steps, however, occurred starting from 2007. R. Tumulka has presented [57] a fully satisfactory and genuinely relativistic invariant dynamical reduction model for a system of noninteracting fermions. Another important contribution [58] came from D. Bedingham. Finally, few month ago, a convincing proof of the viability of the Collapse theories in the relativistic domain has been presented [59]. The nice fact is that the conceptual attitude which underlies this attempt is that what the theory assumes to be true of the world around us is the mass density of the whole universe. In this way one recovers a unified, general picture of a quantum universe both at the micro and macro levels.

I believe that the best way to conclude this Chapter, which has dealt in detail with the compatibility of quantum effects and relativistic requirements, is to quote a clarifying sentence from R. Tumulka [57], who has studied in great detail both the Bohmian as well as the Collapse approaches to this fundamental problem:

*A somewhat surprising feature of the present situation is that we seem to arrive at the following alternative: Bohmian mechanics shows that one can explain quantum mechanics, exactly and completely, if one is willing to pay with using a preferred reference slicing of space-time; our model suggests that one should be able to avoid a preferred slicing if one is willing to pay with a certain deviation from quantum mechanics.*

### **Acknowledgements**

We thank Dr. R. Romano for an accurate critical reading of the manuscript.

### **Author details**

GianCarlo Ghirardi

Professor Emeritus, University of Triest, Italy

### **8. References**

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**Acknowledgements**

Professor Emeritus, University of Triest, Italy

**Author details** GianCarlo Ghirardi

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**Chapter 25**

**Provisional chapter**

**The Husimi Distribution: Development and**

**The Husimi Distribution: Development and**

The Husimi distribution, introduced by Kôdi Husimi in 1940 [1], is a quasi-probability distribution commonly used to study the correspondence between quantum and classical dynamics [2]. Also, it is employed to describe systems in different areas of physics such as Quantum Mechanics, Quantum Optics, Information Theory [3–8]. Additionally, in nanotechnology it is possible to obtain a clear description of localization –which corresponds to classicality– and is crucial to determine correctly the size of systems when the particle dynamics takes into account mobility boundaries [9]. Among its properties, it is always positive definite and unique, conversely it cannot be considered as a true probability distribution over the quantum-mechanical phase space, reason why it is often considered as a quasi probability distribution. Although it possesses no correct marginal properties, its usefulness is to allow the assessment of the expectation values in quantum mechanics in a way similar to the classical case [10]. The semiclassical Husimi probability distribution refers to a special type of probability, this

The Husimi distribution may be obtained in several ways; the strategy that we choose here is to derive it as the expectation value of the density operator in a basis of coherent states [11]. Therefore, the line

> Coherent states ⇓ Husimi Distribution ⇓ Information measures,

©2012 Curilef and Pennini, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Curilef and Pennini; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2013 Curilef and Pennini; licensee InTech. This is a paper distributed under the terms of the Creative Commons

is for simultaneous but approximate location of position and momentum in phase space.

of working in this chapter is illustrated in the following sequence:

**Applications**

10.5772/53846

1. Introduction

**Applications**

Sergio Curilef and Flavia Pennini

Sergio Curilef and Flavia Pennini

http://dx.doi.org/10.5772/53846

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

**Provisional chapter**

### **The Husimi Distribution: Development and Applications Applications**

**The Husimi Distribution: Development and**

Sergio Curilef and Flavia Pennini Additional information is available at the end of the chapter

Sergio Curilef and Flavia Pennini

Additional information is available at the end of the chapter 10.5772/53846

http://dx.doi.org/10.5772/53846

### 1. Introduction

30 Quantum Mechanics <sup>594</sup> Advances in Quantum Mechanics

The Husimi distribution, introduced by Kôdi Husimi in 1940 [1], is a quasi-probability distribution commonly used to study the correspondence between quantum and classical dynamics [2]. Also, it is employed to describe systems in different areas of physics such as Quantum Mechanics, Quantum Optics, Information Theory [3–8]. Additionally, in nanotechnology it is possible to obtain a clear description of localization –which corresponds to classicality– and is crucial to determine correctly the size of systems when the particle dynamics takes into account mobility boundaries [9]. Among its properties, it is always positive definite and unique, conversely it cannot be considered as a true probability distribution over the quantum-mechanical phase space, reason why it is often considered as a quasi probability distribution. Although it possesses no correct marginal properties, its usefulness is to allow the assessment of the expectation values in quantum mechanics in a way similar to the classical case [10]. The semiclassical Husimi probability distribution refers to a special type of probability, this is for simultaneous but approximate location of position and momentum in phase space.

The Husimi distribution may be obtained in several ways; the strategy that we choose here is to derive it as the expectation value of the density operator in a basis of coherent states [11]. Therefore, the line of working in this chapter is illustrated in the following sequence:

Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Curilef and Pennini; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Curilef and Pennini; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

©2012 Curilef and Pennini, licensee InTech. This is an open access chapter distributed under the terms of the

where the transcendence of defining correctly a set of coherent states and the Husimi distribution is evident, being the calculation of measures as Wehrl entropy and/or Fisher information a consequence of this procedure.

10.5772/53846

597

http://dx.doi.org/10.5772/53846

d2*z µ*(*z*) = 1, (3)

The Husimi Distribution: Development and Applications

*<sup>e</sup>*−β*En* |�*z*|*n*�|2, (4)

d2*z µ*(*z*) ln*µ*(*z*), (5)

� = δ*n*,*n*′ (7)

��*n*<sup>|</sup> = 1,<sup>ˆ</sup> (8)


†*a*ˆ + 1/2]– its set of

where {|*z*�} denotes the set of coherent states, which are the eigenstates of the annihilation operator *a*ˆ,

where the integration is carried out over the complex *z* plane and the differential is a real element of

For an arbitrary Hamiltonian *<sup>H</sup>*<sup>ˆ</sup> , with the discrete spectra {*En*}, being *<sup>n</sup>* a positive integer, the Husimi

The Wehrl entropy is a direct application that we introduce here, which is a useful measure of

The uncertainty principle manifests itself through the inequality *W* ≥ 1 which was first conjectured by

<sup>2</sup>/<sup>2</sup> <sup>∞</sup> ∑ *n*=0

*zn* √ *n*!

*Z* ∑*n*

i.e., *a*ˆ|*z*� = *z*|*z*� defined for all *z* ∈ **C** [11]. This distribution is normalized to unity according to

1 π 

*<sup>µ</sup>*(*z*) = <sup>1</sup>

*<sup>W</sup>* <sup>=</sup> <sup>−</sup> <sup>1</sup> π 

In the special case of the Harmonic Oscillator –whose Hamiltonian is *H*ˆ = *h*¯ω[*a*ˆ

<sup>|</sup>*z*� = *<sup>e</sup>*−|*z*<sup>|</sup>

�*n*|*n*′

vectors forms a basis in the state space. This can be expressed by the closure relation

where 1 stands for the identity operator in the space formed by eigenvectors. ˆ

∞ ∑ *n*=0 |*n*′

where δ*n*,*n*′ is the Kronecker delta function, and the energy-spectrum is given by *En* = *h*¯ω(*n* + 1/2), with *n* = 0, 1,... By definition, Hermitian operator *H*ˆ is an observable if this orthonormal system of

where {|*n*�} are a complete orthonormal set of phonon-eigenstates, that is,

area proportional to phase space element given by d2*z* = d*x*d*p*/2¯*h*.

where {|*n*�|} is the set of energy eigenstates with eigenvalues *En* [4, 5].

localization in phase-space [21, 22], whose pertinent definition reads

Wehrl [21] and later proved by Lieb (see, for instance Ref. [4]).

Glauber's coherent states is defined in the form [14]

distribution takes the form

Coherent states provide a close connection between classical and quantum formulations of a given system. They were introduced early by Erwin Schrödinger in 1926 [12], but the name *coherent state* appeared for first time in Glauber's papers [13, 14]– see a detailed study about this in Ref. [15]. It is known that is difficult to construct coherent states for arbitrary quantum mechanical systems. Klauder shows an elegant method for construct it in Ref. [16]. Furthermore, in Ref. [11] Gazeau and Klauder consider essential, among other things, to discuss what an appropriate formulation of coherent states needs [11]. For instance, they suggest a suitable set of requirements. Then, the main interest in this chapter is to discuss, starting from a well defined set of coherent states, some interesting problems related to the Husimi distribution applied to important systems in physics, such as, the harmonics oscillator [5], the Landau diamagnetism model [17, 18] and, the rigid rotator [6, 18]. Also, we will discuss some properties related to systems with continuous spectrum [19]. In each case, the Wehrl entropy is calculated as a possible application.

This chapter is organized as follows. In section 2 we start presenting the background material and methodology that will be employed in the following chapters. In section 3 we revise the Husimi distribution and the Wehrl entropy for the problem of a particle in a magnetic field. In section 4 we discuss phase space delocalization for the rigid rotator within a semiclassical context by recourse to the Husimi distributions of both the linear and the 3*D*−anisotropic instances. In section 5 we propose a procedure to generalize the Husimi distribution to systems with continuous spectrum. We start examining a pioneering work, by Gazeau and Klauder, where the concept of coherent states for systems with discrete spectrum was extended to systems with continuous one. Finally, some concluding remarks and open problems are commented in section 6 .

#### 2. Background material and methodology

In this section we center our attention in 3 topics that we consider relevant to understand the problems that will be discussed in the following sections. These are *i)* the Husimi distribution and the most direct application, i.e., Wehrl entropy, *ii)* a special basis to formulate a suitable set of coherent states and *iii)* a generalization of this concepts to systems with continuous spectrum.

#### 2.1. Husimi distribution and Wehrl entropy

The standard statistical mechanics starts conventionally using the Gibbs's canonical distribution, whose thermal density matrix is represented by

$$
\mathfrak{d} = Z^{-1} e^{-\mathfrak{H}\hat{\mathcal{H}}},
\tag{1}
$$

where *<sup>Z</sup>* <sup>=</sup> Tr(*e*−β*H*<sup>ˆ</sup> ) is the partition function, *<sup>H</sup>*<sup>ˆ</sup> is the Hamiltonian of the system, <sup>β</sup> <sup>=</sup> <sup>1</sup>/*kBT* the inverse temperature *T*, and *kB* the Boltzmann constant [20].

The Husimi distribution is obtained as the expectation value of the density operator in a basis of coherent states as follows [1]

$$
\mu(z) = \langle z|\mathfrak{d}|z\rangle,\tag{2}
$$

where {|*z*�} denotes the set of coherent states, which are the eigenstates of the annihilation operator *a*ˆ, i.e., *a*ˆ|*z*� = *z*|*z*� defined for all *z* ∈ **C** [11]. This distribution is normalized to unity according to

$$\frac{1}{\pi} \int \mathrm{d}^2 z \mu(z) = 1,\tag{3}$$

where the integration is carried out over the complex *z* plane and the differential is a real element of area proportional to phase space element given by d2*z* = d*x*d*p*/2¯*h*.

For an arbitrary Hamiltonian *<sup>H</sup>*<sup>ˆ</sup> , with the discrete spectra {*En*}, being *<sup>n</sup>* a positive integer, the Husimi distribution takes the form

$$\mu(z) = \frac{1}{Z} \sum\_{n} e^{-\beta E\_n} |\langle z|n\rangle|^2,\tag{4}$$

where {|*n*�|} is the set of energy eigenstates with eigenvalues *En* [4, 5].

2 Quantum Mechanics

of this procedure.

entropy is calculated as a possible application.

and open problems are commented in section 6 .

2. Background material and methodology

2.1. Husimi distribution and Wehrl entropy

inverse temperature *T*, and *kB* the Boltzmann constant [20].

thermal density matrix is represented by

where *<sup>Z</sup>* <sup>=</sup> Tr(*e*−β*H*<sup>ˆ</sup>

states as follows [1]

generalization of this concepts to systems with continuous spectrum.

where the transcendence of defining correctly a set of coherent states and the Husimi distribution is evident, being the calculation of measures as Wehrl entropy and/or Fisher information a consequence

Coherent states provide a close connection between classical and quantum formulations of a given system. They were introduced early by Erwin Schrödinger in 1926 [12], but the name *coherent state* appeared for first time in Glauber's papers [13, 14]– see a detailed study about this in Ref. [15]. It is known that is difficult to construct coherent states for arbitrary quantum mechanical systems. Klauder shows an elegant method for construct it in Ref. [16]. Furthermore, in Ref. [11] Gazeau and Klauder consider essential, among other things, to discuss what an appropriate formulation of coherent states needs [11]. For instance, they suggest a suitable set of requirements. Then, the main interest in this chapter is to discuss, starting from a well defined set of coherent states, some interesting problems related to the Husimi distribution applied to important systems in physics, such as, the harmonics oscillator [5], the Landau diamagnetism model [17, 18] and, the rigid rotator [6, 18]. Also, we will discuss some properties related to systems with continuous spectrum [19]. In each case, the Wehrl

This chapter is organized as follows. In section 2 we start presenting the background material and methodology that will be employed in the following chapters. In section 3 we revise the Husimi distribution and the Wehrl entropy for the problem of a particle in a magnetic field. In section 4 we discuss phase space delocalization for the rigid rotator within a semiclassical context by recourse to the Husimi distributions of both the linear and the 3*D*−anisotropic instances. In section 5 we propose a procedure to generalize the Husimi distribution to systems with continuous spectrum. We start examining a pioneering work, by Gazeau and Klauder, where the concept of coherent states for systems with discrete spectrum was extended to systems with continuous one. Finally, some concluding remarks

In this section we center our attention in 3 topics that we consider relevant to understand the problems that will be discussed in the following sections. These are *i)* the Husimi distribution and the most direct application, i.e., Wehrl entropy, *ii)* a special basis to formulate a suitable set of coherent states and *iii)* a

The standard statistical mechanics starts conventionally using the Gibbs's canonical distribution, whose

<sup>ρ</sup><sup>ˆ</sup> <sup>=</sup> *<sup>Z</sup>*−1*e*−β*H*<sup>ˆ</sup>

The Husimi distribution is obtained as the expectation value of the density operator in a basis of coherent

) is the partition function, *<sup>H</sup>*<sup>ˆ</sup> is the Hamiltonian of the system, <sup>β</sup> <sup>=</sup> <sup>1</sup>/*kBT* the

, (1)

*µ*(*z*) = �*z*|ρˆ|*z*�, (2)

The Wehrl entropy is a direct application that we introduce here, which is a useful measure of localization in phase-space [21, 22], whose pertinent definition reads

$$W = -\frac{1}{\pi} \int \mathrm{d}^2 z \,\mu(z) \, \ln \mu(z),\tag{5}$$

The uncertainty principle manifests itself through the inequality *W* ≥ 1 which was first conjectured by Wehrl [21] and later proved by Lieb (see, for instance Ref. [4]).

In the special case of the Harmonic Oscillator –whose Hamiltonian is *H*ˆ = *h*¯ω[*a*ˆ †*a*ˆ + 1/2]– its set of Glauber's coherent states is defined in the form [14]

$$|z\rangle = e^{-|z|^2/2} \sum\_{n=0}^{\infty} \frac{z^n}{\sqrt{n!}} |n\rangle,\tag{6}$$

where {|*n*�} are a complete orthonormal set of phonon-eigenstates, that is,

$$<\langle n|n'\rangle = \mathfrak{S}\_{n,n'} \tag{7}$$

where δ*n*,*n*′ is the Kronecker delta function, and the energy-spectrum is given by *En* = *h*¯ω(*n* + 1/2), with *n* = 0, 1,... By definition, Hermitian operator *H*ˆ is an observable if this orthonormal system of vectors forms a basis in the state space. This can be expressed by the closure relation

$$\sum\_{n=0}^{\infty} |n'\rangle\langle n| = \hat{1},\tag{8}$$

where 1 stands for the identity operator in the space formed by eigenvectors. ˆ

In this situation one conveniently resorts to

$$
\mu\_{HO}(z) = \left(\hat{\mathbf{1}} - e^{-\beta \hbar \mathbf{a}}\right) e^{-\left(1 - e^{-\beta \hbar \mathbf{a}}\right)|z|^2},\tag{9}
$$

$$W\_{HO} = 1 - \ln(1 - e^{-\beta \hbar \alpha}).\tag{10}$$

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), (15)

The Husimi Distribution: Development and Applications


, (18)

), (19)

. (20)

<sup>d</sup><sup>ε</sup> *<sup>e</sup>*−βωε |�*s*, <sup>γ</sup>|ε�|2, (21)

��ε<sup>|</sup> = 1,<sup>ˆ</sup> (16)

where {|ε�} stands for a basis of eigenstates, which we can generalize replacing suitably discrete parameters by continuous ones, sums by integrals and Kronecker by Dirac delta function [46]. In such a case, we can always chose a normalized basis of eigenvectors to rephrase Eqs. (7) and (8) in the

� = <sup>δ</sup>(ε−ε′

 <sup>ε</sup>*<sup>M</sup>* 0

 <sup>ε</sup>*<sup>M</sup>* 0

Coherent states (17) must satisfy resolution of identity. In this case, it was introduced in Ref. [11] the

where *s*′ is a variable of integration with 0 ≤ *s*′ < *s* ≤ ∞. In addition, a non-negative weight function

<sup>d</sup>τ(*s*, <sup>γ</sup>) = <sup>σ</sup>(*s*)*M*(*s*)<sup>2</sup> <sup>d</sup>*<sup>s</sup>* <sup>d</sup><sup>γ</sup>

Gazeau and Klauder shown that resolution of unity is satisfied for systems with continuous spectrum in the present formulation of coherent states [11]. In Ref. [19] the authors have proposed a continuous appearance of Eq. (4), replacing the discrete form by the continuous version of variables, functions and operators involved in the formalism. Hence, we are ready to define the Husimi distribution for systems

2π

) ≥ 0 was introduced in order to satisfy the second requirement. Then, the measure of integration

<sup>d</sup><sup>ε</sup> *<sup>s</sup>*ε*e*−*i*γε ρ(ε)

<sup>d</sup><sup>ε</sup> *<sup>s</sup>*2<sup>ε</sup> ρ(ε)

�ε|ε′

 <sup>ε</sup>*<sup>M</sup>* 0 |ε′

<sup>2</sup>/<sup>2</sup> and *z* = *se*−*i*γε into coherent states (6), we find

<sup>|</sup>*s*, <sup>γ</sup>� = *<sup>M</sup>*(*s*)−<sup>1</sup>

where *s* > 0. Since {|*s*, γ�} are orthonormals, the normalization factor *M*(*s*) is given by

*M*(*s*)<sup>2</sup> =

<sup>ρ</sup>(ε) = *<sup>s</sup>* 0 d*s* ′ *s* ′2ε σ(*s* ′

where ε*<sup>M</sup>* ≤ ∞ [11]. In the section 5 and here we use units in which ¯*h* = 1.

following manner [46]

If we set *M*(*s*) = *e*|*z*<sup>|</sup>

for *M*(*s*)<sup>2</sup> < ∞.

following relation

takes the form [11]

with continuous spectrum in the following manner:

*µQ*(*s*, <sup>γ</sup>) = <sup>1</sup>

*Z* <sup>ε</sup>*<sup>M</sup>* 0

σ(*s*′

and

which respectively are the useful analytical expressions for Husimi distribution and Wehrl entropy [4].

