**Classical and Quantum Conjugate Dynamics – The Interplay Between Conjugate Variables**

Gabino Torres-Vega

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/53598

### **1. Introduction**

There are many proposals for writing Classical and Quantum Mechanics in the same lan‐ guage. Some approaches use complex functions for classical probability densities [1] and other define functions of two variables from single variable quantum wave functions [2,3]. Our approach is to use the same concepts in both types of dynamics but in their own realms, not using foreign unnatural objects. In this chapter, we derive many inter relationships be‐ tween conjugate variables.

#### **1.1. Conjugate variables**

An important object in Quantum Mechanics is the eigenfunctions set {|*n* >} *n*=0 *<sup>∞</sup>* of a Hermi‐ tian operator *F* ^ . These eigenfunctions belong to a Hilbert space and can have several repre‐ sentations, like the coordinate representation *ψn*(*q*)= *q* | *n* . The basis vector used to provide the coordinate representation, |*q* >, of the wave function are themselves eigenfunctions of the coordinate operator *Q* ^ We proceed to define the classical analogue of both objects, the eigenfunction and its support.

Classical motion takes place on the associated cotangent space *T \* Q* with variable *z* =(*q*, *p*), where *q* and *p* are *n* dimensional vectors representing the coordinate and momentum of point particles. We can associate to a dynamical variable *F* (*z*) its eigensurface, i.e. the level set

$$\Sigma\_F(f) = \{ z \in T \: \mathbb{Q} \mid F(z) = f \} \tag{1}$$

© 2013 Torres-Vega; 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 Torres-Vega; 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.

Where *f* is a constant, one of the values that *F* (*z*) can take. This is the set of points in phase space such that when we evaluate *F* (*z*), we obtain the value *f* . Examples of these eigensurfa‐ ces are the constant coordinate surface, *q* = *X* , and the energy shell, *H* (*z*)=*E*, the surface on which the evolution of classical systems take place. These level sets are the classical analogues of the support of quantum eigenfunctions in coordinate or momentum representations.

Many dynamical variables come in pairs. These pairs of dynamical variables are related through the Poisson bracket. For a pair of conjugate variables, the Poisson bracket is equal to one. This is the case for coordinate and momentum variables, as well as for energy and time. In fact, according to Hamilton's equations of motion, and the chain rule, we have that

$$\{\{t\_{\prime}\}\}=\sum\_{i}\left(\frac{\partial \, t}{\partial \, q^{i}}\frac{\partial \, H}{\partial \, p\_{i}}-\frac{\partial \, H}{\partial \, q^{i}}\frac{\partial \, t}{\partial \, p\_{i}}\right) = \sum\_{i}\left(\frac{dt}{dq^{i}}\frac{d \, q^{i}}{dt}+\frac{d \, \, p\_{i}}{dt}\frac{dt}{dq\_{i}}\right) = \frac{dt}{dt} = \mathbf{1} \tag{2}$$

A similar situation is found with the conjugate pair energy-time. Usually the energy is well defined in phase space but time is not. In a previous work, we have developed a method for defining a time coordinate in phase space [4]. The method takes the hypersurface *q* <sup>1</sup> = *X* , where *X* is fixed, as the zero time eigensurface and propagates it forward and backward in

Classical and Quantum Conjugate Dynamics – The Interplay Between Conjugate Variables

Now, recall that any phase space function *G*(*z*) generates a motion in phase space through a

where *f* is a variable with the same units as the conjugate variable *F* (*z*). You can think of *G*(*z*) as the Hamiltonian for a mechanical system and that *f* is the time. For classical sys‐ tems, we are considering conjugate pairs leading to conjugate motions associated to each variable with the conjugate variable serving as the evolution parameter (see below). This will be applied to the energy-time conjugate pair. Let us derive some properties in which the

Some relationships between a pair of conjugate variables are derived in this section. We will deal with general *F* (*z*) and *G*(*z*) conjugate variables, but the results can be applied to coordi‐

is the classical analogue of the corresponding quantum eigenstate in coordinate *q* | *g* and momentum *p* | *g* representations. When *G*(*z*) is evaluated at the points of the support of (*z* | *g*), we get the value *g*. We use a bra-ket like notation to emphasise the similarity with

The magnitude of the vector field | *XG* | is the change of length along the *f* direction

<sup>∂</sup> *<sup>p</sup>* , - <sup>∂</sup> *<sup>G</sup>*

<sup>∂</sup> *<sup>q</sup>* ), (4)

http://dx.doi.org/10.5772/53598

5

*df* , (5)

(*z*|*g*)=*δ*(*z* - *v*), *v* ∈Σ*G*(*g*) (6)

^ or *G*

^ provides margin‐

time generating that way a coordinate system for time in phase space.

*df* <sup>=</sup> *XG*, *XG* =( <sup>∂</sup> *<sup>G</sup>*

nate and momentum or energy and time or to any other conjugate pair.

is the length element.

The overlap between a probability density with an eigenfunction of *F*

<sup>|</sup>*XG*|= *<sup>d</sup> <sup>q</sup> <sup>i</sup> df d q <sup>i</sup> df* + *d pi df d pi df* <sup>=</sup> *<sup>d</sup> lF*

A unit density with the eigensurface Σ*G*(*g*) as support

set of symplectic system of equations, a dynamical system,

*dz*

two conjugate variables participate.

where *dlF* <sup>=</sup> (*dq <sup>i</sup>*)<sup>2</sup> <sup>+</sup> (*<sup>d</sup> <sup>p</sup> <sup>i</sup>*)<sup>2</sup>

the quantum concepts.

al representations of a probability density,

**1.3. The interplay between conjugate variables**

Now, a point in cotangent space can be specified as the intersection of 2*n* hypersurfaces. A set of 2*n* independent, intersecting, hypersurfaces can be seen as a coordinate system in co‐ tangent space, as is the case for the hyper surfaces obtained by fixing values of coordinate and momentum, i.e. the phase space coordinate system with an intersection at *z* =(*q*, *p*). We can think of alternative coordinate systems by considering another set of conjugate dynami‐ cal variables, as is the case of energy and time.