#### 2.2. Gazeau and Klauder's coherent states

Now, we go back to the set of coherent states defined in Eq. (6). Certainly, it is known that coherent states can be constructed in several ways by recourse to different techniques being its formulation of a not unique character. Nevertheless, contrary to this idea and in order to get a unifying perspective, Gazeau and Klauder have suggested that a suitable formalism for coherent states should satisfy at least the following requirements [11]:


$$\int\_{\mathcal{L}} |z\rangle\langle z| d\mathfrak{T}(z) = 1,\tag{11}$$

where |*z*��*z*| denotes a projector, which takes a state vector into a multiple of the vector |*z*�.

3. *Temporal Stability*: the evolution of any coherent state |*z*� always remains a coherent state, which leads to a relation of the form

$$|z(t)\rangle = e^{-i\hat{H}t/\hbar}|z\rangle,\tag{12}$$

where *z*(0) = *z*, for all *z* ∈ *L* and *t*.

4. *Action Identity*: this property requires that

$$
\langle z|\hat{H}|z\rangle = \hbar \mathbf{o} |z|^2. \tag{13}
$$

At this point, we remark that requirements (3) and (4) are directly satisfied when the spectrum of the Hamiltonian *<sup>H</sup>*<sup>ˆ</sup> of the system, has the form *En* <sup>∼</sup> *nh*¯ω, where *<sup>n</sup>* is the quantum number and <sup>ω</sup> is the frequency of the oscillator [11]. In addition, there are some shortcomings about these requirements; for instance, Gazeau and Klauder states cannot be used for degenerate systems. Furthermore, it is questionable that action identity leads to the classical action-angle variable interpretation [23].

#### 2.3. Continuous spectrum

Gazeau and Klauder proposed in Ref. [11] a formulation of coherent states for systems with continuous spectrum. They introduced a Hamiltonian *H*ˆ > 0, with a non-degenerate continuous spectrum, thus

$$
\hat{H}|\mathbf{c}\rangle = \mathfrak{osc}|\mathbf{c}\rangle, \quad 0 < \mathfrak{e} < \mathfrak{e}\_M \tag{14}
$$

where {|ε�} stands for a basis of eigenstates, which we can generalize replacing suitably discrete parameters by continuous ones, sums by integrals and Kronecker by Dirac delta function [46]. In such a case, we can always chose a normalized basis of eigenvectors to rephrase Eqs. (7) and (8) in the following manner [46]

$$
\langle \mathfrak{e} | \mathfrak{e}' \rangle = \mathfrak{d}(\mathfrak{e} - \mathfrak{e}'),
\tag{15}
$$

and

4 Quantum Mechanics

In this situation one conveniently resorts to

2.2. Gazeau and Klauder's coherent states

the following requirements [11]:

representation

means that the expression �|*z*′

leads to a relation of the form

2.3. Continuous spectrum

where *z*(0) = *z*, for all *z* ∈ *L* and *t*. 4. *Action Identity*: this property requires that

*µHO*(*z*)=(1<sup>ˆ</sup> <sup>−</sup>*e*−β*h*¯ω) *<sup>e</sup>*−(1−*e*−β*h*¯<sup>ω</sup>)|*z*<sup>|</sup>

which respectively are the useful analytical expressions for Husimi distribution and Wehrl entropy [4].

Now, we go back to the set of coherent states defined in Eq. (6). Certainly, it is known that coherent states can be constructed in several ways by recourse to different techniques being its formulation of a not unique character. Nevertheless, contrary to this idea and in order to get a unifying perspective, Gazeau and Klauder have suggested that a suitable formalism for coherent states should satisfy at least

1. *Continuity of labeling* refers to the map from the label space *L* into Hilbert space. This condition

�−|*z*�� → 0 whenever *z*′ → *z*. 2. *Resolution of Unity*: a positive measure τ(*z*) on *L* exists such that the unity operator admits the

where |*z*��*z*| denotes a projector, which takes a state vector into a multiple of the vector |*z*�. 3. *Temporal Stability*: the evolution of any coherent state |*z*� always remains a coherent state, which

<sup>|</sup>*z*(*t*)� <sup>=</sup> *<sup>e</sup>*−*iHt* <sup>ˆ</sup> /*h*¯

�*z*|*H*ˆ |*z*� = *h*¯ω|*z*|

At this point, we remark that requirements (3) and (4) are directly satisfied when the spectrum of the Hamiltonian *<sup>H</sup>*<sup>ˆ</sup> of the system, has the form *En* <sup>∼</sup> *nh*¯ω, where *<sup>n</sup>* is the quantum number and <sup>ω</sup> is the frequency of the oscillator [11]. In addition, there are some shortcomings about these requirements; for instance, Gazeau and Klauder states cannot be used for degenerate systems. Furthermore, it is

Gazeau and Klauder proposed in Ref. [11] a formulation of coherent states for systems with continuous spectrum. They introduced a Hamiltonian *H*ˆ > 0, with a non-degenerate continuous spectrum, thus

questionable that action identity leads to the classical action-angle variable interpretation [23].

 *L* 2

*WHO* <sup>=</sup> <sup>1</sup>−ln(1−*e*−β*h*¯ω). (10)



2. (13)

*<sup>H</sup>*<sup>ˆ</sup> <sup>|</sup>ε� <sup>=</sup> ωε|ε�, 0 <sup>&</sup>lt; <sup>ε</sup> <sup>&</sup>lt; <sup>ε</sup>*<sup>M</sup>* (14)

, (9)

$$\int\_0^{\mathfrak{e}\_M} |\mathfrak{e}'\rangle\langle\mathfrak{e}| = \hat{1},\tag{16}$$

where ε*<sup>M</sup>* ≤ ∞ [11]. In the section 5 and here we use units in which ¯*h* = 1. If we set *M*(*s*) = *e*|*z*<sup>|</sup> <sup>2</sup>/<sup>2</sup> and *z* = *se*−*i*γε into coherent states (6), we find

$$\langle s, \mathfrak{y} \rangle = M(s)^{-1} \int\_0^{\mathfrak{e}\_M} d\mathfrak{e} \, \frac{s^{\mathfrak{E}} e^{-l \mathfrak{p} \mathfrak{e}}}{\sqrt{\mathfrak{p}(\mathfrak{e})}} \, |\mathfrak{e}\rangle,\tag{17}$$

where *s* > 0. Since {|*s*, γ�} are orthonormals, the normalization factor *M*(*s*) is given by

$$\left(M(\mathbf{s})\right)^{2} = \int\_{0}^{\mathbf{c}\_{M}} \mathbf{d}\mathbf{\varepsilon} \frac{\mathbf{s}^{2\varepsilon}}{\mathfrak{p}(\varepsilon)},\tag{18}$$

for *M*(*s*)<sup>2</sup> < ∞.

Coherent states (17) must satisfy resolution of identity. In this case, it was introduced in Ref. [11] the following relation

$$\mathfrak{p}(\mathfrak{e}) = \int\_0^s \mathrm{d}s' \, s'^{2\mathfrak{e}} \sigma(s'),\tag{19}$$

where *s*′ is a variable of integration with 0 ≤ *s*′ < *s* ≤ ∞. In addition, a non-negative weight function σ(*s*′ ) ≥ 0 was introduced in order to satisfy the second requirement. Then, the measure of integration takes the form [11]

$$\operatorname{d}\pi(s,\gamma) = \sigma(s)\mathcal{M}(s)^2 \operatorname{ds}\frac{\operatorname{d}\gamma}{2\pi}.\tag{20}$$

Gazeau and Klauder shown that resolution of unity is satisfied for systems with continuous spectrum in the present formulation of coherent states [11]. In Ref. [19] the authors have proposed a continuous appearance of Eq. (4), replacing the discrete form by the continuous version of variables, functions and operators involved in the formalism. Hence, we are ready to define the Husimi distribution for systems with continuous spectrum in the following manner:

$$
\mu\_{\mathcal{Q}}(s,\boldsymbol{\eta}) = \frac{1}{Z} \int\_0^{\varepsilon\_{\mathcal{M}}} \mathrm{d}\mathfrak{e} \, e^{-\beta \mathrm{d}\mathfrak{e}} \, |\langle s, \boldsymbol{\eta}|\boldsymbol{\varepsilon} \rangle|^2,\tag{21}
$$

where ε stands for a continuous parameter. The Husimi distribution is normalized according to

$$\int\_{0}^{\infty} \int\_{-\infty}^{\infty} \mathrm{d}\tau(s,\eta)\mu\_{\mathcal{Q}}(s,\eta) = 1,\tag{22}$$

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the authors focussed only in the transverse motion of a particle. For this reason, it was necessary to normalize the Husimi distribution in order to arrive to a consistent expression for semiclassical measures

Certainly, because the relevant effects seem to come only from the transverse motion, several efforts are made to describe this problem in two dimensions [7, 8, 27, 28, 32-34]. Furthermore, since the discovery of interesting phenomena, as the quantum Hall effect, there has been much interest in understanding the dynamics of electrons confined to move in two dimensions in the presence of a magnetic field perpendicular to the motion plane [31]. The confinement is possible at the *interface* between two materials, typically a semiconductor and an insulator, where a quantum well that traps the particles is formed, forbidding their motion in the direction perpendicular to the interface plane at low energies. However, we propose here to discuss this problem in the most complete form (three dimensions), some results related to the behavior of the Wehrl entropy. From the present line of reasoning, it is concluded that the two-dimensional formulation is sufficient unto itself to explain the problem whenever the length of the cylindrical geometry of the system is large enough. Nevertheless, as suggested before, electronic devices are based in interfaces. Thus, this fact theoretically imposes a natural lower temperature bound

We enter the present application by revisiting the complete set of coherent states of a spinless charged

−→<sup>π</sup> <sup>=</sup> −→*<sup>p</sup>* <sup>+</sup> *<sup>q</sup>*

*H* =

πˆ<sup>±</sup> = *p*ˆ*<sup>x</sup>* ±*ip*ˆ*<sup>y</sup>* ±

of a particle of charge *<sup>q</sup>*, mass *mq*, and linear momentum −→*<sup>p</sup>* , subject to the action of a vector potential −→*<sup>A</sup>* . These are the essential ingredients of the well-known Landau model for diamagnetism: a spinless charged particle in a magnetic field *B* (we follow the presentation of Feldman et al. [28]). The

> −→π · −→π 2*mq*

−→*<sup>A</sup>* . The vector potential is chosen in the symmetric gauge as −→*<sup>A</sup>* = (−*By*/2,*Bx*/2, 0), which corresponds to a uniform magnetic field along the *<sup>z</sup>*−direction.

By using the quantum formulation of the step-ladder operators [28], one needs to define the step

*ih*¯ 2ℓ<sup>2</sup> B

*c*

−→*<sup>A</sup>* , (27)

The Husimi Distribution: Development and Applications

, (28)

(*x*ˆ±*iy*ˆ), (29)

<sup>ℓ</sup><sup>B</sup> = (*hc*¯ /*qB*)1/<sup>2</sup> (30)

that emerges from the analysis when three dimensions are considered [18].

particle in a uniform magnetic field. Consider the classical kinetic momentum

3.1. The model of one charged particle in a magnetic field

∇ ×

[7, 8, 32].

Hamiltonian reads [28]

operators as follows [28]

where the length

and the magnetic field is −→*<sup>B</sup>* <sup>=</sup> −→

where the measure dτ(*s*, γ) is given by Eq. (20).

We see easily from Eq. (17) that, the projection of eingensatates of the Hamiltonian over coherent states, is given by

$$
\langle \mathfrak{s}, \mathfrak{y} | \mathfrak{e} \rangle = \mathcal{M}(\mathfrak{s})^{-1} \frac{s^{\mathfrak{e}} e^{-i \mathfrak{y} \mathfrak{e}}}{\sqrt{\mathfrak{p}(\mathfrak{e})}}, \tag{23}
$$

where we have considered from Eq. (15) the orthogonality of the continuous states {|ε�}. Introducing the above expression into Eq. (21) we finally arrive to [19]

$$
\mu\_{\mathcal{Q}}(\mathbf{s}) = \frac{M(\mathbf{s})^{-2}}{Z} \int\_0^{\mathbf{e}\_M} d\mathbf{e} \, \frac{e^{-\beta \text{ae}\_S 2\varepsilon}}{\mathfrak{p}(\mathbf{e})},\tag{24}
$$

where we have dropped out the dependence on γ. The continuous partition function obviously is [20]

$$Z = \int\_0^{\mathfrak{E}\_M} \mathrm{d}\mathfrak{E} \, e^{-\beta \mathrm{oc}}.\tag{25}$$

It is important to note that Eq. (24) is consistently normalized in accordance with

$$\int\_0^\infty \mathrm{d}\mathfrak{r}(s)\,\mu\_{\mathcal{Q}}(s) = 1,\tag{26}$$

and in this case, the measure is dτ(*s*) = σ(*s*)*M*(*s*)<sup>2</sup> d*s*.

#### 3. Landau diamagnetism: Charged particle in a uniform magnetic field

Diamagnetism was a problem firstly appointed by Landau who showed the discreteness of energy levels for a charged particle in a magnetic field [24]. By the observation of the diverse scenarios in the framework provided by the Landau diamagnetism we can study some relevant physical properties [25– 27] as thermodynamic limit, role of boundaries, decoherence induced by the environment. The main motivation for several specialists work even today it is to make an accurate description of its theoretical and practical consequences.

In the past the appropriate partition function for this problem was calculated by Feldman and Kahn appealing to the concept of Glauber's coherent states as a set of basis states [28]. This formulation allows the use of classical concepts to describe electron orbits, even containing all quantum effects [28]. In a previous effort, this approach was used to obtain the Wehrl entropy [21, 22] and Fisher information [29] with the purpose of studying the thermodynamics of the Landau diamagnetism problem, namely, a free spinless charged particle in a uniform magnetic field [7]. In such contribution the authors focussed only in the transverse motion of a particle. For this reason, it was necessary to normalize the Husimi distribution in order to arrive to a consistent expression for semiclassical measures [7, 8, 32].

Certainly, because the relevant effects seem to come only from the transverse motion, several efforts are made to describe this problem in two dimensions [7, 8, 27, 28, 32-34]. Furthermore, since the discovery of interesting phenomena, as the quantum Hall effect, there has been much interest in understanding the dynamics of electrons confined to move in two dimensions in the presence of a magnetic field perpendicular to the motion plane [31]. The confinement is possible at the *interface* between two materials, typically a semiconductor and an insulator, where a quantum well that traps the particles is formed, forbidding their motion in the direction perpendicular to the interface plane at low energies.

However, we propose here to discuss this problem in the most complete form (three dimensions), some results related to the behavior of the Wehrl entropy. From the present line of reasoning, it is concluded that the two-dimensional formulation is sufficient unto itself to explain the problem whenever the length of the cylindrical geometry of the system is large enough. Nevertheless, as suggested before, electronic devices are based in interfaces. Thus, this fact theoretically imposes a natural lower temperature bound that emerges from the analysis when three dimensions are considered [18].

#### 3.1. The model of one charged particle in a magnetic field

We enter the present application by revisiting the complete set of coherent states of a spinless charged particle in a uniform magnetic field. Consider the classical kinetic momentum

$$
\overrightarrow{\pi} = \overrightarrow{p} + \frac{q}{c}\overrightarrow{A},
\tag{27}
$$

of a particle of charge *<sup>q</sup>*, mass *mq*, and linear momentum −→*<sup>p</sup>* , subject to the action of a vector potential −→*<sup>A</sup>* . These are the essential ingredients of the well-known Landau model for diamagnetism: a spinless charged particle in a magnetic field *B* (we follow the presentation of Feldman et al. [28]). The Hamiltonian reads [28]

$$H = \frac{\overrightarrow{\pi} \cdot \overrightarrow{\pi}}{2m\_q},\tag{28}$$

and the magnetic field is −→*<sup>B</sup>* <sup>=</sup> −→ ∇ × −→*<sup>A</sup>* . The vector potential is chosen in the symmetric gauge as −→*<sup>A</sup>* = (−*By*/2,*Bx*/2, 0), which corresponds to a uniform magnetic field along the *<sup>z</sup>*−direction.