Thus, in general, the *T \* Q* points can be represented as the intersection of the eigensurfaces of the pair of conjugate variables *F* and *G*,

$$\Sigma\_{\rm FG}(f,\,\,\,\mathbf{g}) = \left\{ \mathbf{z} \in T \, \, \, \middle| \, \, \, \, \mathcal{F}(\mathbf{z}) = f, \,\, \, \mathcal{G}(\mathbf{z}) = \mathbf{g} \right\}.\tag{3}$$

A point in this set will be denoted as an abstract bra ( *f* , *g* |, such that ( *f* , *g* |*u*) means the function *u*( *f* , *g*).

We can also have marginal representations of functions in phase space by using the eigen‐ surfaces of only one of the functions,

$$\Sigma\_{\Gamma}(f) = \left\{ z \in T \: \bigotimes^\* F(z) = f \right\}, \quad \text{and} \quad \Sigma\_{G}(g) = \left\{ z \in T \: \bigotimes^\* Q \mid G(z) = g \right\}.$$

A point in the set Σ*<sup>F</sup>* ( *f* ) Σ*G*(*g*) will be denoted by the bra ( *f* | (*g* | and an object like ( *f* | *u*) (*g* |*u*) will mean the *f g* dependent function *u*( *f* ) *u*(*g*) .

#### **1.2. Conjugate coordinate systems**

It is usual that the origin of one of the variables of a pair of conjugate variables is not well defined. This happens, for instance, with the pair of conjugate variables *q* and *p*. Even though the momentum can be well defined, the origin of the coordinate is arbitrary on the trajectory of a point particle, and it can be different for each trajectory. A coordinate system fixes the origin of coordinates for all of the momentum eigensurfaces.

A similar situation is found with the conjugate pair energy-time. Usually the energy is well defined in phase space but time is not. In a previous work, we have developed a method for defining a time coordinate in phase space [4]. The method takes the hypersurface *q* <sup>1</sup> = *X* , where *X* is fixed, as the zero time eigensurface and propagates it forward and backward in time generating that way a coordinate system for time in phase space.

Now, recall that any phase space function *G*(*z*) generates a motion in phase space through a set of symplectic system of equations, a dynamical system,

$$\frac{dz}{d\bar{f}} = X\_{G'} \qquad \qquad X\_G = \begin{pmatrix} \frac{\partial G}{\partial p} \; \; \; \; \; \; \; \; \frac{\partial G}{\partial q} \; \; \; \; \; \; \tag{4}$$

where *f* is a variable with the same units as the conjugate variable *F* (*z*). You can think of *G*(*z*) as the Hamiltonian for a mechanical system and that *f* is the time. For classical sys‐ tems, we are considering conjugate pairs leading to conjugate motions associated to each variable with the conjugate variable serving as the evolution parameter (see below). This will be applied to the energy-time conjugate pair. Let us derive some properties in which the two conjugate variables participate.

#### **1.3. The interplay between conjugate variables**

Where *f* is a constant, one of the values that *F* (*z*) can take. This is the set of points in phase space such that when we evaluate *F* (*z*), we obtain the value *f* . Examples of these eigensurfa‐ ces are the constant coordinate surface, *q* = *X* , and the energy shell, *H* (*z*)=*E*, the surface on which the evolution of classical systems take place. These level sets are the classical analogues of the support of quantum eigenfunctions in coordinate or momentum representations.

Many dynamical variables come in pairs. These pairs of dynamical variables are related through the Poisson bracket. For a pair of conjugate variables, the Poisson bracket is equal to one. This is the case for coordinate and momentum variables, as well as for energy and time. In fact, according to Hamilton's equations of motion, and the chain rule, we have that

Now, a point in cotangent space can be specified as the intersection of 2*n* hypersurfaces. A set of 2*n* independent, intersecting, hypersurfaces can be seen as a coordinate system in co‐ tangent space, as is the case for the hyper surfaces obtained by fixing values of coordinate and momentum, i.e. the phase space coordinate system with an intersection at *z* =(*q*, *p*). We can think of alternative coordinate systems by considering another set of conjugate dynami‐

A point in this set will be denoted as an abstract bra ( *f* , *g* |, such that ( *f* , *g* |*u*) means the

We can also have marginal representations of functions in phase space by using the eigen‐

A point in the set Σ*<sup>F</sup>* ( *f* ) Σ*G*(*g*) will be denoted by the bra ( *f* | (*g* | and an object like

It is usual that the origin of one of the variables of a pair of conjugate variables is not well defined. This happens, for instance, with the pair of conjugate variables *q* and *p*. Even though the momentum can be well defined, the origin of the coordinate is arbitrary on the trajectory of a point particle, and it can be different for each trajectory. A coordinate system

*Q* points can be represented as the intersection of the eigensurfaces

*Q* | *G*(*z*)= *g*}.

*Q* | *F*(*z*)= *f* , *G*(*z*)= *g*}. (3)

*dt* =1 (2)

{*t*, *H* }=∑ *i* ( <sup>∂</sup> *<sup>t</sup>* ∂ *q <sup>i</sup>* ∂ *H* <sup>∂</sup> *pi* - <sup>∂</sup> *<sup>H</sup>* ∂ *q <sup>i</sup>* ∂ *t* ∂ *pi* ) =∑ *i* ( *dt d q <sup>i</sup> d q <sup>i</sup> dt* + *d pi dt dt d pi* ) <sup>=</sup> *dt*

cal variables, as is the case of energy and time.

<sup>Σ</sup>*FG*( *<sup>f</sup>* , *<sup>g</sup>*)={*<sup>z</sup>* <sup>∈</sup>*<sup>T</sup>* \*

*<sup>Q</sup>* <sup>|</sup> *<sup>F</sup>* (*z*)= *<sup>f</sup>* } , and <sup>Σ</sup>*G*(*g*)={*<sup>z</sup>* <sup>∈</sup>*<sup>T</sup>* \*

( *f* | *u*) (*g* |*u*) will mean the *f g* dependent function *u*( *f* ) *u*(*g*) .

fixes the origin of coordinates for all of the momentum eigensurfaces.

of the pair of conjugate variables *F* and *G*,

surfaces of only one of the functions,

**1.2. Conjugate coordinate systems**

Thus, in general, the *T \**

4 Advances in Quantum Mechanics

function *u*( *f* , *g*).

<sup>Σ</sup>*<sup>F</sup>* ( *<sup>f</sup>* )={*<sup>z</sup>* <sup>∈</sup>*<sup>T</sup>* \*

Some relationships between a pair of conjugate variables are derived in this section. We will deal with general *F* (*z*) and *G*(*z*) conjugate variables, but the results can be applied to coordi‐ nate and momentum or energy and time or to any other conjugate pair.

The magnitude of the vector field | *XG* | is the change of length along the *f* direction

$$\mathbb{I}\mid \mathbf{X}\_{\leq \mathbf{f}}\mid = \sqrt{\frac{d\,q^i}{df}\,\frac{d\,q^i}{df} + \frac{d\,p\_i}{df}\frac{d\,p\_i}{df}} = \frac{d\,l\_F}{df}\,,\tag{5}$$

where *dlF* <sup>=</sup> (*dq <sup>i</sup>*)<sup>2</sup> <sup>+</sup> (*<sup>d</sup> <sup>p</sup> <sup>i</sup>*)<sup>2</sup> is the length element.

A unit density with the eigensurface Σ*G*(*g*) as support

$$(z \mid \mathbf{g}) = \delta(z \text{ - } v), \quad v \in \Sigma\_G(\mathbf{g}) \tag{6}$$

is the classical analogue of the corresponding quantum eigenstate in coordinate *q* | *g* and momentum *p* | *g* representations. When *G*(*z*) is evaluated at the points of the support of (*z* | *g*), we get the value *g*. We use a bra-ket like notation to emphasise the similarity with the quantum concepts.

The overlap between a probability density with an eigenfunction of *F* ^ or *G* ^ provides margin‐ al representations of a probability density,

$$\log(f) \coloneqq \{ f \mid \rho \} \coloneqq \mathfrak{f}(f \mid \mathbf{z}) \langle \mathbf{z} \mid \rho \rangle dz = \mathfrak{f} \delta(\mathbf{z} \cdot f) \rho(\mathbf{z}) dz \,, \quad f \in \sum\_{\mathbf{F}} \langle f \rangle. \tag{7}$$

$$\mathfrak{g}(\emptyset) \coloneqq \mathfrak{g} \mid \mathfrak{\rho} \rangle = \mathfrak{f}(\mathfrak{g} \mid \mathfrak{z})(\mathfrak{z} \mid \mathfrak{\rho})dz = \mathfrak{f}\mathfrak{f}(\mathfrak{z} \cdot \mathfrak{g})\mathfrak{\rho}(\mathfrak{z})dz \,, \quad \mathfrak{g} \in \sum\_{\mathbb{C}} \mathfrak{k}\mathfrak{g} \,. \tag{8}$$

Note that *F* is at the same time a parameter in terms of which the motion of points in phase

*dG* <sup>=</sup> - <sup>∂</sup> *<sup>F</sup>*

*dF* <sup>=</sup> *XG* , *XG* =( <sup>∂</sup> *<sup>G</sup>*

Classical and Quantum Conjugate Dynamics – The Interplay Between Conjugate Variables

Now, *G* is the shift parameter besides of being the conjugate variable to *F*. This also renders

*d q <sup>i</sup> dG* ∂ *G* <sup>∂</sup> *<sup>q</sup> <sup>i</sup>* <sup>=</sup> *dG*

Then, the motion along one of the *F* or *G* directions is determined by the corresponding

If the motion of phase space points is governed by the vector field (15), *F* remains constant

In contrast, when motion occurs in the *F* direction, by means of Eq. (16), it is the *G* variable

∂ *pi* ∂ *F* ∂ *q <sup>i</sup>* <sup>∂</sup> *<sup>F</sup>* +

Hence, motion originated by the conjugate variables *F* (*z*) and *G*(*z*) occurs on the shells of

<sup>∂</sup> *<sup>q</sup> <sup>i</sup>* =0 , <sup>∇</sup> <sup>∙</sup> *XG* <sup>=</sup> <sup>∂</sup>

∂ *q <sup>i</sup>* ∂ *F* ∂ *pi*

> ∂ *q <sup>i</sup>* ∂ *G* <sup>∂</sup> *pi* - <sup>∂</sup> ∂ *pi* ∂ *G*

<sup>∂</sup> *pi* . (13)

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7

*dG* =1 . (14)

, - <sup>∂</sup> *<sup>G</sup>*

<sup>∂</sup> *<sup>G</sup>* =0 . (16)

<sup>∂</sup> *<sup>F</sup>* =0 . (17)

<sup>∂</sup> *<sup>q</sup> <sup>i</sup>* =0 . (18)

<sup>∂</sup> *<sup>q</sup> <sup>i</sup>* ) (15)

∂ *pi*

<sup>∂</sup> *<sup>q</sup> <sup>i</sup>* , *<sup>d</sup> <sup>q</sup> <sup>i</sup>*

space is written, and also the conjugate variable to *G*.

*d pi dG* <sup>=</sup> <sup>∂</sup> *<sup>F</sup>*

{*<sup>F</sup>* , *<sup>G</sup>*}= *<sup>d</sup> pi dG* ∂ *G* <sup>∂</sup> *pi* +

> ∂ *F* ∂ *pi* , <sup>∂</sup> *<sup>F</sup>*

The dynamical systems and vector fields for the motions just defined are

<sup>∂</sup> *<sup>q</sup> <sup>i</sup>* ) , and *dz*

conjugate variable. These vector fields in general are not orthogonal, nor parallel.