By using the quantum formulation of the step-ladder operators [28], one needs to define the step operators as follows [28]

$$\hbar\_{\pm} = \hat{p}\_{\chi} \pm i \hat{p}\_{\chi} \pm \frac{i\hbar}{2\ell\_{\text{B}}^{2}} (\hat{\chi} \pm i\hat{\chi}),\tag{29}$$

where the length

6 Quantum Mechanics

is given by

where ε stands for a continuous parameter. The Husimi distribution is normalized according to

We see easily from Eq. (17) that, the projection of eingensatates of the Hamiltonian over coherent states,

�*s*, <sup>γ</sup>|ε� <sup>=</sup> *<sup>M</sup>*(*s*)−<sup>1</sup> *<sup>s</sup>*ε*e*−*i*γε

where we have considered from Eq. (15) the orthogonality of the continuous states {|ε�}. Introducing

 <sup>ε</sup>*<sup>M</sup>* 0

where we have dropped out the dependence on γ. The continuous partition function obviously is [20]

ρ(ε)

<sup>d</sup><sup>ε</sup> *<sup>e</sup>*−βωε*s*2<sup>ε</sup>

dτ(*s*, γ)*µQ*(*s*, γ) = 1, (22)

, (23)

<sup>ρ</sup>(ε) , (24)

<sup>d</sup><sup>ε</sup> *<sup>e</sup>*−βωε. (25)

dτ(*s*)*µQ*(*s*) = 1, (26)

 <sup>∞</sup> 0

where the measure dτ(*s*, γ) is given by Eq. (20).

the above expression into Eq. (21) we finally arrive to [19]

and in this case, the measure is dτ(*s*) = σ(*s*)*M*(*s*)<sup>2</sup> d*s*.

and practical consequences.

 <sup>∞</sup> −∞

*µQ*(*s*) = *<sup>M</sup>*(*s*)−<sup>2</sup>

*Z* = <sup>ε</sup>*<sup>M</sup>* 0

It is important to note that Eq. (24) is consistently normalized in accordance with

 <sup>∞</sup> 0

3. Landau diamagnetism: Charged particle in a uniform magnetic field

Diamagnetism was a problem firstly appointed by Landau who showed the discreteness of energy levels for a charged particle in a magnetic field [24]. By the observation of the diverse scenarios in the framework provided by the Landau diamagnetism we can study some relevant physical properties [25– 27] as thermodynamic limit, role of boundaries, decoherence induced by the environment. The main motivation for several specialists work even today it is to make an accurate description of its theoretical

In the past the appropriate partition function for this problem was calculated by Feldman and Kahn appealing to the concept of Glauber's coherent states as a set of basis states [28]. This formulation allows the use of classical concepts to describe electron orbits, even containing all quantum effects [28]. In a previous effort, this approach was used to obtain the Wehrl entropy [21, 22] and Fisher information [29] with the purpose of studying the thermodynamics of the Landau diamagnetism problem, namely, a free spinless charged particle in a uniform magnetic field [7]. In such contribution

*Z*

$$\ell\_{\mathbb{B}} = (\hbar c/q\mathbb{B})^{1/2} \tag{30}$$

is the classical radius of the ground-state Landau orbit [28]. Motion along the *z*−axis is free [28]. For the transverse motion, the Hamiltonian specializes to [28]

$$
\hat{H}\_{\rm l} = \frac{\hbar\_{+}\hbar\_{-}}{2m\_{q}} + \frac{1}{2}\hbar\Omega\hat{1},\tag{31}
$$

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The Husimi Distribution: Development and Applications

where β = 1/*kBT*, *kB* the Boltzmann constant and *T* the temperature. Besides, *Z* is the partition function for the particle total motion. If *Z* is separated in a similar way as other physical properties are separated, it is possible to assure that *Z* = *ZlZt* , where *Zt* is the contribution for the transverse motion and *Zl* the

<sup>e</sup>−β*h*¯Ω(*n*+1/2)

being *L* the length of the cylinder, *A* = π*R*<sup>2</sup> the area for cylindrical geometry [28]. In addition, the matrix element |�*n*,*m*|α,ξ�|<sup>2</sup> represents the probability of finding the charged particle in the coherent

where η has been separated as a function of two distributions, namely, η*<sup>l</sup>* = η*l*(*pz*) and η*<sup>t</sup>* = η*t*(*x*, *px*;*y*, *py*). The dependence on the variable *z* has been missed due to the explicit form of the

*<sup>A</sup>mq*<sup>Ω</sup> (1−e−β*h*¯Ω) <sup>e</sup>−(1−e−β*h*¯<sup>Ω</sup>)|α<sup>|</sup>

where the length ℓ<sup>B</sup> is defined by the Eq. (30). From expressions (41) and (42), we emphasize again that η*l*(*pz*) describes the free motion of the particle in the magnetic field direction and η*t*(*x*, *px*;*y*, *py*) the Landau levels due to the circular motion in a transverse plane to the magnetic field, similar to the

<sup>2</sup>/2ℓ<sup>2</sup>

contain the complete description of the system. We noticed both distributions are naturally normalized

*Zl* = (*L*/*h*)(2π*mqkBT* )1/<sup>2</sup> and (38) *Zt* = *Amq*Ω/(4π*h*¯ sinh(β*h*¯Ω/2)), (39)

η = η*l*(*pz*)η*t*(*x*, *px*;*y*, *py*), (40)

, (41)

B. Consequently Eqs. (40), (41) and (42) together

*<sup>h</sup>* <sup>η</sup>*l*(*pz*) = 1, (43)

η*t*(*x*, *px*;*y*, *py*) = 1. (44)

<sup>B</sup> , (42)

<sup>2</sup>/2ℓ<sup>2</sup>

contribution for the one-dimensional free motion. Thus, the Husimi function [1] is written as

*<sup>z</sup>* /2*mq ZlZt* <sup>∑</sup>*<sup>n</sup>*,*<sup>m</sup>*

<sup>η</sup> <sup>=</sup> <sup>e</sup>−β*p*<sup>2</sup>

state |α,ξ� and we can find its expression as defined previously [34]. It should be noticed that the distribution η can be written as follows

hamiltonian *<sup>H</sup>*ˆ*l*. Accordingly, after summing in Eq. (37) we find

<sup>η</sup>*<sup>l</sup>* <sup>=</sup> <sup>e</sup>−β*p*<sup>2</sup>

<sup>η</sup>*<sup>t</sup>* <sup>=</sup> <sup>2</sup>π*h*¯

harmonic oscillator of Eq. (9) since |*z*|

in a standard form, i.e.,

and

*<sup>z</sup>* /2*mq Zl*

<sup>2</sup> → |α|

d*z*d*pz*

 d2αd2ξ 4π2ℓ<sup>4</sup> B

where

where an important quantity characterizes the problem, namely,

$$
\Omega = qB / m\_q c,\tag{32}
$$

the cyclotron frequency [33]. The eigenstates |*N*,*m*� are determined by two quantum numbers: *N* (associated to the energy) and *m* (to the *z*− projection of the angular momentum). As a consequence, they are simultaneously eigenstates of both *<sup>H</sup>*ˆ*<sup>t</sup>* and the angular momentum operator *<sup>L</sup>*ˆ*<sup>z</sup>* [28], so that

$$
\hat{H}\_{\text{I}}|N,m\rangle = \left(N + \frac{1}{2}\right) \hbar \Omega |N,m\rangle = E\_N |N,m\rangle \tag{33}
$$

and

$$
\hat{L}\_{\mathbb{C}}|N,m\rangle = m\hbar|N,m\rangle. \tag{34}
$$

We note that the eigenvalues of *<sup>L</sup>*ˆ*<sup>z</sup>* are not bounded by below (*<sup>m</sup>* takes the values <sup>−</sup>∞,...,−1, 0, 1,...,*N*) [28]. This agrees with the fact that the energies (*N* + 1/2)*h*¯Ω are infinitely degenerate [33]. Such a fact diminishes the physical relevance of phase-space localization for estimation purposes, as we shall see below. Moreover, *Lz* is not an independent constant of the motion [33].

There exists a analogous formulation of an charged particle in a magnetic field by Kowalski that takes into account the geometry of a circle [30] (and for a comparison with the Feldman formulation see Ref.[8]), but at this point, we choose the Feldman formulation to work because the measure is easily defined and the normalization condition and other semiclassical measures are well described.

#### 3.2. Husimi distribution and Wehrl entropy

We will start our present endeavor defining the Hamiltonian *<sup>H</sup>*<sup>ˆ</sup> <sup>=</sup> *<sup>H</sup>*ˆ*<sup>t</sup>* <sup>+</sup>*H*ˆ*<sup>l</sup>* for a particle of mass *mq* and charge *<sup>q</sup>* in a magnetic field *<sup>B</sup>*, where *<sup>H</sup>*ˆ*<sup>t</sup>* <sup>=</sup> *<sup>h</sup>*¯Ω(*N*<sup>ˆ</sup> <sup>+</sup>1/2) describes the transverse motion, being <sup>Ω</sup> the cyclotron frequency as defined by the Eq. (32) and *N*ˆ the number operator. In addition, the Hamiltonian *<sup>H</sup>*ˆ*<sup>l</sup>* <sup>=</sup> *<sup>p</sup>*<sup>ˆ</sup> 2 *<sup>z</sup>* /2*mq* represents a longitudinal one-dimensional free motion. After constructing a coherent state basis, a possible way to define the Husimi function η, for the complete motion, is given by

$$
\mathfrak{n}(\mathfrak{x}, p\_{\mathfrak{x}}; \mathfrak{y}, p\_{\mathfrak{y}}; p\_{\mathfrak{z}}) = \langle \mathfrak{a}, \mathfrak{f}, k\_{\mathfrak{z}} | \mathfrak{d} | \mathfrak{a}, \mathfrak{f}, k\_{\mathfrak{z}} \rangle,\tag{35}
$$

where ρˆ is the thermal density operator and the set {|α,ξ,*kz*�} represents the coherent states for the motion in three dimensions. Taking the direct product |α,ξ,*kz*�≡|α,ξ� |*kz*�, the set {|α,ξ�} corresponds to the coherent states of the transverse motion and {|*kz*�} to the longitudinal motion. Therefore, the thermal density operator is given by

$$\Phi = \frac{1}{Z} \mathbf{e}^{-\beta(\hat{H}\_l + \hat{H}\_l)},\tag{36}$$

where β = 1/*kBT*, *kB* the Boltzmann constant and *T* the temperature. Besides, *Z* is the partition function for the particle total motion. If *Z* is separated in a similar way as other physical properties are separated, it is possible to assure that *Z* = *ZlZt* , where *Zt* is the contribution for the transverse motion and *Zl* the contribution for the one-dimensional free motion. Thus, the Husimi function [1] is written as

$$\mathfrak{m} = \frac{\mathbf{e}^{-\beta p\_c^2 / 2m\_q}}{Z\_l Z\_l} \sum\_{n,m} \mathbf{e}^{-\beta \hbar \Omega(n+1/2)} |\langle n,m|\alpha,\mathfrak{k}\rangle|^2. \tag{37}$$

where

8 Quantum Mechanics

and

*<sup>H</sup>*ˆ*<sup>l</sup>* <sup>=</sup> *<sup>p</sup>*<sup>ˆ</sup> 2

is the classical radius of the ground-state Landau orbit [28]. Motion along the *z*−axis is free [28]. For

the cyclotron frequency [33]. The eigenstates |*N*,*m*� are determined by two quantum numbers: *N* (associated to the energy) and *m* (to the *z*− projection of the angular momentum). As a consequence, they are simultaneously eigenstates of both *<sup>H</sup>*ˆ*<sup>t</sup>* and the angular momentum operator *<sup>L</sup>*ˆ*<sup>z</sup>* [28], so that

We note that the eigenvalues of *<sup>L</sup>*ˆ*<sup>z</sup>* are not bounded by below (*<sup>m</sup>* takes the values <sup>−</sup>∞,...,−1, 0, 1,...,*N*) [28]. This agrees with the fact that the energies (*N* + 1/2)*h*¯Ω are infinitely degenerate [33]. Such a fact diminishes the physical relevance of phase-space localization for estimation purposes, as we shall

There exists a analogous formulation of an charged particle in a magnetic field by Kowalski that takes into account the geometry of a circle [30] (and for a comparison with the Feldman formulation see Ref.[8]), but at this point, we choose the Feldman formulation to work because the measure is easily

We will start our present endeavor defining the Hamiltonian *<sup>H</sup>*<sup>ˆ</sup> <sup>=</sup> *<sup>H</sup>*ˆ*<sup>t</sup>* <sup>+</sup>*H*ˆ*<sup>l</sup>* for a particle of mass *mq* and charge *<sup>q</sup>* in a magnetic field *<sup>B</sup>*, where *<sup>H</sup>*ˆ*<sup>t</sup>* <sup>=</sup> *<sup>h</sup>*¯Ω(*N*<sup>ˆ</sup> <sup>+</sup>1/2) describes the transverse motion, being <sup>Ω</sup> the cyclotron frequency as defined by the Eq. (32) and *N*ˆ the number operator. In addition, the Hamiltonian

where ρˆ is the thermal density operator and the set {|α,ξ,*kz*�} represents the coherent states for

corresponds to the coherent states of the transverse motion and {|*kz*�} to the longitudinal motion.

<sup>e</sup>−β(*H*ˆ*l*+*H*ˆ*t*)

<sup>ρ</sup><sup>ˆ</sup> <sup>=</sup> <sup>1</sup> *Z*

state basis, a possible way to define the Husimi function η, for the complete motion, is given by

the motion in three dimensions. Taking the direct product |α,ξ,*kz*�≡|α,ξ�

*<sup>z</sup>* /2*mq* represents a longitudinal one-dimensional free motion. After constructing a coherent

η(*x*, *px*;*y*, *py*; *pz*) = �α,ξ,*kz*|ρˆ|α,ξ,*kz*�, (35)


, (36)

defined and the normalization condition and other semiclassical measures are well described.

ˆ (31)

Ω = *qB*/*mqc*, (32)

*h*¯Ω|*N*,*m*� = *EN*|*N*,*m*� (33)

*<sup>L</sup>*ˆ*z*|*N*,*m*� <sup>=</sup> *mh*¯|*N*,*m*�. (34)

*<sup>H</sup>*ˆ*<sup>t</sup>* <sup>=</sup> <sup>π</sup>ˆ+πˆ<sup>−</sup> 2*mq* + 1 2 *h*¯Ω1,

the transverse motion, the Hamiltonian specializes to [28]

where an important quantity characterizes the problem, namely,

*<sup>H</sup>*ˆ*<sup>t</sup>* <sup>|</sup>*N*,*m*� <sup>=</sup>

see below. Moreover, *Lz* is not an independent constant of the motion [33].

3.2. Husimi distribution and Wehrl entropy

Therefore, the thermal density operator is given by

 *N* + 1 2 

$$Z\_l = \left(\angle/h\right) \left(2\pi m\_q k\_B T\right)^{1/2} \qquad \text{and} \tag{38}$$

$$Z\_t = \mathcal{A}m\_q \Omega / (4\pi\hbar \sinh(\mathfrak{H}\Omega/2)),\tag{39}$$

being *L* the length of the cylinder, *A* = π*R*<sup>2</sup> the area for cylindrical geometry [28]. In addition, the matrix element |�*n*,*m*|α,ξ�|<sup>2</sup> represents the probability of finding the charged particle in the coherent state |α,ξ� and we can find its expression as defined previously [34].