We can also define other dynamical system as

the Poisson bracket to the identity

*dG* = *XF* , *XF* =(-

*dF dG* <sup>=</sup> <sup>∂</sup> *<sup>F</sup>* ∂ *q <sup>i</sup>* ∂ *q <sup>i</sup>* <sup>∂</sup> *<sup>G</sup>* + ∂ *F* ∂ *pi* ∂ *pi* <sup>∂</sup> *<sup>G</sup>* <sup>=</sup> <sup>∂</sup> *pi* ∂ *G* ∂ *q<sup>i</sup>* <sup>∂</sup> *<sup>G</sup>* - <sup>∂</sup> *<sup>q</sup> <sup>i</sup>* ∂ *G* ∂ *pi*

the one that remains constant because

*dG dF* <sup>=</sup> <sup>∂</sup> *<sup>G</sup>* ∂ *q <sup>i</sup>* ∂ *q <sup>i</sup>* <sup>∂</sup> *<sup>F</sup>* + ∂ *G* ∂ *pi* ∂ *pi* <sup>∂</sup> *<sup>F</sup>* = -

constant *F* (*z*) or of constant *G*(*z*), respectively.

The divergence of these vector fields is zero,

∂ *q <sup>i</sup>* ∂ *F* <sup>∂</sup> *pi* +

∂ ∂ *pi* ∂ *F*

<sup>∇</sup> <sup>∙</sup> *XF* <sup>=</sup> - <sup>∂</sup>

*dz*

because

But, a complete description of a function in *T \* Q* is obtained by using the two dimensions unit density (*z* | *f* , *g*)=*δ*(*z* - ( *f* , *g*)), the eigenfunction of a location in phase space,

$$\mathfrak{g}(f,\,\,\mathbf{g}) = \{f,\,\,\mathbf{g} \mid \rho\} = \mathfrak{f}(f,\,\,\mathbf{g} \mid \mathbf{z}) \langle \mathbf{z} \mid \rho \rangle dz = \mathfrak{f} \delta(z \,\,\mathbf{z} \,\, \mathbf{f} \,\, \mathbf{g}) \rho(\mathbf{z}) dz \,, \,, \quad \left(f,\,\,\mathbf{g}\right) \in \sum\_{\mathcal{C}} \langle f,\,\,\mathbf{g} \rangle. \tag{9}$$

In this way, we have the classical analogue of the quantum concepts of eigenfunctions of op‐ erators and the projection of vectors on them.

#### **1.4. Conjugate motions**

Two dynamical variables with a constant Poisson bracket between them induce two types of complementary motions in phase space. Let us consider two real functions *F*(*z*) and *G*(*z*) of points in cotangent space *z* ∈ *T\* Q* of a mechanical system, and a unit Poisson bracket be‐ tween them,

$$\{F, \,\,\,\mathbf{G}\} = \frac{\partial F}{\partial q^i}, \frac{\partial G}{\partial p\_i} - \frac{\partial G}{\partial q^i}, \frac{\partial F}{\partial p\_i} = \mathbf{1} \,\,\,\,\tag{10}$$

valid on some domain D=D( <sup>∂</sup> *<sup>F</sup>* <sup>∂</sup> *<sup>q</sup> <sup>i</sup>* ) <sup>∩</sup>D( <sup>∂</sup> *<sup>F</sup>* <sup>∂</sup> *<sup>q</sup> <sup>i</sup>* ) <sup>∩</sup>D( <sup>∂</sup> *<sup>F</sup>* <sup>∂</sup> *<sup>q</sup> <sup>i</sup>* ) <sup>∩</sup>D( <sup>∂</sup> *<sup>F</sup>* <sup>∂</sup> *<sup>q</sup> <sup>i</sup>* ), according to the considered func‐ tions *F* and *G*. The application of the chain rule to functions of *p* and *q*, and Eq. (10), sug‐ gests two ways of defining dynamical systems for functions *F* and *G* that comply with the unit Poisson bracket. One of these dynamical systems is

$$\frac{d^i p\_i^i}{dF^i} = 1 \cdot \frac{\partial \, G}{\partial \, q^i} \,, \qquad \frac{d \, q^i}{dF^i} = \frac{\partial \, G}{\partial \, p\_i^i} \,, \tag{11}$$

With these replacements, the Poisson bracket becomes the derivative of a function with re‐ spect to itself

$$\{F\_{\nu}, G\} = \frac{\partial \, F}{\partial q^{i}} \frac{\partial q^{i}}{\partial F} + \frac{\partial \, p\_{i}}{\partial F} \frac{\partial F}{\partial p\_{i}} = \frac{\mathrm{d}F}{\mathrm{d}F} = \mathbf{1} \,\tag{12}$$

Note that *F* is at the same time a parameter in terms of which the motion of points in phase space is written, and also the conjugate variable to *G*.

We can also define other dynamical system as

ρ( *f* )≔( *f* | *ρ*)≔*∫*( *f* | *z*)(*z* | *ρ*)*dz* =*∫δ*(*z* - *f* )*ρ*(*z*)*dz* , *f* ∈∑

ρ(*g*)≔(*g* | *ρ*)≔*∫*(*g* | *z*)(*z* | *ρ*)*dz* =*∫δ*(*z* - *g*)*ρ*(*z*)*dz* , *g* ∈∑

unit density (*z* | *f* , *g*)=*δ*(*z* - ( *f* , *g*)), the eigenfunction of a location in phase space,

In this way, we have the classical analogue of the quantum concepts of eigenfunctions of op‐

Two dynamical variables with a constant Poisson bracket between them induce two types of complementary motions in phase space. Let us consider two real functions *F*(*z*) and *G*(*z*) of

<sup>ρ</sup>( *<sup>f</sup>* , g)≔( *<sup>f</sup>* , *<sup>g</sup>* <sup>|</sup> *<sup>ρ</sup>*)≔*∫*( *<sup>f</sup>* , *<sup>g</sup>* <sup>|</sup> *<sup>z</sup>*)(*<sup>z</sup>* <sup>|</sup> *<sup>ρ</sup>*)*dz* <sup>=</sup>*∫δ*(*<sup>z</sup>* - ( *<sup>f</sup>* , *<sup>g</sup>*))*ρ*(*z*)*dz* , ( *<sup>f</sup>* , *<sup>g</sup>*)<sup>∈</sup> <sup>∑</sup>

{*<sup>F</sup>* , *<sup>G</sup>*}= <sup>∂</sup> *<sup>F</sup>* ∂ *q <sup>i</sup>* ∂ *G* <sup>∂</sup> *pi* - <sup>∂</sup> *<sup>G</sup>* ∂ *q <sup>i</sup>* ∂ *F*

<sup>∂</sup> *<sup>q</sup> <sup>i</sup>* ) <sup>∩</sup>D( <sup>∂</sup> *<sup>F</sup>*

unit Poisson bracket. One of these dynamical systems is

*d pi dF* <sup>=</sup> - <sup>∂</sup> *<sup>G</sup>*

{*<sup>F</sup>* , *<sup>G</sup>*}= <sup>∂</sup> *<sup>F</sup>* ∂ *q <sup>i</sup>* ∂ *q <sup>i</sup>* <sup>∂</sup> *<sup>F</sup>* + ∂ *pi* ∂ *F* ∂ *F* <sup>∂</sup> *pi* <sup>=</sup> <sup>d</sup>*<sup>F</sup>*

<sup>∂</sup> *<sup>q</sup> <sup>i</sup>* ) <sup>∩</sup>D( <sup>∂</sup> *<sup>F</sup>*

<sup>∂</sup> *<sup>q</sup> <sup>i</sup>* , *<sup>d</sup> <sup>q</sup> <sup>i</sup>*

With these replacements, the Poisson bracket becomes the derivative of a function with re‐

tions *F* and *G*. The application of the chain rule to functions of *p* and *q*, and Eq. (10), sug‐ gests two ways of defining dynamical systems for functions *F* and *G* that comply with the

<sup>∂</sup> *<sup>q</sup> <sup>i</sup>* ) <sup>∩</sup>D( <sup>∂</sup> *<sup>F</sup>*

*dF* <sup>=</sup> <sup>∂</sup> *<sup>G</sup>*

But, a complete description of a function in *T \**

erators and the projection of vectors on them.

**1.4. Conjugate motions**

6 Advances in Quantum Mechanics

tween them,

spect to itself

points in cotangent space *z* ∈ *T\**

valid on some domain D=D( <sup>∂</sup> *<sup>F</sup>*

*F*

*G*

*Q* is obtained by using the two dimensions

*FG*

<sup>∂</sup> *pi* =1 , (10)

<sup>∂</sup> *<sup>q</sup> <sup>i</sup>* ), according to the considered func‐

<sup>∂</sup> *pi* . (11)

<sup>d</sup>*<sup>F</sup>* =1 . (12)

*Q* of a mechanical system, and a unit Poisson bracket be‐

( *<sup>f</sup>* ). (7)

(*g*). (8)

( *<sup>f</sup>* , *<sup>g</sup>*). (9)

$$\frac{d^i p\_i^i}{d\overline{G}} = \frac{\partial \, F}{\partial q^i} \,, \qquad \frac{d^i q^i}{d\overline{G}} = \text{ - } \frac{\partial \, F}{\partial p\_i} \,. \tag{13}$$

Now, *G* is the shift parameter besides of being the conjugate variable to *F*. This also renders the Poisson bracket to the identity

$$\{F,\,\,\,G\} = \frac{d}{dG}\frac{p\_i}{\partial \,p\_i} + \frac{d\,q^i}{dG}\frac{\partial G}{\partial q^i} = \frac{dG}{dG} = \mathbf{1}\,\,. \tag{14}$$

The dynamical systems and vector fields for the motions just defined are

$$\frac{dz}{dG} = X\_F \, \, \, \, \quad \, X\_F = \begin{pmatrix} \frac{\partial F}{\partial p\_i \, \, \, \, \, \, \partial \, \, F} \end{pmatrix} \, \, \, \quad \text{and} \quad \, \frac{dz}{dF} = X\_G \, \, \, \, \, \, \, \, X\_G = \begin{pmatrix} \frac{\partial G}{\partial p\_i \, \, \, \, \, \, \, \, \, \, \partial G} \end{pmatrix} \, \, \, \, \tag{15}$$

Then, the motion along one of the *F* or *G* directions is determined by the corresponding conjugate variable. These vector fields in general are not orthogonal, nor parallel.

If the motion of phase space points is governed by the vector field (15), *F* remains constant because

$$\frac{dF}{d\mathbf{G}} = \frac{\partial \, F}{\partial q^i} \frac{\partial q^i}{\partial \, G} + \frac{\partial \, F}{\partial \, p\_i} \frac{\partial \, p\_i}{\partial \, G} = \frac{\partial \, p\_i}{\partial \, G} \frac{\partial q^i}{\partial \, G} - \frac{\partial q^i}{\partial \, G} \frac{\partial \, p\_i}{\partial \, G} = \mathbf{0} \, \, \tag{16}$$

In contrast, when motion occurs in the *F* direction, by means of Eq. (16), it is the *G* variable the one that remains constant because

$$\frac{dG}{dF} = \frac{\partial G}{\partial q^i}, \frac{\partial q^i}{\partial F} + \frac{\partial G}{\partial p\_i}\frac{\partial p\_i}{\partial F} = \dots \\ \frac{\partial p\_i}{\partial F}\frac{\partial q^i}{\partial F} + \frac{\partial q^i}{\partial F}\frac{\partial p\_i}{\partial F} = 0 \dots \tag{17}$$

Hence, motion originated by the conjugate variables *F* (*z*) and *G*(*z*) occurs on the shells of constant *F* (*z*) or of constant *G*(*z*), respectively.

The divergence of these vector fields is zero,

$$
\nabla \bullet X\_F = -\frac{\partial}{\partial q^i} \frac{\partial F}{\partial p\_i} + \frac{\partial}{\partial p\_i} \frac{\partial F}{\partial q^i} = 0 \ \ \ \ \ \ \nabla \bullet X\_G = \frac{\partial}{\partial q^i} \frac{\partial G}{\partial p\_i} - \frac{\partial}{\partial p\_i} \frac{\partial G}{\partial q^i} = \mathbf{0} \ \ \tag{18}
$$

Thus, the motions associated to each of these conjugate variables preserve the phase space area.