It should be noticed that the distribution η can be written as follows

$$
\mathfrak{h} = \mathfrak{n}\_{\varGamma}(p\_{\mathfrak{z}}) \mathfrak{n}\_{\varGamma}(\mathfrak{x}, p\_{\mathfrak{x}}; \mathfrak{y}, p\_{\mathfrak{y}}), \tag{40}
$$

where η has been separated as a function of two distributions, namely, η*<sup>l</sup>* = η*l*(*pz*) and η*<sup>t</sup>* = η*t*(*x*, *px*;*y*, *py*). The dependence on the variable *z* has been missed due to the explicit form of the hamiltonian *<sup>H</sup>*ˆ*l*. Accordingly, after summing in Eq. (37) we find

$$
\eta\_l = \frac{\mathbf{e}^{-\|\rho\_\varepsilon^2/2m\_q}}{Z\_l},
\tag{41}
$$

$$\mathfrak{m}\_{\mathsf{I}} = \frac{2\pi\hbar}{\mathcal{A}m\_{\mathsf{q}}\Omega} \left(1 - \mathbf{e}^{-\left|\mathsf{H}\Omega\right|}\right) \mathbf{e}^{-\left(1 - \mathbf{e}^{-\left|\mathsf{H}\Omega\right|}\right) |\mathsf{a}|^{2} / 2\ell\_{\mathsf{B}}^{2}},\tag{42}$$

where the length ℓ<sup>B</sup> is defined by the Eq. (30). From expressions (41) and (42), we emphasize again that η*l*(*pz*) describes the free motion of the particle in the magnetic field direction and η*t*(*x*, *px*;*y*, *py*) the Landau levels due to the circular motion in a transverse plane to the magnetic field, similar to the harmonic oscillator of Eq. (9) since |*z*| <sup>2</sup> → |α| <sup>2</sup>/2ℓ<sup>2</sup> B. Consequently Eqs. (40), (41) and (42) together contain the complete description of the system. We noticed both distributions are naturally normalized in a standard form, i.e.,

$$\int \frac{\mathrm{d}z \mathrm{d}p\_z}{h} \mathfrak{n}\_l(p\_z) = 1,\tag{43}$$

and

$$\int \frac{d^2 \text{ad}^2 \xi}{4\pi^2 \ell\_\mathbf{B}^4} \eta\_\mathbf{l}(\mathbf{x}, p\_\mathbf{x}; \mathbf{y}, p\_\mathbf{y}) = 1. \tag{44}$$

In consequence, both Eqs. (41) and (42), under conditions (43) and (44), bring a promising way to get the exact form of the Wehrl entropy. Furthermore, using the additivity as the most basic property of the entropy, we can state *W*total = *Wl* +*Wt* . Hence,

$$\mathcal{W}\_l = -\int \frac{\mathrm{d}\boldsymbol{\varepsilon} \mathrm{d}p\_{\boldsymbol{\varepsilon}}}{h} \boldsymbol{\eta}\_l(p\_{\boldsymbol{\varepsilon}}) \ln \boldsymbol{\eta}\_l(p\_{\boldsymbol{\varepsilon}}),\tag{45}$$

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The Husimi Distribution: Development and Applications

internal physical parameters such as transverse area, external magnetic field, charge of the particle, etc. If the system is large then the minimum temperature is low. However, modern electronic systems has junctions where *L* is practically zero. In such case the required minimum temperature to make

Nevertheless, the entropy associated with transverse motion satisfies *Wt* ≥ 1+ln(*g*) for all temperatures in the system of a particle in a magnetic field where the symmetry is polar, which is almost the Lieb condition for systems in one dimension [37] with an additional term associated with the degeneracy *g*. Roughly speaking, the transverse motion is bi-dimensional, but in the Landau approach the quantum motion of the particle in a magnetic field is reduced to a degenerate spectrum in one dimension. This degeneracy essentially recovers the physics of the missing dimension. Resuming the discussion of the behavior of the Wehrl entropy, it is not plausible to adventure any conclusion about the applicability of the present treatment because the Lieb condition is always satisfied. This is the main problem stems from the restricted vision presented in other contributions over this topic which only put its emphasis on the transverse motion [8, 28, 30] and represent the main difference from the vision obtained in that other contributions that discuss this topic. From the combined reasoning of both motions we conclude that the present description, this is the calculation of *Wt* , has sense when the imposition over the temperature is satisfied. Under *T*<sup>0</sup> the behavior is intrinsically anomalous and the present proposal is not applicable. If we consider *kBT* <sup>≫</sup> *<sup>h</sup>*¯Ω, we can apply the first order of approximation as ln(*g*/(<sup>1</sup> <sup>−</sup> *<sup>e</sup>*−β*h*¯Ω)) <sup>≈</sup> ln(*AT*/*T*0*L*<sup>2</sup>). Indeed, taking into account that the thermal wave length can be rewritten in terms of the temperature *<sup>T</sup>*<sup>0</sup> this way <sup>λ</sup> = *L*(*eT*0/*<sup>T</sup>* )1/2, the expression (49) after a bit of algebra reduces to

> <sup>+</sup>ln *<sup>A</sup> L*<sup>2</sup>

> > *<sup>T</sup> <sup>g</sup>*+*W*(1)

<sup>2</sup> <sup>+</sup>ln *<sup>V</sup>* λ3 

This is a particular expression for the entropy of a free particle in three dimensions related to the motion of a charged particle into a region of the magnetic field making mention of some geometrical properties

In second order of approximation for high temperatures, considering the special condition *A* ∼ *L*2,

As explained before, the Wehrl entropy takes values that are permitted by the Lieb condition, namely, *W* ≥ 1. According to Eq. (53) the slope decreases as temperature increases. This fact illustrates why the disorder slowly increases as the magnetic field increases too. Consequently, at extremely high temperatures as expected, the slope of the present linear dependence tends to zero apparently taking a constant value close to the corresponding classical entropy of the free particle in three dimensions.

. (51)

. (52)

total. (53)

applicable our description is numerically high enough [39].

*W*(1) total <sup>≈</sup> <sup>3</sup> 2 ln *<sup>T</sup> T*0 

of the system.

Wehrl entropy is expressed as follows

*W*(2) total <sup>≈</sup> *<sup>T</sup>*<sup>0</sup>

Considering that *V* = *AL* in Eq. (51), the total Wehrl entropy can be expressed as follows

*W*(1) total <sup>=</sup> <sup>3</sup>

*<sup>T</sup> <sup>g</sup>*<sup>+</sup> 3 2 + 3 2 ln *<sup>T</sup> T*0 <sup>=</sup> *<sup>T</sup>*<sup>0</sup>

$$W\_l = -\int \frac{\mathrm{d}^2 \mathrm{ad}^2 \xi}{4\pi^2 \ell\_\mathrm{B}^4} \eta\_l(\mathbf{x}, p\_\mathbf{x}; \mathbf{y}, p\_\mathbf{y}) \ln \eta\_l(\mathbf{x}, p\_\mathbf{x}; \mathbf{y}, p\_\mathbf{y}), \tag{46}$$

where, as before, the subindex *l* stands for the longitudinal motion and *t* the transverse.

After evaluating the respective integrals in Eqs. (45) and (46), it is feasible to identify the two particular entropies

$$W\_l = \frac{1}{2} + \ln\left(\frac{\mathcal{L}}{\overline{\lambda}}\right),\tag{47}$$

$$W\_l = 1 - \ln\left(1 - e^{-\beta \hbar \Omega}\right) + \ln\left(g\right),\tag{48}$$

where <sup>λ</sup> = *<sup>h</sup>*/(2π*mqk*B*<sup>T</sup>* )1/<sup>2</sup> is the mean thermal wavelength of the particle and *<sup>g</sup>* = *A*/2πℓ<sup>2</sup> <sup>B</sup> stands for the degeneracy of a Landau level [35]. Indeed, Eq. (47) coincides with the classical entropy for a free particle in one dimension. Eq. (48) is the Wehrl entropy for the transverse motion and possesses a form for the one close to the harmonic oscillator entropy given by the Eq. (10), with the exception of a term associated with the degeneracy.

#### 3.3. Semiclassical behavior and consequences

Although the total Wehrl entropy is expressed simply as follows

$$W\_{\text{total}} = \frac{3}{2} - \ln(1 - \mathbf{e}^{-\beta \hbar \Omega}) + \ln(g) + \ln\left(\frac{\mathcal{L}}{\lambda}\right),\tag{49}$$

we notice that some of its properties are directly derived from Eqs. (47) and (48). First, as we commented before, *Wl* coincides with the classical entropy for the free motion in one dimension. From this glance, we can add that *Wl* has to be nonnegative, *Wl* ≥ 0 at all temperatures. This last condition imposes a minimum temperature, given by

$$T\_0 = \frac{h^2}{2\pi m\_q e k\_B \mathcal{L}^2},\tag{50}$$

where *e* = 2.718281828. The standard behavior of *Wl* obligates the system to take high values of temperature, wherever the temperature *T* ought to be greater than *T*0, in such case the conduct of the system is classical. This is equivalent to assert that, if *T*/*T*<sup>0</sup> ≥ 1, the length of a thermal wave λ lower than the average of the spacing among particles and quantum considerations are not relevant [36]. In addition, *T*<sup>0</sup> only depends on the size of the system and does not depend on other external or internal physical parameters such as transverse area, external magnetic field, charge of the particle, etc. If the system is large then the minimum temperature is low. However, modern electronic systems has junctions where *L* is practically zero. In such case the required minimum temperature to make applicable our description is numerically high enough [39].

10 Quantum Mechanics

entropies

entropy, we can state *W*total = *Wl* +*Wt* . Hence,

term associated with the degeneracy.

imposes a minimum temperature, given by

3.3. Semiclassical behavior and consequences

Although the total Wehrl entropy is expressed simply as follows

*<sup>W</sup>*total <sup>=</sup> <sup>3</sup>

*Wl*=−

*Wt*=−

d*z*d*pz*

where, as before, the subindex *l* stands for the longitudinal motion and *t* the transverse.

<sup>2</sup> <sup>+</sup>ln

 d2 αd2 ξ 4π2ℓ<sup>4</sup> B

*Wl* <sup>=</sup> <sup>1</sup>

*Wt* = 1−ln

In consequence, both Eqs. (41) and (42), under conditions (43) and (44), bring a promising way to get the exact form of the Wehrl entropy. Furthermore, using the additivity as the most basic property of the

After evaluating the respective integrals in Eqs. (45) and (46), it is feasible to identify the two particular

1−e−β*h*¯<sup>Ω</sup>

*L* λ 

where <sup>λ</sup> = *<sup>h</sup>*/(2π*mqk*B*<sup>T</sup>* )1/<sup>2</sup> is the mean thermal wavelength of the particle and *<sup>g</sup>* = *A*/2πℓ<sup>2</sup>

for the degeneracy of a Landau level [35]. Indeed, Eq. (47) coincides with the classical entropy for a free particle in one dimension. Eq. (48) is the Wehrl entropy for the transverse motion and possesses a form for the one close to the harmonic oscillator entropy given by the Eq. (10), with the exception of a

<sup>2</sup> <sup>−</sup>ln(1−e−β*h*¯Ω) +ln(*g*) <sup>+</sup>ln

we notice that some of its properties are directly derived from Eqs. (47) and (48). First, as we commented before, *Wl* coincides with the classical entropy for the free motion in one dimension. From this glance, we can add that *Wl* has to be nonnegative, *Wl* ≥ 0 at all temperatures. This last condition

*<sup>T</sup>*<sup>0</sup> <sup>=</sup> *<sup>h</sup>*<sup>2</sup>

where *e* = 2.718281828. The standard behavior of *Wl* obligates the system to take high values of temperature, wherever the temperature *T* ought to be greater than *T*0, in such case the conduct of the system is classical. This is equivalent to assert that, if *T*/*T*<sup>0</sup> ≥ 1, the length of a thermal wave λ lower than the average of the spacing among particles and quantum considerations are not relevant [36]. In addition, *T*<sup>0</sup> only depends on the size of the system and does not depend on other external or

*<sup>h</sup>* <sup>η</sup>*l*(*pz*)lnη*l*(*pz*), (45)

*L* λ 

<sup>2</sup>π*mqekBL*<sup>2</sup> , (50)

η*t*(*x*,*px*;*y*,*py*)lnη*t*(*x*,*px*;*y*,*py*), (46)

, (47)

+ln(*g*), (48)

, (49)

<sup>B</sup> stands

Nevertheless, the entropy associated with transverse motion satisfies *Wt* ≥ 1+ln(*g*) for all temperatures in the system of a particle in a magnetic field where the symmetry is polar, which is almost the Lieb condition for systems in one dimension [37] with an additional term associated with the degeneracy *g*. Roughly speaking, the transverse motion is bi-dimensional, but in the Landau approach the quantum motion of the particle in a magnetic field is reduced to a degenerate spectrum in one dimension. This degeneracy essentially recovers the physics of the missing dimension. Resuming the discussion of the behavior of the Wehrl entropy, it is not plausible to adventure any conclusion about the applicability of the present treatment because the Lieb condition is always satisfied. This is the main problem stems from the restricted vision presented in other contributions over this topic which only put its emphasis on the transverse motion [8, 28, 30] and represent the main difference from the vision obtained in that other contributions that discuss this topic. From the combined reasoning of both motions we conclude that the present description, this is the calculation of *Wt* , has sense when the imposition over the temperature is satisfied. Under *T*<sup>0</sup> the behavior is intrinsically anomalous and the present proposal is not applicable.

If we consider *kBT* <sup>≫</sup> *<sup>h</sup>*¯Ω, we can apply the first order of approximation as ln(*g*/(<sup>1</sup> <sup>−</sup> *<sup>e</sup>*−β*h*¯Ω)) <sup>≈</sup> ln(*AT*/*T*0*L*<sup>2</sup>). Indeed, taking into account that the thermal wave length can be rewritten in terms of the temperature *<sup>T</sup>*<sup>0</sup> this way <sup>λ</sup> = *L*(*eT*0/*<sup>T</sup>* )1/2, the expression (49) after a bit of algebra reduces to

$$W\_{\text{total}}^{(1)} \approx \frac{3}{2} \ln\left(\frac{T}{T\_0}\right) + \ln\left(\frac{\mathcal{A}}{\mathcal{L}^2}\right). \tag{51}$$

Considering that *V* = *AL* in Eq. (51), the total Wehrl entropy can be expressed as follows

$$W\_{\text{total}}^{(1)} = \frac{3}{2} + \ln\left(\frac{\mathcal{V}}{\lambda^3}\right). \tag{52}$$

This is a particular expression for the entropy of a free particle in three dimensions related to the motion of a charged particle into a region of the magnetic field making mention of some geometrical properties of the system.

In second order of approximation for high temperatures, considering the special condition *A* ∼ *L*2, Wehrl entropy is expressed as follows

$$W\_{\text{total}}^{(2)} \approx \frac{T\_0}{T} \mathbf{g} + \frac{3}{2} + \frac{3}{2} \ln\left(\frac{T}{T\_0}\right) = \frac{T\_0}{T} \mathbf{g} + W\_{\text{total}}^{(1)}.\tag{53}$$

As explained before, the Wehrl entropy takes values that are permitted by the Lieb condition, namely, *W* ≥ 1. According to Eq. (53) the slope decreases as temperature increases. This fact illustrates why the disorder slowly increases as the magnetic field increases too. Consequently, at extremely high temperatures as expected, the slope of the present linear dependence tends to zero apparently taking a constant value close to the corresponding classical entropy of the free particle in three dimensions.

The lower bound of temperature is related to *T*/*T*<sup>0</sup> → 1+, because this approach does not consider temperature values under *T*0. The total Wehrl entropy is reduced to logarithm behavior of the magnetic field.

To study what occurs close to zero temperature, in accordance with Eq. (50), we need to take systems with *L* → ∞ and after this consideration the transverse entropy of Eq. (48) can be seen as follows

$$W\_l^{T \to 0^+} = 1 + \ln\left(g\right). \tag{54}$$

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equal to ν, which must be greater than 1 due to the inequality (55) obtaining an infinite family of Wehrl

Again, Eq. (55) provides the limiting value of ν and, as before, the transverse entropy always satisfies the Lieb bound for all temperatures and large enough systems when the quantum Hall effect is manifested at least for the integer quantum Hall effect. Conversely, fractional values of ν less than 1 are

The rigid rotator is a system of a single particle whose quantum spectrum of energy is exactly known. Therefore, the study of typical thermodynamic properties can be analytically derived [40]. Applications lead to the treatment of important aspects of molecular systems [41] and several applications to

We start the present study by exploring a simple model, the linear rigid rotator, based on the excellent discussion concerning the coherent states for angular momenta given in Ref. [43]. The Hamiltonian of

> *<sup>H</sup>*<sup>ˆ</sup> <sup>=</sup> *<sup>L</sup>*<sup>ˆ</sup> <sup>2</sup> 2*Ixy*

inertia. We have assumed that *Ixy* ≡ *Ix* = *Iy*. Calling |*IK*� the set of *H*-eigenstates, we recall that they

*L*ˆ <sup>2</sup>|*IK*� = *I*(*I* +1)*h*¯ <sup>2</sup>|*IK*�

<sup>ε</sup>*<sup>I</sup>* <sup>=</sup> *<sup>I</sup>*(*<sup>I</sup>* <sup>+</sup>1)*h*¯ <sup>2</sup> 2*Ixy*

Coherent states are constructed in Ref. [44, 45] for the lineal rigid rotator, using Schwinger's oscillator

(*I* +*K*)!(*I* −*K*)!

† +)*<sup>I</sup>*+*K*(*a*<sup>ˆ</sup> † <sup>−</sup>)*I*−*<sup>K</sup>*

with *I* = 0, 1, 2..., for −*I* ≤ *K* ≤ *I*, the eigenstates' energy spectrum being given by

<sup>|</sup>*IK*� <sup>=</sup> (*a*<sup>ˆ</sup>

*<sup>y</sup>* is the angular momentum operator and *Ix* and *Iy* are the associated moments of

*<sup>L</sup>*ˆ*z*|*IK*� <sup>=</sup> *Kh*¯|*IK*�, (60)

4. Description of the molecular rotation: Rigid rotator

*Wt* <sup>=</sup> <sup>1</sup>−ln(1−*e*−β*h*¯Ω) +lnν. (58)

The Husimi Distribution: Development and Applications

, (59)

. (61)


entropies

materials [42].

where *L*ˆ <sup>2</sup> = *L*ˆ <sup>2</sup>

verify the relations

left out the present approach.

4.1. Linear rigid rotator

the linear rigid rotator is [20]

*<sup>x</sup>* <sup>+</sup> *<sup>L</sup>*<sup>ˆ</sup> <sup>2</sup>

model of angular momentum, in the fashion

As we discussed before, this Wehrl entropy is also a kind of harmonic oscillator entropy and the lower bound complies with being greater than a bound limiting value of the temperature, which has been suggested by Wehrl and shown by Lieb, *W* ≥ 1 [37]. Starting from this condition it must arrive to the following inequality for the magnetic field

$$g \ge 1,\tag{55}$$

where *g* = *qAB*/*hc* also accounts for the ratio between the flux of the magnetic field *AB* and the quantum of the magnetic flux given by *hc*/*q* = 4.14×10−7[*gauss*/*cm*2] [17]. Then the inequality (55) adopts the form

$$B \ge \frac{1}{\mathcal{A}} \frac{hc}{q} = B\_0. \tag{56}$$

Therefore, the quantity *<sup>B</sup>*<sup>0</sup> = *hc*/*A<sup>q</sup>* becomes a bound limiting field that represents the minimum value for the external magnetic field. To study what occurs close to zero magnetic field we need to take systems with *A* → ∞.

For finite values of *A* and *<sup>B</sup>* lower than *<sup>B</sup>*<sup>0</sup> is manifested the Haas-van Alphen effect, which describes oscillations in the magnetization because at temperatures low enough the particles will tend to occupy the lowest energy states. Whereas if the value of the magnetic field decreases a less number of particles can be in the lowest state due to degeneracy is directly proportional to *B* [35]. Then, the transverse Wehrl entropy *Wt* is well defined for values of the magnetic field over *B*0, this is *B*/*B*<sup>0</sup> ≥ 1 and/or *g* → 1+.

We can assert that this description of the system is not quantum, we say that it is semiclassical; for instance, it does not contain the Haas-van Alphen effect, the same condition marks the beginning of one description and the ending of the other.

Other relevant effect that emerges from the Landau quantization [38] is the quantum Hall effect [39] which is a quantum-mechanical version of the Hall effect [31], observed in two-dimensional electron systems subjected to low temperatures and strong magnetic fields. The degeneracy is given by [17]

$$
\boldsymbol{\phi} = \mathbf{v} \boldsymbol{\phi}\_0,\tag{57}
$$

where φ<sup>0</sup> = *hc*/*q* is the quantum of the magnetic flux. The factor ν is related to the "filling factor" that takes integer values (ν = 1, 2, 3, . . . ). The discovery of the fractional quantum Hall effect [32] extend these values to rational fractions (ν = 1/3, 1/5, 5/2, 12/5, . . . ). The integer quantum Hall effect is simply explained in terms of the conductivity quantization σ = ν*q*2/*h*. However, the fractional quantum Hall effect relies on other phenomena related to interactions. Consistently, we see that the degeneracy is equal to ν, which must be greater than 1 due to the inequality (55) obtaining an infinite family of Wehrl entropies

$$W\_l = 1 - \ln\left(1 - e^{-\beta \hbar \Omega}\right) + \ln \mathbf{v}.\tag{58}$$

Again, Eq. (55) provides the limiting value of ν and, as before, the transverse entropy always satisfies the Lieb bound for all temperatures and large enough systems when the quantum Hall effect is manifested at least for the integer quantum Hall effect. Conversely, fractional values of ν less than 1 are left out the present approach.

#### 4. Description of the molecular rotation: Rigid rotator

The rigid rotator is a system of a single particle whose quantum spectrum of energy is exactly known. Therefore, the study of typical thermodynamic properties can be analytically derived [40]. Applications lead to the treatment of important aspects of molecular systems [41] and several applications to materials [42].

#### 4.1. Linear rigid rotator

12 Quantum Mechanics

adopts the form

systems with *A* → ∞.

description and the ending of the other.

*g* → 1+.

following inequality for the magnetic field

field.

The lower bound of temperature is related to *T*/*T*<sup>0</sup> → 1+, because this approach does not consider temperature values under *T*0. The total Wehrl entropy is reduced to logarithm behavior of the magnetic

To study what occurs close to zero temperature, in accordance with Eq. (50), we need to take systems with *L* → ∞ and after this consideration the transverse entropy of Eq. (48) can be seen as follows

As we discussed before, this Wehrl entropy is also a kind of harmonic oscillator entropy and the lower bound complies with being greater than a bound limiting value of the temperature, which has been suggested by Wehrl and shown by Lieb, *W* ≥ 1 [37]. Starting from this condition it must arrive to the

where *g* = *qAB*/*hc* also accounts for the ratio between the flux of the magnetic field *AB* and the quantum of the magnetic flux given by *hc*/*q* = 4.14×10−7[*gauss*/*cm*2] [17]. Then the inequality (55)

Therefore, the quantity *<sup>B</sup>*<sup>0</sup> = *hc*/*A<sup>q</sup>* becomes a bound limiting field that represents the minimum value for the external magnetic field. To study what occurs close to zero magnetic field we need to take

For finite values of *A* and *<sup>B</sup>* lower than *<sup>B</sup>*<sup>0</sup> is manifested the Haas-van Alphen effect, which describes oscillations in the magnetization because at temperatures low enough the particles will tend to occupy the lowest energy states. Whereas if the value of the magnetic field decreases a less number of particles can be in the lowest state due to degeneracy is directly proportional to *B* [35]. Then, the transverse Wehrl entropy *Wt* is well defined for values of the magnetic field over *B*0, this is *B*/*B*<sup>0</sup> ≥ 1 and/or

We can assert that this description of the system is not quantum, we say that it is semiclassical; for instance, it does not contain the Haas-van Alphen effect, the same condition marks the beginning of one

Other relevant effect that emerges from the Landau quantization [38] is the quantum Hall effect [39] which is a quantum-mechanical version of the Hall effect [31], observed in two-dimensional electron systems subjected to low temperatures and strong magnetic fields. The degeneracy is given by [17]

where φ<sup>0</sup> = *hc*/*q* is the quantum of the magnetic flux. The factor ν is related to the "filling factor" that takes integer values (ν = 1, 2, 3, . . . ). The discovery of the fractional quantum Hall effect [32] extend these values to rational fractions (ν = 1/3, 1/5, 5/2, 12/5, . . . ). The integer quantum Hall effect is simply explained in terms of the conductivity quantization σ = ν*q*2/*h*. However, the fractional quantum Hall effect relies on other phenomena related to interactions. Consistently, we see that the degeneracy is

*<sup>t</sup>* = 1+ln(*g*). (54)

*g* ≥ 1, (55)

*<sup>q</sup>* <sup>=</sup> *<sup>B</sup>*0. (56)

φ = νφ0, (57)

*<sup>W</sup>T*→0<sup>+</sup>

*B* ≥ 1 *A hc* We start the present study by exploring a simple model, the linear rigid rotator, based on the excellent discussion concerning the coherent states for angular momenta given in Ref. [43]. The Hamiltonian of the linear rigid rotator is [20]

$$
\hat{H} = \frac{\hat{L}^2}{2I\_{\text{xy}}},
\tag{59}
$$

where *L*ˆ <sup>2</sup> = *L*ˆ <sup>2</sup> *<sup>x</sup>* <sup>+</sup> *<sup>L</sup>*<sup>ˆ</sup> <sup>2</sup> *<sup>y</sup>* is the angular momentum operator and *Ix* and *Iy* are the associated moments of inertia. We have assumed that *Ixy* ≡ *Ix* = *Iy*. Calling |*IK*� the set of *H*-eigenstates, we recall that they verify the relations

$$\begin{array}{rcl} \hat{L}^2|IK\rangle &= I(I+1)\hbar^2|IK\rangle\\ \hat{L}\_{\varepsilon}|IK\rangle &= K\hbar|IK\rangle, \end{array} \tag{60}$$

with *I* = 0, 1, 2..., for −*I* ≤ *K* ≤ *I*, the eigenstates' energy spectrum being given by

$$\mathfrak{e}\_I = \frac{I(I+1)\hbar^2}{2I\_{\rm xy}}.\tag{61}$$

Coherent states are constructed in Ref. [44, 45] for the lineal rigid rotator, using Schwinger's oscillator model of angular momentum, in the fashion

$$|IK\rangle = \frac{(\mathring{a}\_+^\dagger)^{I+K} (\mathring{a}\_-^\dagger)^{I-K}}{\sqrt{(I+K)!(I-K)!}} |0\rangle,\tag{62}$$

with *a*ˆ+, *a*ˆ<sup>−</sup> the pertinent creation and annihilation operators, respectively, and |0�≡|0, 0� the vacuum state. The states |*IK*� are orthogonal and satisfy the closure relation, i.e.,

$$
\langle I^{'}K^{'}|IK\rangle = \mathfrak{S}\_{I',I}\mathfrak{S}\_{K',K},
\tag{63}
$$

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*<sup>n</sup>*−! . (73)

The Husimi Distribution: Development and Applications

<sup>π</sup> <sup>|</sup>*z*1*z*2��*z*1*z*2<sup>|</sup> <sup>=</sup> 1. (74)

*µ*(*z*1,*z*2) = �*z*1,*z*2|ρˆ|*z*1,*z*2�, (75)

*<sup>K</sup>*=−*<sup>I</sup>* . Inserting now the closure relation into Eq. (75), and using

*T*

*T*

<sup>π</sup> *<sup>µ</sup>*(*z*1,*z*2) = 1, (79)

*<sup>I</sup>*=<sup>0</sup> (2*<sup>I</sup>* <sup>+</sup>1) *<sup>e</sup>*−*I*(*I*+1) <sup>Θ</sup>

<sup>2</sup>)4*<sup>I</sup>* firstly and then integrate over the whole complex plane (in two dimensions)

<sup>2</sup>*<sup>D</sup>* exp(−β*H*ˆ). (76)

*<sup>T</sup>* , (77)

. (78)

where *n*<sup>+</sup> = *I* +*K* and *n*<sup>−</sup> = *I* −*K*. Therefore, the probability of observing the state |*IK*� in the coherent

<sup>2</sup> |*z*1| 2*n*<sup>+</sup> *n*+!



 d2*z*<sup>1</sup> π

d2*z*<sup>2</sup>

<sup>ρ</sup><sup>ˆ</sup> = *<sup>Z</sup>*−<sup>1</sup>

∞ ∑ *I*=0

(2*<sup>I</sup>* <sup>+</sup>1) *<sup>e</sup>*−*I*(*I*+1) <sup>Θ</sup>

with Θ = *h*¯ <sup>2</sup>/(2*IxykB*). Remark that in the present context, speaking of the "trace operation" entails

<sup>2</sup> ∑∞ *I*=0 |*z*| 4*I* (2*I*)! *<sup>e</sup>*−*I*(*I*+1) <sup>Θ</sup>

∑∞

d2*z*<sup>2</sup>

where *z*<sup>1</sup> and *z*<sup>2</sup> are given by Eqs. (66), (67), and (71). Note that we must deal with the binomial

in order to verify the normalization condition (79). The differential element of area in the *z*1(*z*2) plane is d2*z*<sup>1</sup> = d*x*d*px*/2¯*h* (d2*z*<sup>2</sup> = d*y*d*py*/2¯*h*) [13]. Moreover, we have the phase-space relationships

Following the procedure developed by Anderson *et al.* [4], we can readily calculate the pertinent Husimi

The present coherent states satisfy resolution of unity

Furthermore, *z*<sup>1</sup> and *z*<sup>2</sup> are continuous variables.

distribution [1]. For our system this is defined, from Eq. (4), as

The concomitant rotational partition function *Z*2*<sup>D</sup>* is given in Ref. [20]

*<sup>I</sup>*=<sup>0</sup> <sup>∑</sup>*<sup>I</sup>*

Eq. (73), we finally get our Husimi distribution in the fashion

It is easy to show that this distribution is normalized to unity

*Z*2*<sup>D</sup>* =

*<sup>µ</sup>*(*z*1,*z*2) = *<sup>e</sup>*−|*z*<sup>|</sup>

 d2*z*<sup>1</sup> π

state |*z*1*z*2� is of the form

where the density operator is

performing the sum Tr ≡ ∑<sup>∞</sup>

expression (|*z*1|

<sup>2</sup> +|*z*2|

$$\sum\_{I=0}^{\infty} \sum\_{K=-I}^{I} |IK\rangle\langle IK| = \hat{1}.\tag{64}$$

Since we deal with two degrees of freedom the ensuing coherent states are of the tensor product form (involving |*z*1� and |*z*2�) [43, 46]

$$
\langle z\_1 z\_2 \rangle = |z\_1\rangle \otimes |z\_2\rangle,\tag{65}
$$

where

$$
\langle \hat{a}\_{+} | z\_{1} z\_{2} \rangle = z\_{1} | z\_{1} z\_{2} \rangle,\tag{66}
$$

$$
\hat{a}\_{-}|z\_{1}z\_{2}\rangle = \varepsilon\_{2}|z\_{1}z\_{2}\rangle. \tag{67}
$$

Therefore, the coherent state |*z*1*z*2� writes [43]

$$\langle z\_1 z\_2 \rangle = e^{-\frac{|z|^2}{2}} e^{z\_1 \mathbf{d}\_+^\uparrow} e^{z\_2 \mathbf{d}\_-^\uparrow} |0\rangle,\tag{68}$$

with

$$|z\_1\rangle = e^{-\frac{|z\_1|^2}{2}} e^{z\_1 \mathfrak{d}\_+^\dagger} |0\rangle,\tag{69}$$

$$|z\_2\rangle = e^{-\frac{|z\_2|^2}{2}} e^{\varepsilon\_2 \mathfrak{d}\_-^\dagger}|0\rangle. \tag{70}$$

We have introduced the convenient notation

$$|z|^2 = |z\_1|^2 + |z\_2|^2. \tag{71}$$

Using Eqs. (62) and (68) we easily calculate |*z*1*z*2� and, after a bit of algebra, find

$$|z\_1 z\_2\rangle = e^{-\frac{|z|^2}{2}} \sum\_{n\_+, n\_-} \frac{z\_1^{n\_+}}{\sqrt{n\_+!}} \frac{z\_2^{n\_-}}{\sqrt{n\_-!}} |IK\rangle \tag{72}$$

where *n*<sup>+</sup> = *I* +*K* and *n*<sup>−</sup> = *I* −*K*. Therefore, the probability of observing the state |*IK*� in the coherent state |*z*1*z*2� is of the form

$$|\langle IK|z\_1 z\_2\rangle|^2 = e^{-|z|^2} \frac{|z\_1|^{2n\_+}}{n\_+!} \frac{|z\_2|^{2n\_-}}{n\_-!}.\tag{73}$$

The present coherent states satisfy resolution of unity

$$\int \frac{\mathbf{d}^2 z\_1}{\pi} \frac{\mathbf{d}^2 z\_2}{\pi} |z\_1 z\_2\rangle \langle z\_1 z\_2| = 1. \tag{74}$$

Furthermore, *z*<sup>1</sup> and *z*<sup>2</sup> are continuous variables.

Following the procedure developed by Anderson *et al.* [4], we can readily calculate the pertinent Husimi distribution [1]. For our system this is defined, from Eq. (4), as

$$
\mu(z\_1, z\_2) = \langle z\_1, z\_2 | \Phi | z\_1, z\_2 \rangle,\tag{75}
$$

where the density operator is

14 Quantum Mechanics

(involving |*z*1� and |*z*2�) [43, 46]

Therefore, the coherent state |*z*1*z*2� writes [43]

We have introduced the convenient notation

where

with

with *a*ˆ+, *a*ˆ<sup>−</sup> the pertinent creation and annihilation operators, respectively, and |0�≡|0, 0� the vacuum

<sup>|</sup>*IK*� = <sup>δ</sup>*<sup>I</sup>* ′ ,*I*δ*K*′

Since we deal with two degrees of freedom the ensuing coherent states are of the tensor product form


,*K*, (63)

ˆ (64)

<sup>−</sup> |0�, (68)

<sup>+</sup> |0�, (69)

<sup>−</sup> |0�. (70)

2. (71)



*a*ˆ+|*z*1*z*2� = *z*1|*z*1*z*2�, (66)

*a*ˆ−|*z*1*z*2� = *z*2|*z*1*z*2�. (67)

state. The states |*IK*� are orthogonal and satisfy the closure relation, i.e.,

�*I* ′ *K*′

∞ ∑ *I*=0

<sup>|</sup>*z*1*z*2� <sup>=</sup> *<sup>e</sup>*<sup>−</sup> <sup>|</sup>*z*<sup>|</sup>

<sup>|</sup>*z*1� <sup>=</sup> *<sup>e</sup>*<sup>−</sup> <sup>|</sup>*z*1<sup>|</sup>

<sup>|</sup>*z*2� <sup>=</sup> *<sup>e</sup>*<sup>−</sup> <sup>|</sup>*z*2<sup>|</sup>


> 2 <sup>2</sup> <sup>∑</sup>*<sup>n</sup>*+,*n*<sup>−</sup>

Using Eqs. (62) and (68) we easily calculate |*z*1*z*2� and, after a bit of algebra, find

<sup>|</sup>*z*1*z*2� <sup>=</sup> *<sup>e</sup>*<sup>−</sup> <sup>|</sup>*z*<sup>|</sup>

2 <sup>2</sup> *ez*1*a*<sup>ˆ</sup> † <sup>+</sup> *ez*2*a*<sup>ˆ</sup> †

> 2 <sup>2</sup> *ez*1*a*<sup>ˆ</sup> †

> 2 <sup>2</sup> *ez*2*a*<sup>ˆ</sup> †

<sup>2</sup> +|*z*2|

*z n*+ √ 1 *n*+!

*z n*<sup>−</sup> √ 2 *n*−!