A constant Poisson bracket is related to the constancy of a cross product because

$$\mathbf{X}\_{G} \wedge \mathbf{X}\_{F} = \frac{dz}{dF} \wedge \frac{dz}{dG} = \begin{vmatrix} \stackrel{\scriptstyle \Omega}{q} & \stackrel{\scriptstyle \Omega}{p} & \stackrel{\scriptstyle \Omega}{n} \\ \frac{\partial G}{\partial p} & -\frac{\partial G}{\partial q} & \mathbf{0} \\ -\frac{\partial F}{\partial p} & \frac{\partial F}{\partial q} & \mathbf{0} \end{vmatrix} = \stackrel{\scriptstyle \Omega}{n} \begin{pmatrix} \frac{\partial G}{\partial p} \frac{\partial F}{\partial q} & -\frac{\partial G}{\partial q} \frac{\partial F}{\partial p} \end{pmatrix} = \stackrel{\scriptstyle \Omega}{n} \begin{pmatrix} F \ \ G \end{pmatrix} \tag{19}$$

and the other is

*d df* <sup>=</sup> *dq df* ∂ ∂ *q* + *dp df* ∂ ∂ *p* + ∂ <sup>∂</sup> *<sup>f</sup>* <sup>=</sup> *dz*

*d dg* <sup>=</sup> *dq dg* ∂ ∂ *q* + *dp dg* ∂ ∂ *p* + ∂ <sup>∂</sup> *<sup>g</sup>* <sup>=</sup> *dz*

and

and

<sup>L</sup>*<sup>F</sup>* <sup>=</sup> <sup>∂</sup> *<sup>F</sup>* ∂ *q* ∂ <sup>∂</sup> *<sup>p</sup>* - <sup>∂</sup> *<sup>F</sup>* ∂ *p* ∂

{*<sup>F</sup>* , *<sup>G</sup>*}=( <sup>∂</sup> *<sup>F</sup>*

With these, we have introduced the Liouville type operators

∂

ators and commutators as in Quantum Mechanics.

ieved with the above Liouvillian operators as

∂ *q* ∂ <sup>∂</sup> *<sup>p</sup>* - <sup>∂</sup> *<sup>F</sup>* ∂ *p* ∂

<sup>∂</sup> *<sup>q</sup>* <sup>=</sup> *XF* ∙∇, and <sup>L</sup>*<sup>G</sup>* <sup>=</sup> <sup>∂</sup> *<sup>G</sup>*

<sup>∂</sup> *<sup>f</sup>* <sup>=</sup> - <sup>L</sup>*G*, and <sup>∂</sup>

∂

∂

Also, note that for any function *u*(*z*) of a phase space point *z*, we have that

tive of functions vanishes, i.e. the total amount of a function is conserved,

*df* ∙∇ +

*dg* ∙∇ +

<sup>L</sup>*<sup>F</sup>* , *<sup>u</sup>*(*z*) <sup>=</sup>L*<sup>F</sup> <sup>u</sup>*(*z*)= *XF* ∙∇*u*(*z*)= dz

<sup>L</sup>*G*, *<sup>u</sup>*(*z*) <sup>=</sup>L*Gu*(*z*)= *XG* ∙∇*u*(*z*)= *dz*

dependent operators. The formal solutions to these equations are

These are the classical analogues of the quantum evolution equation *<sup>d</sup>*

∂ *p* ∂ <sup>∂</sup> *<sup>q</sup>* - <sup>∂</sup> *<sup>G</sup>* ∂ *q* ∂

Classical and Quantum Conjugate Dynamics – The Interplay Between Conjugate Variables

These are Lie derivatives in the directions of *XF* and *XG*, respectively. These operators gen‐ erate complementary motion of functions in phase space. Note that now, we also have oper‐

Conserved motion of phase space functions moving along the *f* or *g* directions can be ach‐

Indeed, with the help these definitions and of the chain rule, we have that the total deriva‐

∂ <sup>∂</sup> *<sup>f</sup>* =L*<sup>G</sup>* +

∂ <sup>∂</sup> *<sup>g</sup>* =L*<sup>F</sup>* +

dG ∙∇u(z)= - <sup>∂</sup>

*dF* ∙∇*u*(*z*)= - <sup>∂</sup>

<sup>∂</sup> *<sup>f</sup>* = *XG* ∙∇ +

<sup>∂</sup> *<sup>g</sup>* = *XF* ∙∇ +

which are the evolution equations for functions along the conjugate directions *f* and *g*.

<sup>∂</sup> *<sup>q</sup>* )*G* = L*<sup>F</sup>* , *G* =1 . (23)

<sup>∂</sup> *<sup>g</sup>* = - L*<sup>F</sup>* . (25)

<sup>∂</sup> *<sup>f</sup>* +

∂

<sup>∂</sup> *<sup>g</sup> u*(*z*), (28)

<sup>∂</sup> *<sup>f</sup> u*(*z*) , (29)

*<sup>i</sup>*<sup>ℏ</sup> , *H*

^ for time

*dt* <sup>=</sup> <sup>1</sup>

<sup>∂</sup> *<sup>f</sup>* =0 , (26)

<sup>∂</sup> *<sup>g</sup>* =0 . (27)

∂ <sup>∂</sup> *<sup>f</sup>* <sup>=</sup> - <sup>∂</sup>

∂ <sup>∂</sup> *<sup>g</sup>* <sup>=</sup> - <sup>∂</sup> ∂ *g* + ∂

<sup>∂</sup> *<sup>p</sup>* = *XG* ∙∇ . (24)

http://dx.doi.org/10.5772/53598

9

where *n* ^ is the unit vector normal to the phase space plane. Then, the magnitudes of the vec‐ tor fields and the angle between them changes in such a way that the cross product remains constant when the Poisson bracket is equal to one, i.e. the cross product between conjugate vector fields is a conserved quantity.