*I* ∑ *K*=−*I*

$$\Phi = Z\_{2D}^{-1} \exp\left(-\beta \varGamma\right). \tag{76}$$

The concomitant rotational partition function *Z*2*<sup>D</sup>* is given in Ref. [20]

$$Z\_{2D} = \sum\_{I=0}^{\infty} \left(2I + 1\right) e^{-I\left(I + 1\right)\frac{\Omega}{I}},\tag{77}$$

with Θ = *h*¯ <sup>2</sup>/(2*IxykB*). Remark that in the present context, speaking of the "trace operation" entails performing the sum Tr ≡ ∑<sup>∞</sup> *<sup>I</sup>*=<sup>0</sup> <sup>∑</sup>*<sup>I</sup> <sup>K</sup>*=−*<sup>I</sup>* . Inserting now the closure relation into Eq. (75), and using Eq. (73), we finally get our Husimi distribution in the fashion

$$\mu(z\_1, z\_2) = e^{-|z|^2} \frac{\sum\_{I=0}^{\infty} \frac{|z|^{4I}}{(2I)!} e^{-I(I+1)\frac{\Theta}{I}}}{\sum\_{I=0}^{\infty} (2I+1) e^{-I(I+1)\frac{\Theta}{I}}}. \tag{78}$$

It is easy to show that this distribution is normalized to unity

$$
\int \frac{\mathbf{d}^2 z\_1}{\pi} \frac{\mathbf{d}^2 z\_2}{\pi} \,\mu(z\_1, z\_2) = 1,\tag{79}
$$

where *z*<sup>1</sup> and *z*<sup>2</sup> are given by Eqs. (66), (67), and (71). Note that we must deal with the binomial expression (|*z*1| <sup>2</sup> +|*z*2| <sup>2</sup>)4*<sup>I</sup>* firstly and then integrate over the whole complex plane (in two dimensions) in order to verify the normalization condition (79). The differential element of area in the *z*1(*z*2) plane is d2*z*<sup>1</sup> = d*x*d*px*/2¯*h* (d2*z*<sup>2</sup> = d*y*d*py*/2¯*h*) [13]. Moreover, we have the phase-space relationships

$$\left|z\_1\right|^2 = \frac{1}{4} \left(\frac{x^2}{\sigma\_x^2} + \frac{p\_x^2}{\sigma\_{p\_x}^2}\right),\tag{80}$$

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<sup>|</sup>*IMK*��*IMK*<sup>|</sup> = 1.<sup>ˆ</sup> (86)

The Husimi Distribution: Development and Applications

, (87)

, (88)


<sup>2</sup>)2. (91)

*<sup>z</sup>* and assume axial symmetry, i.e., *Ixy* <sup>≡</sup> *Ix* = *Iy*, we can recast the

*I*!(*I* + *M*)!(*I* − *M*)!(*I* +*K*)!(*I* −*K*)!, (89)

*d*Γ|*z*1*z*2*z*3��*z*1*z*2*z*3| = 1 (92)

∞ ∑ *I*=0

If we take *L*ˆ <sup>2</sup> = *L*ˆ <sup>2</sup>

and then write [47]

relation to the last property we add

Hamiltonian as

*<sup>x</sup>* <sup>+</sup> *<sup>L</sup>*<sup>ˆ</sup> <sup>2</sup>

concomitant spectrum of energy becomes

*<sup>y</sup>* <sup>+</sup> *<sup>L</sup>*<sup>ˆ</sup> <sup>2</sup>

*I* ∑ *M*=−*I*

*<sup>H</sup>*<sup>ˆ</sup> <sup>=</sup> <sup>1</sup> 2*Ixy*

<sup>ε</sup>*I*,*<sup>K</sup>* <sup>=</sup> *<sup>h</sup>*¯ <sup>2</sup> 2*Ixy*

*4.2.1. Coherent states for the rigid rotator in three dimensions*

*XI*,*M*,*<sup>K</sup>* =

2 <sup>2</sup> ∑ *IMK*

where the following supplementary variable were introduced by Morales *et al.* in Ref. [47]

<sup>2</sup>(1+|*z*1|

All coherent states share at least two requirements. Continuity of labeling and resolution of unity. In

<sup>|</sup>*z*1*z*2*z*3� <sup>=</sup> *<sup>e</sup>*<sup>−</sup> <sup>|</sup>*u*<sup>|</sup>


foundations. First, they introduced the auxiliary quantity

*Ixy*/*Iz* → ∞, that correspond to the extremely oblate- and prolate cases, respectively.

*I* ∑ *K*=−*I*

*L*ˆ <sup>2</sup> +

where *<sup>L</sup>*<sup>ˆ</sup> <sup>2</sup> is the angular momentum operator and *<sup>L</sup>*ˆ*<sup>z</sup>* is its projection on the rotation axis *<sup>z</sup>*. The

*<sup>I</sup>*(*<sup>I</sup>* <sup>+</sup>1) +*Ixy*

where *I* = 0, 1, 2,··· and it represents the eigenvalue of the angular momentum operator *L*ˆ 2, the numbers *m* = −*I*,···,−1, 0, 1,···,*I* stand for the projections on the intrinsic rotation axis of the rotator. All states exhibit a (2*I* +1)−degeneracy. The parameters *Ix* = *Iy* ≡ *Ixy* and *Iz* are the inertia momenta. Different "geometrical" instances are characterized through the *Ixy*/*Iz*−ratio. For example, the value *Ixy*/*Iz* = 1 corresponds to the spherical rotator. Limiting cases can also be considered. This is, *Ixy*/*Iz* = 1/2 and

In order to obtain the Husimi distribution for this problem we need first of all to have the associated coherent states. Morales *et al.* have constructed them in Ref. [47] and discussed their mathematical

> [(2*I*)!] 2*z* (*I*+*M*) <sup>1</sup> *zI* 2*z* (*I*+*K*) 3

> > *XI*,*M*,*<sup>K</sup>*

<sup>2</sup>)2(1+|*z*3|

*Iz* −1 *K*2 

*Ixy Iz* −1 *L*ˆ 2 *z* 

$$|z\_2|^2 = \frac{1}{4} \left( \frac{\text{y}^2}{\sigma\_\text{y}^2} + \frac{p\_\text{y}^2}{\sigma\_{p\_\text{y}}^2} \right), \tag{81}$$

where <sup>σ</sup>*<sup>x</sup>* <sup>≡</sup> <sup>σ</sup>*<sup>y</sup>* <sup>=</sup> <sup>√</sup>*h*¯/2*m*<sup>ω</sup> and <sup>σ</sup>*px* <sup>≡</sup> <sup>σ</sup>*py* <sup>=</sup> <sup>√</sup>*m*ω*h*¯/2.

The profile of the Husimi function is similar to that of a Gaussian distribution.

The Wehrl entropy is a semiclassical measure of localization [21] (so is Fisher's one [5] as well). Indeed, Wehrl's measure is simply a logarithmic Shannon measure built up with Husimi distributions. For the present bi-dimensional model this entropy reads

$$\mathcal{W} = -\int \frac{\mathbf{d}^2 z\_1}{\pi} \frac{\mathbf{d}^2 z\_2}{\pi} \,\mu(z\_1, z\_2) \,\ln \mu(z\_1, z\_2), \tag{82}$$

where *µ*(*z*1,*z*2) is given by Eq. (78).

#### 4.2. Rigid rotator in three dimensions

In the present section we consider a more general problem, the model of the rigid rotator in three dimensions, whose Hamiltonian writes [47]

$$
\hat{H} = \frac{\hat{L}\_\chi^2}{2I\_\chi} + \frac{\hat{L}\_\chi^2}{2I\_\chi} + \frac{\hat{L}\_\chi^2}{2I\_\chi},
\tag{83}
$$

where *Ix*, *Iy*, and *Iz* are the associated moments of inertia. A complete set of rotator eigenstates is {|*IMK*�}. The following relations apply

$$\begin{aligned} \langle \hat{L}^2 | IMK \rangle &= I(I+1)\hbar^2 |IMK\rangle \\ \langle \hat{L}\_{\mathbb{Z}} | IMK \rangle &= K\hbar |IMK\rangle \\ \langle \hat{J}\_{\mathbb{Z}} | IMK \rangle &= M\hbar |IMK\rangle, \end{aligned} \tag{84}$$

where *I* = 0,...,∞,−*I* ≤ *K* ≤ *I*, and −*I* ≤ *M* ≤ *I*. The states |*IMK*� satisfy orthogonality and closure relations [47]

$$
\langle \,^{\prime}M^{'}K^{'} \vert IMK \rangle = \delta\_{I^{'},I}\delta\_{M^{'},M}\delta\_{K^{'},K} \tag{85}
$$

<sup>610</sup> Advances in Quantum Mechanics The Husimi Distribution: Development and Applications 17 10.5772/53846 The Husimi Distribution: Development and Applications http://dx.doi.org/10.5772/53846 611

$$\sum\_{I=0}^{\infty} \sum\_{M=-I}^{I} \sum\_{K=-I}^{I} |IMK\rangle\langle IMK| = \hat{1}. \tag{86}$$

If we take *L*ˆ <sup>2</sup> = *L*ˆ <sup>2</sup> *<sup>x</sup>* <sup>+</sup> *<sup>L</sup>*<sup>ˆ</sup> <sup>2</sup> *<sup>y</sup>* <sup>+</sup> *<sup>L</sup>*<sup>ˆ</sup> <sup>2</sup> *<sup>z</sup>* and assume axial symmetry, i.e., *Ixy* <sup>≡</sup> *Ix* = *Iy*, we can recast the Hamiltonian as

$$
\hat{H} = \frac{1}{2I\_{\text{xy}}} \left[ \mathcal{L}^2 + \left( \frac{I\_{\text{xy}}}{I\_{\text{z}}} - 1 \right) \mathcal{L}\_{\text{z}}^2 \right], \tag{87}
$$

where *<sup>L</sup>*<sup>ˆ</sup> <sup>2</sup> is the angular momentum operator and *<sup>L</sup>*ˆ*<sup>z</sup>* is its projection on the rotation axis *<sup>z</sup>*. The concomitant spectrum of energy becomes

$$\mathfrak{e}\_{I,K} = \frac{\hbar^2}{2I\_{\rm xy}} \left[ I(I+1) + \left( \frac{I\_{\rm xy}}{I\_{\rm z}} - 1 \right) K^2 \right],\tag{88}$$

where *I* = 0, 1, 2,··· and it represents the eigenvalue of the angular momentum operator *L*ˆ 2, the numbers *m* = −*I*,···,−1, 0, 1,···,*I* stand for the projections on the intrinsic rotation axis of the rotator. All states exhibit a (2*I* +1)−degeneracy. The parameters *Ix* = *Iy* ≡ *Ixy* and *Iz* are the inertia momenta. Different "geometrical" instances are characterized through the *Ixy*/*Iz*−ratio. For example, the value *Ixy*/*Iz* = 1 corresponds to the spherical rotator. Limiting cases can also be considered. This is, *Ixy*/*Iz* = 1/2 and *Ixy*/*Iz* → ∞, that correspond to the extremely oblate- and prolate cases, respectively.

#### *4.2.1. Coherent states for the rigid rotator in three dimensions*

In order to obtain the Husimi distribution for this problem we need first of all to have the associated coherent states. Morales *et al.* have constructed them in Ref. [47] and discussed their mathematical foundations. First, they introduced the auxiliary quantity

$$X\_{I,M,K} = \sqrt{I!(I+M)!(I-M)!(I+K)!(I-K)!},\tag{89}$$

and then write [47]

16 Quantum Mechanics



The profile of the Husimi function is similar to that of a Gaussian distribution.

 d2*z*<sup>1</sup> π

> *<sup>H</sup>*<sup>ˆ</sup> <sup>=</sup> *<sup>L</sup>*<sup>ˆ</sup> <sup>2</sup> *x* 2*Ix* + *L*ˆ 2 *y* 2*Iy* + *L*ˆ 2 *z* 2*Iz*

*J*ˆ

�*I* ′ *M*′ *K*′

*W* = −

where <sup>σ</sup>*<sup>x</sup>* <sup>≡</sup> <sup>σ</sup>*<sup>y</sup>* <sup>=</sup> <sup>√</sup>*h*¯/2*m*<sup>ω</sup> and <sup>σ</sup>*px* <sup>≡</sup> <sup>σ</sup>*py* <sup>=</sup> <sup>√</sup>*m*ω*h*¯/2.

present bi-dimensional model this entropy reads

where *µ*(*z*1,*z*2) is given by Eq. (78).

4.2. Rigid rotator in three dimensions

dimensions, whose Hamiltonian writes [47]

{|*IMK*�}. The following relations apply

relations [47]

<sup>+</sup> *<sup>p</sup>*<sup>2</sup> *x* σ2 *px*

<sup>+</sup> *<sup>p</sup>*<sup>2</sup> *y* σ2 *py*

The Wehrl entropy is a semiclassical measure of localization [21] (so is Fisher's one [5] as well). Indeed, Wehrl's measure is simply a logarithmic Shannon measure built up with Husimi distributions. For the

d2*z*<sup>2</sup>

In the present section we consider a more general problem, the model of the rigid rotator in three

where *Ix*, *Iy*, and *Iz* are the associated moments of inertia. A complete set of rotator eigenstates is

*L*ˆ <sup>2</sup>|*IMK*� = *I*(*I* +1)*h*¯ <sup>2</sup>|*IMK*�

where *I* = 0,...,∞,−*I* ≤ *K* ≤ *I*, and −*I* ≤ *M* ≤ *I*. The states |*IMK*� satisfy orthogonality and closure

′ ,*I*δ*M*′ ,*M*δ*K*′

*<sup>z</sup>*|*IMK*� = *Mh*¯|*IMK*�,

<sup>|</sup>*IMK*� <sup>=</sup> <sup>δ</sup>*<sup>I</sup>*

, (80)

, (81)

<sup>π</sup> *<sup>µ</sup>*(*z*1,*z*2) ln*µ*(*z*1,*z*2), (82)

*<sup>L</sup>*ˆ*z*|*IMK*� <sup>=</sup> *Kh*¯|*IMK*� (84)

, (83)

,*<sup>K</sup>* (85)

$$|z\_{1}z\_{2}z\_{3}\rangle = e^{-\frac{|\mathbf{z}|^{2}}{2}} \sum\_{IMK} \frac{[(2I)!]^{2} z\_{1}^{(I+M)} z\_{2}^{I} z\_{3}^{(I+K)}}{X\_{I,M,K}} |IMK\rangle,\tag{90}$$

where the following supplementary variable were introduced by Morales *et al.* in Ref. [47]

$$|\mu|^2 = |z\_2|^2 (1 + |z\_1|^2)^2 (1 + |z\_3|^2)^2. \tag{91}$$

All coherent states share at least two requirements. Continuity of labeling and resolution of unity. In relation to the last property we add

$$
\int d\Gamma |z\_1 z\_2 z\_3\rangle \langle z\_1 z\_2 z\_3| = 1 \tag{92}
$$

where dΓ is the measure of integration given by [47]

$$\mathrm{d}\Gamma = \mathrm{d}\tau \left\{ 4[(1+|z\_1|^2)(1+|z\_3|^2)]^4 |z\_2|^4 - 8[(1+|z\_1|^2)(1+|z\_3|^2)]^2 |z\_2|^2 + 1 \right\} \tag{93}$$

with

$$\mathbf{d}\mathbf{\bar{\pi}} = \frac{\mathbf{d}^2 z\_1}{\pi} \frac{\mathbf{d}^2 z\_2}{\pi} \frac{\mathbf{d}^2 z\_3}{\pi},\tag{94}$$

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<sup>2</sup>)2. (101)

The Husimi Distribution: Development and Applications

dΓ*µ*(*z*1,*z*2,*z*3) = 1, (102)

dΓ*µ*(*z*1,*z*2,*z*3) ln*µ*(*z*1,*z*2,*z*3). (103)

*T*

*T*

. (104)

) =

*<sup>I</sup>*=<sup>0</sup> (2*<sup>I</sup>* <sup>+</sup>1)<sup>2</sup> *<sup>e</sup>*−*I*(*I*+1) <sup>Θ</sup>


In the special instance *Ixy*/*Iz* = 1, that corresponds to the spherical rotator, we explicitly obtain

<sup>2</sup> ∑<sup>∞</sup> *I*=0 |*u*| 2*I <sup>I</sup>*! *<sup>e</sup>*−*I*(*I*+1) <sup>Θ</sup>

∑∞

In order to emphasize some special cases associated to possible applications we consider several

4. The extremely prolate rotator is equivalent to the linear case (all diatomic molecules, *Iz* = 0, this is

In this section we propose a procedure to generalize the Husimi distribution to systems with continuous spectrum. We start extending the concept of coherent states for systems with discrete spectrum to systems with continuous one. In the present section, we see the Husimi distribution as a representation of the density operator in terms of a basis of coherent states. We specially discuss the problem of the

From the ρ(ε) definition expressed in Eq. (19), we can take a non-negative weight function like σ(*s*′

). However, this choice is not fully arbitrary, because it relies on , at least, two reasons: 1) it is related to the harmonic oscillator and, 2) it is a useful function that permits exactly to solve the integral

We can easily verify that *µ*(*z*1,*z*2,*z*3) is normalized in the fashion

*W* = 

*<sup>µ</sup>*(*z*1,*z*2,*z*3) = *<sup>e</sup>*−|*u*<sup>|</sup>

1. The spherical rotator *Ixy* = *Ix* = *Iy* = *Iz*, thus *Ixy*/*Iz* = 1 (*e.g. CH*4).

Having the Husimi functions the Wehrl entropy is straightforwardly computed.

2. The oblate rotator *Ixy* = *Ix* = *Iy* < *Iz*, specifically 1/2 ≤ *Ixy*/*Iz* < 1 (*e.g. C*6*H*6). 3. The prolate rotator *Ixy* = *Ix* = *Iy* > *Iz*, which corresponds to *Ixy*/*Iz* > 1 (*e.g. PCl*5).

5. Husimi distribution for systems with continuous spectrum

5.1. The exponential weight function: Harmonic oscillator

(19). The latter reason allows to express such integral in the following way

We compute now (i) the Wehrl entropy in the form

possibilities.

exp(−*s*′

*Ixy*/*Iz* → ∞ (*e.g. CO*2, *C*2*H*2).

continuous harmonic oscillator [20].

<sup>2</sup>(1+|*z*3|

and, of course, in this case we have three degrees of freedom. The present formulation satisfy the weaker version of the second requirement, because the measure is defined non positive [47].

#### *4.2.2. Husimi function, Wehrl entropy*

Using now Eq. (90) we find

$$|\langle IMK|z\_1z\_2z\_3\rangle|^2 = \frac{e^{-|u|^2}}{X\_{I,M,K}^2} [(2I)!]^2 |z\_1|^{2(I+M)} |z\_2|^{2I} |z\_3|^{2(I+K)}\tag{95}$$

and determine that, in this case, the rotational partition function reads

$$Z\_{\mathfrak{3}D} = \sum\_{I=0}^{\infty} \sum\_{K=-I}^{I} \sum\_{M=-I}^{I} e^{-\beta \mathfrak{E}\_{I,K}},\tag{96}$$

i.e.,

$$Z\_{3D} = \sum\_{I=0}^{\infty} \left(2I + 1\right) e^{-I\left(I + 1\right)\frac{\Theta}{T}} \sum\_{K=-I}^{I} e^{-\left(\frac{l\_V}{T\_c} - 1\right)K^2 \frac{\Theta}{T}}.\tag{97}$$

Remark that if we take the "extremely prolate" limiting case *Ixy*/*Iz* → ∞ just one term that survives in the right sum of the right side in Eq. (97), that for *K* = 0, while all terms for *K* �= 0 vanish. In this special instance case *Z*2*<sup>D</sup>* is recovered from *Z*3*D*. The pertinent Husimi distribution becomes

$$\mu(z\_1, z\_2, z\_3) = \frac{e^{-|\boldsymbol{\mu}|^2}}{Z\_{\text{3D}}} \sum\_{I=0}^{\infty} \frac{(2I)!}{I!} |\boldsymbol{\nu}|^{2I} e^{-I(I+1)\frac{\Theta}{T}} \times \mathbf{g}(I),\tag{98}$$

where

$$\log(I) = \sum\_{K=-I}^{I} \frac{|z\_{\Im}|^{2(I+K)}}{(I+K)!(I-K)!} e^{-\left(\frac{l\varepsilon}{l\_{\mathbb{C}}}-1\right)K^{2}\varPhi},\tag{99}$$

with

$$|\nu|^2 = (1 + |z\_1|^2)^2 |z\_2|^2,\tag{100}$$

<sup>612</sup> Advances in Quantum Mechanics The Husimi Distribution: Development and Applications 19 10.5772/53846 The Husimi Distribution: Development and Applications http://dx.doi.org/10.5772/53846 613

$$|\mu|^2 = |\nu|^2 (1 + |z\_3|^2)^2. \tag{101}$$

We can easily verify that *µ*(*z*1,*z*2,*z*3) is normalized in the fashion

$$\int \mathrm{d}\Gamma \,\mu(z\_1, z\_2, z\_3) = 1,\tag{102}$$

We compute now (i) the Wehrl entropy in the form

18 Quantum Mechanics

with

i.e.,

where

with

where dΓ is the measure of integration given by [47]

4[(1+|*z*1|

<sup>2</sup>)(1+|*z*3|


and determine that, in this case, the rotational partition function reads

∞ ∑ *I*=0

*<sup>µ</sup>*(*z*1,*z*2,*z*3) = *<sup>e</sup>*−|*u*<sup>|</sup>

*g*(*I*) =

*Z*3*<sup>D</sup>* =

*Z*3*<sup>D</sup>* =

<sup>2</sup>)]4|*z*2|

<sup>d</sup><sup>τ</sup> <sup>=</sup> d2*z*<sup>1</sup> π

weaker version of the second requirement, because the measure is defined non positive [47].

2

[(2*I*)!] <sup>2</sup>|*z*1| 2(*I*+*M*) |*z*2| 2*I* |*z*3|

*X*2 *I*,*M*,*K*

> ∞ ∑ *I*=0

(2*<sup>I</sup>* <sup>+</sup>1) *<sup>e</sup>*−*I*(*I*+1) <sup>Θ</sup>

special instance case *Z*2*<sup>D</sup>* is recovered from *Z*3*D*. The pertinent Husimi distribution becomes

∞ ∑ *I*=0


<sup>2</sup> = (1+|*z*1|

2

*Z*3*<sup>D</sup>*

*I* ∑ *K*=−*I*


*I* ∑ *K*=−*I*

*I* ∑ *M*=−*I*

> *T I* ∑ *K*=−*I e* − *Ixy Iz* <sup>−</sup><sup>1</sup> *K*<sup>2</sup> <sup>Θ</sup>

Remark that if we take the "extremely prolate" limiting case *Ixy*/*Iz* → ∞ just one term that survives in the right sum of the right side in Eq. (97), that for *K* = 0, while all terms for *K* �= 0 vanish. In this

> (2*I*)! *<sup>I</sup>*! <sup>|</sup>*v*<sup>|</sup>

<sup>2</sup>*<sup>I</sup> <sup>e</sup>*−*I*(*I*+1) <sup>Θ</sup>

*e* − *Ixy Iz* <sup>−</sup><sup>1</sup> *K*<sup>2</sup> <sup>Θ</sup>

<sup>2</sup>)2|*z*2|

<sup>4</sup> −8[(1+|*z*1|

d2*z*<sup>3</sup>

d2*z*<sup>2</sup> π

and, of course, in this case we have three degrees of freedom. The present formulation satisfy the

<sup>2</sup>)(1+|*z*3|

<sup>2</sup>)]2|*z*2|

<sup>π</sup> , (94)

<sup>2</sup> +1 

<sup>2</sup>(*I*+*K*) (95)

*<sup>T</sup>* . (97)

*<sup>T</sup>* ×*g*(*I*), (98)

*<sup>T</sup>* , (99)

2, (100)

*<sup>e</sup>*−βε*I*,*<sup>K</sup>* , (96)

(93)

dΓ = dτ

*4.2.2. Husimi function, Wehrl entropy*

Using now Eq. (90) we find

$$\mathcal{W} = \int \mathrm{d}\Gamma \,\mu(z\_1, z\_2, z\_3) \, \ln \mu(z\_1, z\_2, z\_3) . \tag{103}$$

In the special instance *Ixy*/*Iz* = 1, that corresponds to the spherical rotator, we explicitly obtain

$$\mu(z\_1, z\_2, z\_3) = e^{-|\mu|^2} \frac{\sum\_{I=0}^{\infty} \frac{|u|^{2I}}{I!} e^{-I(I+1)\frac{\Theta}{T}}}{\sum\_{I=0}^{\infty} (2I+1)^2 e^{-I(I+1)\frac{\Theta}{T}}}. \tag{104}$$

Having the Husimi functions the Wehrl entropy is straightforwardly computed.

In order to emphasize some special cases associated to possible applications we consider several possibilities.


#### 5. Husimi distribution for systems with continuous spectrum

In this section we propose a procedure to generalize the Husimi distribution to systems with continuous spectrum. We start extending the concept of coherent states for systems with discrete spectrum to systems with continuous one. In the present section, we see the Husimi distribution as a representation of the density operator in terms of a basis of coherent states. We specially discuss the problem of the continuous harmonic oscillator [20].

#### 5.1. The exponential weight function: Harmonic oscillator

From the ρ(ε) definition expressed in Eq. (19), we can take a non-negative weight function like σ(*s*′ ) = exp(−*s*′ ). However, this choice is not fully arbitrary, because it relies on , at least, two reasons: 1) it is related to the harmonic oscillator and, 2) it is a useful function that permits exactly to solve the integral (19). The latter reason allows to express such integral in the following way

$$\begin{split} \mathfrak{p}(\mathfrak{e}) &= \int\_0^s \mathrm{d}s' s'^{2\mathfrak{e}} \exp(-s'), \\ &= e^{-s/2} \frac{s^{\mathfrak{e}}}{2\mathfrak{e}+1} \mathcal{M}(\mathfrak{e}, \mathfrak{e} + 1/2, s) \end{split} \tag{105}$$

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Then, the substitution of the Eq. (111) into (110) leads us to the appearance

In the high temperature limit, this becomes

instance, ε*<sup>M</sup>* → 1), we find *µQ*(0) = 1−βω/12.

If we replace this result into Eq. (124) we obtain

and, from Eq. (24) we write

maximum [48]

5.3. Asymptotic behavior of the Husimi function

of the Whittaker function [48] defined for *s* → ∞, as follows:

lim *s*→∞

*µQ*(0) = (2*e*−βωε*<sup>M</sup>* βωε*<sup>M</sup>* <sup>+</sup>2*e*−βωε*<sup>M</sup>* <sup>+</sup>*e*−βωε*<sup>M</sup>* βω−2−βω)

*µQ*(0) <sup>≈</sup> <sup>1</sup>

ε*M*

If we take into account a kind of particles filling a band in the lowest continuous levels of energy (for

In this part of the work, we are considering a particular range for ε; *i.e.,* 0 ≤ ε ≤ ε*<sup>M</sup>* = 1 and we study the asymptotic behavior of the Husimi distribution. This trend might be obtained from the limiting case

> <sup>ε</sup>*<sup>M</sup>* 0

*<sup>Z</sup> es*/<sup>2</sup>

1 <sup>Γ</sup>(2ε+1) <sup>≈</sup>

where *A*<sup>0</sup> = .9963530195, *A*<sup>1</sup> = 1.221909147, *A*<sup>2</sup> = −3.108524622, and *A*<sup>3</sup> = 1.333217620.

3 ∑ *n*=0 *An*

Now, we follow expanding to third order the inverse of the gamma function, 1/Γ(2ε+1), around its

<sup>d</sup><sup>ε</sup> *<sup>s</sup>*2<sup>ε</sup> Γ(2ε+1)

 <sup>ε</sup>*<sup>M</sup>* 0

3 ∑ *n*=0

*M <sup>n</sup>* 2 , *n* <sup>2</sup> <sup>+</sup> <sup>1</sup>

<sup>d</sup><sup>ε</sup> *<sup>e</sup>*−ωβε*s*2<sup>ε</sup> Γ(2ε+1)

<sup>2</sup> ,−2ln(*s*)

 (*n*+1)(−2ln(*s*))1+*n*/<sup>2</sup> , (118)

*e*−*s*/2*s*<sup>ε</sup>*M* (ε, ε+1/2,*s*)

*<sup>M</sup>*(*s*)<sup>2</sup> = *es*/<sup>2</sup>

*µQ*(*s*) = *<sup>M</sup>*(*s*)−<sup>2</sup>

From Eq. (115), we derive a approximate result for *M*(*s*)2, which is given by

2

*<sup>M</sup>*(*s*)<sup>2</sup> <sup>=</sup> *es*/<sup>2</sup> *<sup>s</sup>*

<sup>−</sup> βωε*<sup>M</sup>* 6(ε*<sup>M</sup>* +1)

βωε*M*(*e*−βωε*<sup>M</sup>* <sup>−</sup>1)(ε*<sup>M</sup>* <sup>+</sup>1) . (112)

The Husimi Distribution: Development and Applications

<sup>2</sup>ε+<sup>1</sup> <sup>=</sup> <sup>Γ</sup>(2ε+1). (114)

. (113)

, (115)

. (116)

*An*ε*n*, (117)

where *M* (*a*,*b*,*x*) is the Whittaker function [48]. Besides, in relation to the first reason, when we consider ε = *n*, where *n* is integer, in the limit *s* → ∞; the Eq. (105) drops into the known quantum result for the harmonic oscillator, ρ(*n*) = *n*! [11].

Moreover, the measure in phase space can be explicitly expressed from Eq. (20) as follows

$$\mathrm{d}\mathfrak{t}(s) = \mathrm{d}s\,e^{-s/2} \int\_0^{\mathfrak{e}\_F} \mathrm{d}\mathfrak{e} \frac{(2\mathfrak{e} + 1)s^{\mathfrak{e}}}{\mathcal{M}(\mathfrak{e}, \mathfrak{e} + 1/2, s)}. \tag{106}$$

Although obtaining this explicit form of the measure, a most general expression for the integral of Eq. (106) strongly depends on the particular spectrum of the system. In the present case, a spectrum like ε∝ω, the harmonic oscillator in the continuous limit, is considered.

### 5.2. *s* → 0 approximation for the Husimi distribution

In order to know the shape of the Husimi distribution in *s* = 0, we need to calculate some important quantities. First, we evaluate ρ(ε) given by Eq. (105) expanding the exponential which appears inside the integral, as follows

$$\mathfrak{p}(\mathfrak{e}) \approx \lim\_{s \to 0} \int\_0^s \mathrm{d}s'^{2\varepsilon} (1 - s' + \cdots),\tag{107}$$

$$\approx \frac{s^{2\mathfrak{c}+1}}{2\mathfrak{c}+1} \left( 1 - \frac{2\mathfrak{c}+1}{2\mathfrak{c}+2} s + \cdots \right). \tag{108}$$

But, we are interested in evaluating the inverse of ρ(ε), therefore

$$\frac{1}{\mathfrak{p}(\mathfrak{e})} \approx \frac{2\mathfrak{e} + 1}{s^{2\mathfrak{e} + 1}} \left( 1 + \frac{2\mathfrak{e} + 1}{2\mathfrak{e} + 2} s + \cdots \right). \tag{109}$$

Second, we show easily that, in the limit *s* → 0, the Husimi distribution is given by

$$
\mu\_{\mathcal{Q}}(0) = \frac{1}{Z} \frac{\int\_0^{\mathfrak{E}\_M} \mathrm{d}\mathfrak{E}\,(2\mathfrak{E}+1)e^{-\beta \mathrm{e}\mathfrak{e}}}{\int\_0^{\mathfrak{E}\_M} \mathrm{d}\mathfrak{E}\,(2\mathfrak{E}+1)}.\tag{110}
$$

Now, after integrating Eq. (25) the partition function is expressed as follows

$$Z = \frac{1 - \exp(-\beta \text{osc}\_M)}{\beta \text{o}}.\tag{111}$$

Then, the substitution of the Eq. (111) into (110) leads us to the appearance

$$\mu\_{\mathcal{Q}}(0) = \frac{(2e^{-\beta \text{osc}\_M} \beta \text{osc}\_M + 2e^{-\beta \text{osc}\_M} + e^{-\beta \text{osc}\_M} \beta \text{osc} - 2 - \beta \text{osc})}{\beta \text{osc}\_M (e^{-\beta \text{osc}\_M} - 1)(\varepsilon\_M + 1)}. \tag{112}$$

In the high temperature limit, this becomes

20 Quantum Mechanics

the integral, as follows

ρ(ε) =

result for the harmonic oscillator, ρ(*n*) = *n*! [11].

 *<sup>s</sup>* 0 d*s* ′ *s*

<sup>=</sup> *<sup>e</sup>*−*s*/<sup>2</sup> *<sup>s</sup>*<sup>ε</sup>

Moreover, the measure in phase space can be explicitly expressed from Eq. (20) as follows

 <sup>ε</sup>*<sup>F</sup>* 0

Although obtaining this explicit form of the measure, a most general expression for the integral of Eq. (106) strongly depends on the particular spectrum of the system. In the present case, a spectrum

In order to know the shape of the Husimi distribution in *s* = 0, we need to calculate some important quantities. First, we evaluate ρ(ε) given by Eq. (105) expanding the exponential which appears inside

 1+

<sup>ε</sup>*<sup>M</sup>*

<sup>ε</sup>*<sup>M</sup>*

*<sup>Z</sup>* <sup>=</sup> <sup>1</sup>−exp(−βωε*M*)

<sup>1</sup><sup>−</sup> <sup>2</sup>ε+<sup>1</sup> 2ε+2

> 2ε+1 2ε+2

<sup>0</sup> <sup>d</sup><sup>ε</sup> (2ε+1)*e*−βωε

*s*+··· 

*s*+··· 

<sup>d</sup>τ(*s*) = <sup>d</sup>*s e*−*s*/<sup>2</sup>

like ε∝ω, the harmonic oscillator in the continuous limit, is considered.