The Jacobian for transformations from phase space coordinates to ( *f* , *g*) variables is one for each type of motion:

$$f = \begin{vmatrix} \frac{\partial q}{\partial f} & \frac{\partial p}{\partial f} \\ \frac{\partial q}{\partial g} & \frac{\partial p}{\partial g} \end{vmatrix} = \begin{vmatrix} \frac{\partial G}{\partial p} & -\frac{\partial G}{\partial q} \\ \frac{\partial q}{\partial g} & \frac{\partial p}{\partial g} \end{vmatrix} = \frac{\partial G}{\partial p}\frac{\partial p}{\partial g} + \frac{\partial G}{\partial q}\frac{\partial q}{\partial g} = \frac{dG}{dg} = \mathbf{1} \tag{20}$$

and

$$J = \begin{vmatrix} \frac{\partial q}{\partial f} & \frac{\partial p}{\partial f} \\ \frac{\partial q}{\partial g} & \frac{\partial p}{\partial g} \end{vmatrix} = \begin{vmatrix} \frac{\partial q}{\partial f} & \frac{\partial p}{\partial f} \\ -\frac{\partial F}{\partial p} & \frac{\partial F}{\partial g} \end{vmatrix} = \frac{\partial F}{\partial q} \frac{\partial q}{\partial f} + \frac{\partial F}{\partial p} \frac{\partial p}{\partial f} = \frac{dF}{df} = 1 \,\tag{21}$$

We have seen some properties related to the motion of phase space points caused by conju‐ gate variables.

#### **1.5. Poisson brackets and commutators**

We now consider the use of commutators in the classical realm.

The Poisson bracket can also be written in two ways involving a *commutator.* One form is

$$\{F, G\} = \left(\frac{\partial \, G}{\partial \, p} \frac{\partial}{\partial q} - \frac{\partial \, G}{\partial q} \frac{\partial}{\partial \, p}\right) F = \left[\mathbb{L}\_{G\prime}, F\right] = \mathbf{1}\_{\prime} \tag{22}$$

Classical and Quantum Conjugate Dynamics – The Interplay Between Conjugate Variables http://dx.doi.org/10.5772/53598 9

and the other is

Thus, the motions associated to each of these conjugate variables preserve the phase space

^ is the unit vector normal to the phase space plane. Then, the magnitudes of the vec‐

tor fields and the angle between them changes in such a way that the cross product remains constant when the Poisson bracket is equal to one, i.e. the cross product between conjugate

The Jacobian for transformations from phase space coordinates to ( *f* , *g*) variables is one for


*dg* =1 , (20)

*df* =1 . (21)

<sup>∂</sup> *<sup>p</sup>* )*<sup>F</sup>* <sup>=</sup> <sup>L</sup>*G*, *<sup>F</sup>* =1 , (22)

A constant Poisson bracket is related to the constancy of a cross product because

^ *<sup>n</sup>* ^

∂ *G* ∂ *q* 0

∂ *F* ∂ *q* 0 |=*n* ^( <sup>∂</sup> *<sup>G</sup>* ∂ *p* ∂ *F* <sup>∂</sup> *<sup>q</sup>* - <sup>∂</sup> *<sup>G</sup>* ∂ *q* ∂ *F* <sup>∂</sup> *<sup>p</sup>* ) =*n*

^ *<sup>p</sup>*

∂ *G* <sup>∂</sup> *<sup>p</sup>* -


area.

where *n*

and

gate variables.

each type of motion:

*XG* <sup>∧</sup> *XF* <sup>=</sup> *dz*

8 Advances in Quantum Mechanics

vector fields is a conserved quantity.

*<sup>J</sup>* =| <sup>∂</sup> *<sup>q</sup>* ∂ *f*

*<sup>J</sup>* =| <sup>∂</sup> *<sup>q</sup>* ∂ *f*

**1.5. Poisson brackets and commutators**

∂ *q* ∂ *g*

∂ *q* ∂ *g*

∂ *p* ∂ *f*



We now consider the use of commutators in the classical realm.

{*<sup>F</sup>* , *<sup>G</sup>*}=( <sup>∂</sup> *<sup>G</sup>*

∂ *p* ∂ <sup>∂</sup> *<sup>q</sup>* - <sup>∂</sup> *<sup>G</sup>* ∂ *q* ∂


∂ *q* ∂ *g* ∂ *p* ∂ *g*

> ∂ *p* ∂ *f*

∂ *F* ∂ *q* |= <sup>∂</sup> *<sup>F</sup>* ∂ *q* ∂ *q* <sup>∂</sup> *<sup>f</sup>* + ∂ *F* ∂ *p* ∂ *p* <sup>∂</sup> *<sup>f</sup>* <sup>=</sup> *dF*

We have seen some properties related to the motion of phase space points caused by conju‐

The Poisson bracket can also be written in two ways involving a *commutator.* One form is

∂ *p* ∂ *g*

∂ *p* ∂ *f*

∂ *p* ∂ *g*

*dF* ∧ *dz dG* =| *q*

$$\{F\_{\nu}, G\} = \begin{pmatrix} \frac{\partial \cdot F}{\partial q} \frac{\partial}{\partial p} - \frac{\partial \cdot F}{\partial p} \frac{\partial}{\partial q} \end{pmatrix} G = \begin{bmatrix} \mathbb{L}\_{F, \nu} \ G \end{bmatrix} = \mathbf{1} \tag{23}$$