ρ(ε) ≈ lim *s*→0 *<sup>s</sup>* 0 d*s* ′ *s* ′2ε (1−*s*

> <sup>≈</sup> *<sup>s</sup>*2ε+<sup>1</sup> 2ε+1

5.2. *s* → 0 approximation for the Husimi distribution

But, we are interested in evaluating the inverse of ρ(ε), therefore

1

<sup>ρ</sup>(ε) <sup>≈</sup> <sup>2</sup>ε+<sup>1</sup> *s*2ε+<sup>1</sup>

Second, we show easily that, in the limit *s* → 0, the Husimi distribution is given by

*Z*

*µQ*(0) = <sup>1</sup>

Now, after integrating Eq. (25) the partition function is expressed as follows

′2<sup>ε</sup> exp(−*<sup>s</sup>*

2ε+1

where *M* (*a*,*b*,*x*) is the Whittaker function [48]. Besides, in relation to the first reason, when we consider ε = *n*, where *n* is integer, in the limit *s* → ∞; the Eq. (105) drops into the known quantum

′ ),

<sup>d</sup><sup>ε</sup> (2ε+1)*s*<sup>ε</sup> *M* (ε, ε+1/2,*s*)

*M* (ε, ε+1/2,*s*) (105)

. (106)

′ +···), (107)

. (108)

. (109)

<sup>0</sup> <sup>d</sup><sup>ε</sup> (2ε+1) . (110)

βω . (111)

$$
\mu\_{\mathcal{Q}}(0) \approx \frac{1}{\mathfrak{e}\_M} - \frac{\mathfrak{Boe}\_M}{6(\mathfrak{e}\_M + 1)}.\tag{113}
$$

If we take into account a kind of particles filling a band in the lowest continuous levels of energy (for instance, ε*<sup>M</sup>* → 1), we find *µQ*(0) = 1−βω/12.

#### 5.3. Asymptotic behavior of the Husimi function

In this part of the work, we are considering a particular range for ε; *i.e.,* 0 ≤ ε ≤ ε*<sup>M</sup>* = 1 and we study the asymptotic behavior of the Husimi distribution. This trend might be obtained from the limiting case of the Whittaker function [48] defined for *s* → ∞, as follows:

$$\lim\_{s \to \infty} \frac{e^{-s/2} s^{\mathfrak{E}} \mathcal{M}(\mathfrak{e}, \mathfrak{e} + 1/2, \mathfrak{s})}{2\mathfrak{e} + 1} = \Gamma(2\mathfrak{e} + 1). \tag{114}$$

If we replace this result into Eq. (124) we obtain

$$\left(\mathcal{M}(\mathbf{s})\right)^2 = e^{s/2} \int\_0^{\mathfrak{E}\_M} \mathrm{d}\mathfrak{E} \frac{\mathfrak{s}^{2\mathfrak{E}}}{\Gamma(2\mathfrak{E} + 1)},\tag{115}$$

and, from Eq. (24) we write

$$
\mu\_{\mathcal{Q}}(s) = \frac{M(s)^{-2}}{Z} e^{s/2} \int\_0^{\mathfrak{e}\_M} d\mathfrak{e} \frac{e^{-\alpha \mathfrak{Re}\_S 2\varepsilon}}{\Gamma(2\varepsilon + 1)}.\tag{116}
$$

Now, we follow expanding to third order the inverse of the gamma function, 1/Γ(2ε+1), around its maximum [48]

$$\frac{1}{\Gamma(2\mathfrak{c}+1)} \approx \sum\_{n=0}^{3} A\_n \mathfrak{c}^n,\tag{117}$$

where *A*<sup>0</sup> = .9963530195, *A*<sup>1</sup> = 1.221909147, *A*<sup>2</sup> = −3.108524622, and *A*<sup>3</sup> = 1.333217620.

From Eq. (115), we derive a approximate result for *M*(*s*)2, which is given by

$$\mathcal{M}(s)^2 = e^{s/2} \frac{s}{2} \sum\_{n=0}^{3} A\_n \frac{\mathcal{M}\left(\frac{n}{2}, \frac{n}{2} + \frac{1}{2}, -2\ln(s)\right)}{(n+1)(-2\ln(s))^{1+n/2}},\tag{118}$$

and combining all above expressions, we have finally found an expression to third order of approximation for Husimi distribution given by

$$\mu\_{\mathcal{Q}}(s) = \frac{M(s)^{-2}}{Z} e^{s/2 - \beta \mathfrak{a}/2} \frac{s}{2} \sum\_{n=0}^{3} A\_n \frac{\mathcal{M}\left(\frac{n}{2}, \frac{n}{2} + \frac{1}{2}, \beta \mathfrak{a} - 2\ln(s)\right)}{(n+1)(\beta \mathfrak{a} - 2\ln(s))^{1+n/2}},\tag{119}$$

where *M* (*a*,*b*,*c*) is again the Whittaker function [48].

In the high temperature approximation, Eq. (119) is given by

$$\mu\_{\mathcal{Q}}(s) \approx \text{Box} \frac{\exp(-\beta \text{ao}/2)}{1 - \exp(-\beta \text{ao})} \approx \exp(-\beta \text{ao}/2). \tag{120}$$

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continuous limit of a particle in a Coulomb potential. Therefore, it is necessary to introduce a density of states *g*(*E*) in the formulation of continuous coherent states (17) and immediately get the following

d*E g*(*E*)

<sup>d</sup>*E g*(*E*)<sup>2</sup> *<sup>s</sup>*2*E*/*<sup>A</sup>*

*sE*/*Ae*<sup>−</sup>*i*γ*E*/*<sup>A</sup>*

<sup>ρ</sup>(*E*) <sup>|</sup>ε�, (123)

The Husimi Distribution: Development and Applications

<sup>ρ</sup>(*E*) (124)

 *EM* 0

> *EM* 0

We have included in the current work some motivational elements to develop possible future applications to information theory and condensed matter. We have focused attention primarily upon Husimi distribution and its analytical results, beyond the numerical, graphical, or approximate calculations. A semiclassical description undertaking can be tackled, (i) trying to estimate phase-space location via measures as Fisher information and (ii) evaluating the semiclassical Wehrl entropy. A crucial point, in such an estimation, is to define the Husimi distribution in a convenient set of coherent states. Hence, we introduce a formal view of general requirements for formulations of coherent states in the context of the Gazeau and Klauder formalism for the harmonic oscillator – we have included some mathematical details in order to make it easy to follow and instructive in courses of quantum mechanics for graduates– we show some practical elements to apply the present formalisms to specific calculations

By using a suitable formulation of coherent states in every case, we show explicitly the form of the Husimi distribution for i) a spinless charged particle in a uniform magnetic field (Landau diamagnetism), (ii) the linear and the three dimensional rotator (molecular rotation) and (iii) a case

In addition, we can calculate the probability by projecting the states over the coherent states as a function of a variable related to the coherent states. We see that the localization of probability, in the phase space decreases as temperature increases. Also, as always, the localization of the Husimi distribution in the phase space decreases as temperature increases. The present derivation of Husimi distributions is based on the evaluation of the mean value of the density operator in the basis of a single-particle coherent state. While the Husimi function takes into account collective and environmental effects, the coherent states are independent-particle states. Thus, if the Husimi distribution is delocalized, we need many wave packets (independent-particle states) to represent the state. Furthermore, the thermodynamics of particles in systems, which come from environmentally induced effects, does not depend on the

formulation of the coherent states. In this manner, we expect this behavior to become general.

In conclusion, quantal distributions in the phase space, such as the Husimi distribution, have long been recognized as powerful tools for studying the quantum-classical correspondence and semi-classical aspects of quantum mechanics, since they provide a phase-space picture of the density matrix. We

<sup>|</sup>*s*, <sup>γ</sup>� = *<sup>M</sup>*(*s*)−<sup>1</sup>

*M*(*s*)<sup>2</sup> =

modification

where the function *M*(*s*)

6. Final remarks

of semi-classical measures.

of the limiting harmonic oscillator (continuous spectrum).

aknowledge partial financial support by FONDECYT 1110827.

represents the normalization factor.

The present result does not depend on the values of the parameter *s*. Furthermore, this approximation is valid whenever 0 ≤ ε ≤ 1. We notice that the asymptotic trend of the Husimi distribution approaches to the Boltzmann weight in the ground state of the harmonic oscillator.

#### 5.4. Some applications and consequences

In Ref. [11], the mean value of energy is obtained from the expectation value of the classical Hamiltonian *H* in a coherent state as follows *H* (*s*) = �*s*, γ|*H* |*s*, γ�, therefore they arrive to the relation *H* (*s*) = *s*∂ln*M*(*s*)/∂*s*.

However, it is our interest here to calculate the mean value of energy in a different way, integrating in the variable *s* with *µQ*(*s*) as a weigh function. Hence, we have

$$
\langle \mathcal{H} \rangle = \int \mathrm{d}\mathfrak{r}(s) \, \mu\_{\mathcal{Q}}(s) \, \mathcal{H}(s), \tag{121}
$$

where *H* , expressed in terms of the variable *s*, denotes the classical Hamiltonian of the system. Inserting the Husimi distribution (24) into Eq. (121) and making use the relation (19) we finally get

$$
\langle \mathcal{H} \rangle = \frac{1}{Z} \int\_0^{\varepsilon\_M} d\mathfrak{e} \, e^{-\beta \mathfrak{e}} \mathcal{H}(\mathfrak{e}), \tag{122}
$$

that is the classical mean energy [20]. We emphasize that the Husimi distribution, for a system with continuous spectrum, conduces in a natural way to the classical mean value of energy. Obviously, this is not true when the spectrum is discrete.

An extra motivation consists in extending the formulation of coherent states to systems with continuum spectrum considering its explicit form; for instance, we can take a spectrum whose appearance is *E* = *A*εν, where *A* and ν are constant. The values ν = ±1 and ν = 2 might define the continuous limit of three remarkable cases in physics. Certainly, in a general study other values of the parameter ν may be conveniently considered as an interesting analytical extension. Thus, for ν = 1 and *A* = ω we have the continuous limit of a particle in a harmonic potential; this case is being in detail discussed in the present work. For ν = 2 we have the continuous limit of a particle in a box. For ν = −1 we have the continuous limit of a particle in a Coulomb potential. Therefore, it is necessary to introduce a density of states *g*(*E*) in the formulation of continuous coherent states (17) and immediately get the following modification

$$|s,\gamma\rangle = M(s)^{-1} \int\_0^{E\_M} \mathrm{d}E \, g(E) \, \frac{s^{E/A} e^{-l\eta E/A}}{\sqrt{\mathfrak{p}(E)}} \, |\varepsilon\rangle,\tag{123}$$

where the function *M*(*s*)

22 Quantum Mechanics

approximation for Husimi distribution given by

*µQ*(*s*) = *<sup>M</sup>*(*s*)−<sup>2</sup>

where *M* (*a*,*b*,*c*) is again the Whittaker function [48].

5.4. Some applications and consequences

is not true when the spectrum is discrete.

*H* (*s*) = *s*∂ln*M*(*s*)/∂*s*.

In the high temperature approximation, Eq. (119) is given by

to the Boltzmann weight in the ground state of the harmonic oscillator.

the variable *s* with *µQ*(*s*) as a weigh function. Hence, we have

�*H* � = 

�*<sup>H</sup>* � <sup>=</sup> <sup>1</sup> *Z* <sup>ε</sup>*<sup>M</sup>* 0

the Husimi distribution (24) into Eq. (121) and making use the relation (19) we finally get

*<sup>Z</sup> es*/2−βω/<sup>2</sup> *<sup>s</sup>*

*µQ*(*s*) <sup>≈</sup> βω exp(−βω/2)

and combining all above expressions, we have finally found an expression to third order of

3 ∑ *n*=0 *An*

The present result does not depend on the values of the parameter *s*. Furthermore, this approximation is valid whenever 0 ≤ ε ≤ 1. We notice that the asymptotic trend of the Husimi distribution approaches

In Ref. [11], the mean value of energy is obtained from the expectation value of the classical Hamiltonian *H* in a coherent state as follows *H* (*s*) = �*s*, γ|*H* |*s*, γ�, therefore they arrive to the relation

However, it is our interest here to calculate the mean value of energy in a different way, integrating in

where *H* , expressed in terms of the variable *s*, denotes the classical Hamiltonian of the system. Inserting

that is the classical mean energy [20]. We emphasize that the Husimi distribution, for a system with continuous spectrum, conduces in a natural way to the classical mean value of energy. Obviously, this

An extra motivation consists in extending the formulation of coherent states to systems with continuum spectrum considering its explicit form; for instance, we can take a spectrum whose appearance is *E* = *A*εν, where *A* and ν are constant. The values ν = ±1 and ν = 2 might define the continuous limit of three remarkable cases in physics. Certainly, in a general study other values of the parameter ν may be conveniently considered as an interesting analytical extension. Thus, for ν = 1 and *A* = ω we have the continuous limit of a particle in a harmonic potential; this case is being in detail discussed in the present work. For ν = 2 we have the continuous limit of a particle in a box. For ν = −1 we have the

*M <sup>n</sup>* 2 , *n* <sup>2</sup> <sup>+</sup> <sup>1</sup>

<sup>2</sup> ,βω−2ln(*s*)

<sup>1</sup>−exp(−βω) <sup>≈</sup> exp(−βω/2). (120)

<sup>d</sup>τ(*s*)*µQ*(*s*)*H* (*s*), (121)

<sup>d</sup>ε*e*−βε*H* (ε), (122)

 (*n*+1)(βω−2ln(*s*))1+*n*/<sup>2</sup> , (119)

2

$$\left(M(s)\right)^{2} = \int\_{0}^{E\_{M}} \mathrm{d}E \, g(E)^{2} \frac{s^{2E/A}}{\mathfrak{p}(E)}\tag{124}$$

represents the normalization factor.

#### 6. Final remarks

We have included in the current work some motivational elements to develop possible future applications to information theory and condensed matter. We have focused attention primarily upon Husimi distribution and its analytical results, beyond the numerical, graphical, or approximate calculations. A semiclassical description undertaking can be tackled, (i) trying to estimate phase-space location via measures as Fisher information and (ii) evaluating the semiclassical Wehrl entropy. A crucial point, in such an estimation, is to define the Husimi distribution in a convenient set of coherent states. Hence, we introduce a formal view of general requirements for formulations of coherent states in the context of the Gazeau and Klauder formalism for the harmonic oscillator – we have included some mathematical details in order to make it easy to follow and instructive in courses of quantum mechanics for graduates– we show some practical elements to apply the present formalisms to specific calculations of semi-classical measures.

By using a suitable formulation of coherent states in every case, we show explicitly the form of the Husimi distribution for i) a spinless charged particle in a uniform magnetic field (Landau diamagnetism), (ii) the linear and the three dimensional rotator (molecular rotation) and (iii) a case of the limiting harmonic oscillator (continuous spectrum).

In addition, we can calculate the probability by projecting the states over the coherent states as a function of a variable related to the coherent states. We see that the localization of probability, in the phase space decreases as temperature increases. Also, as always, the localization of the Husimi distribution in the phase space decreases as temperature increases. The present derivation of Husimi distributions is based on the evaluation of the mean value of the density operator in the basis of a single-particle coherent state. While the Husimi function takes into account collective and environmental effects, the coherent states are independent-particle states. Thus, if the Husimi distribution is delocalized, we need many wave packets (independent-particle states) to represent the state. Furthermore, the thermodynamics of particles in systems, which come from environmentally induced effects, does not depend on the formulation of the coherent states. In this manner, we expect this behavior to become general.

In conclusion, quantal distributions in the phase space, such as the Husimi distribution, have long been recognized as powerful tools for studying the quantum-classical correspondence and semi-classical aspects of quantum mechanics, since they provide a phase-space picture of the density matrix. We aknowledge partial financial support by FONDECYT 1110827.

#### Author details

Sergio Curilef1 and Flavia Pennini1,2

1 Departamento de Física, Universidad Católica del Norte, Antofagasta, Chile

2 Instituto de Física La Plata–CCT-CONICET, Fac. de Ciencias Exactas, Universidad Nacional de La Plata, La Plata, Argentina

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The Husimi Distribution: Development and Applications

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24 Quantum Mechanics

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1 Departamento de Física, Universidad Católica del Norte, Antofagasta, Chile

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**Section 7**

**Quantum Information and Related Topics**


**Quantum Information and Related Topics**

26 Quantum Mechanics

620 Advances in Quantum Mechanics

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**Chapter 26**

**The Quantum Mechanics Aspect of Structural**

Theoretical and experimental investigations of size effects have made a substantial contribu‐ tion to the development of nanophysics and nanochemistry. However, a great deal needs to be done in this field. Experimental results are not necessarily consistent with the traditional concepts. In particular, the melting temperature of small nanoparticles unexpectedly turned out to be higher than the melting temperature of a macroscopic sample of the same chemical

It is this chemical composition of a macroscopic system that determines its melting temperature *Tm*, the specific heat of melting *Qm*, and the entropy of melting *Sm*. These quantities do not depend on the number *M* of atoms (in the limit, *M*→*∞*). This statement ceases to be valid for relatively small systems. In the given case, it is necessary to take into account the dependences of the quantities *Tm*, *Qm*, and *Sm* on the number *M* of atoms. Moreover, as the size of the system decreases, the interpretation of the physical quantities *Tm*, *Qm*, and *Sm* should be refined. Indeed, one molecule, for example, the hydrogen molecule, cannot melt, because its dissociation occurs with an increase in the temperature. In this respect, it is advisable to analyze the structural transformations in nanosystems within a unified approach of the first principles of quantum

In the framework of classical physics each structural modification is set by the vector

( ) 1 2 , , ... , ... , **R rr r r** = *i M* (1)

© 2013 Bal'makov; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

© 2013 Bal'makov; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

distribution, and reproduction in any medium, provided the original work is properly cited.

**Transformations in Nanosystems**

Additional information is available at the end of the chapter

M. D. Bal'makov

**1. Introduction**

composition [1].

mechanics and statistical physics.

**2. Quasiclosed ensembles**

http://dx.doi.org/10.5772/53704