With these, we have introduced the Liouville type operators

$$\mathsf{L}\_{F} = \frac{\partial F}{\partial q} \frac{\partial}{\partial p} \text{ } \frac{\partial}{\partial p} \text{ } \frac{\partial}{\partial q} \text{ } = \mathsf{X}\_{F} \bullet \ \nabla \mathsf{L} \quad \text{ and } \mathsf{L}\_{G} = \frac{\partial G}{\partial p} \frac{\partial}{\partial q} \text{ } \frac{\partial G}{\partial q} \frac{\partial}{\partial p} \text{ } = \mathsf{X}\_{G} \bullet \ \nabla \mathsf{L} \quad \text{and } \nabla \cdot \mathsf{L} = \mathsf{L}\_{F} \bullet \ \nabla \mathsf{L} \text{ } \frac{\partial}{\partial q} \text{ } \frac{\partial}{\partial p} \text{ } \frac{\partial}{\partial q} \text{ } \frac{\partial}{\partial q} \text{ } \frac{\partial}{\partial p} \text{ } \frac{\partial}{\partial q} \text{ } \frac{\partial}{\partial q} \text{ } \frac{\partial}{\partial p} \text{ } \frac{\partial}{\partial q} \text{ } \frac{\partial}{\partial q} \text{ } \frac{\partial}{\partial p} \text{ } \frac{\partial}{\partial q} \text{ } \frac{\partial}{\partial q} \text{ } \frac{\partial}{\partial p} \text{ } \frac{\partial}{\partial q} \text{ } \frac{\partial}{\partial p} \text{ } \frac{\partial}{\partial q} \text{ } \frac{\partial}{\partial q} \text{ } \frac{\partial}{\partial p} \text{ } \frac{\partial}{\partial q} \text{ } \frac{\partial}{\partial q} \text{ } \frac{\partial}{\partial p} \text{ } \frac{\partial}{\partial q} \text{ } \frac{\partial}{\partial q} \text{ } \frac{\partial}{\partial p} \text{ } \frac{\partial}{\partial q} \text{ } \frac{\partial}{\partial p} \text{ } \frac{\partial$$

These are Lie derivatives in the directions of *XF* and *XG*, respectively. These operators gen‐ erate complementary motion of functions in phase space. Note that now, we also have oper‐ ators and commutators as in Quantum Mechanics.

Conserved motion of phase space functions moving along the *f* or *g* directions can be ach‐ ieved with the above Liouvillian operators as

$$\frac{\partial}{\partial f} = -\mathsf{L}\_{G\prime} \quad \text{and} \qquad \frac{\partial}{\partial g} = -\mathsf{L}\_{F} \quad . \tag{25}$$

Indeed, with the help these definitions and of the chain rule, we have that the total deriva‐ tive of functions vanishes, i.e. the total amount of a function is conserved,

$$\frac{d}{d\boldsymbol{f}} = \frac{dq}{d\boldsymbol{f}} \frac{\partial}{\partial q} + \frac{dp}{d\boldsymbol{f}} \frac{\partial}{\partial p} + \frac{\partial}{\partial f} = \frac{dz}{d\boldsymbol{f}} \bullet \nabla + \frac{\partial}{\partial f} = \boldsymbol{X}\_G \bullet \nabla + \frac{\partial}{\partial f} = \boldsymbol{\L}\_G + \frac{\partial}{\partial f} = \boldsymbol{-\frac{\partial}{\partial f}} + \frac{\partial}{\partial f} = \boldsymbol{0} \,,\tag{26}$$

and

$$\frac{d}{d\mathbf{g}} = \frac{d\mathbf{q}}{d\mathbf{g}} \frac{\partial}{\partial q} + \frac{d\mathbf{p}}{d\mathbf{g}} \frac{\partial}{\partial p} + \frac{\partial}{\partial g} = \frac{dz}{d\mathbf{g}} \bullet \nabla + \frac{\partial}{\partial g} = X\_F \bullet \nabla + \frac{\partial}{\partial g} = \mathsf{L}\_F + \frac{\partial}{\partial g} = \mathsf{L}\_{\overline{\partial g}} + \frac{\partial}{\partial g} = \mathsf{0} \ . \tag{27}$$

Also, note that for any function *u*(*z*) of a phase space point *z*, we have that

$$\mathbb{L}\_{F}\left[\mathbb{L}\_{F}\left.\mu(z)\right]\right] = \mathbb{L}\_{F}\mu(z) = X\_{F}\bullet\nabla\,\mu(z) = \frac{dx}{dG}\bullet\nabla\,\mathbf{u}(z) = -\frac{\partial}{\partial g}\mu(z),\tag{28}$$

and

$$\mathbb{L}\_{G\prime}\ u(z)\mathbb{I} = \mathbb{L}\_{G}u(z) = X\_{G} \bullet \nabla\ u(z) = \frac{dz}{d\mathbb{F}} \bullet \nabla\ u(z) = -\frac{\partial}{\partial f}u(z) \,,\tag{29}$$

which are the evolution equations for functions along the conjugate directions *f* and *g*. These are the classical analogues of the quantum evolution equation *<sup>d</sup> dt* <sup>=</sup> <sup>1</sup> *<sup>i</sup>*<sup>ℏ</sup> , *H* ^ for time dependent operators. The formal solutions to these equations are

$$u(z; \mathcal{g}) = e^{\cdot \mathcal{g} \mathcal{L}\_F} u(z) \quad \quad \text{and} \quad u(z; f) = e^{-f^{\perp\_{\mathcal{L}\_G}}} u(z) \,. \tag{30}$$

This is a relationship that indicates how to translate the function *F* (*z*) as an operator. When this equality is acting on the number one, we arrive at the translation property for *F* as a function

i.e., up to an additive constant, *f* is the value of *F* (*z*) itself, one can be replaced by the other and actually they are the same object, with *f* the classical analogue of the spectrum of a

Continuing in a similar way, we can obtain the relationships shown in the following diagram

*d*

This implies that

quantum operator.

**Diagram 1.**

*F* (*z*; *f* )=(*e <sup>f</sup>* <sup>L</sup>*<sup>G</sup> F* (*z*)) = *F* (*z*) *e <sup>f</sup>* <sup>L</sup>*G* 1+ *f e <sup>f</sup>* <sup>L</sup>*G*1= *F*(*z*) + *f* . (37)

Classical and Quantum Conjugate Dynamics – The Interplay Between Conjugate Variables

*df F*(*z*; *f* )=1 , (38)

http://dx.doi.org/10.5772/53598

11

With these equations, we can now move a function *u*(*z*) on *T \* Q* in such a way that the points of their support move according to the dynamical systems Eqs. (15) and the total amount of *u* is conserved.

#### **1.6. The commutator as a derivation and its consequences**

As in quantum theory, we have found commutators and there are many properties based on them, taking advantage of the fact that a commutator is a derivation.

Since the commutator is a derivation, for conjugate variables *F* (*z*) and *G*(*z*) we have that, for integer *n*,

$$\left[\mathsf{L}\_{G,}^{\tiny\tiny\raisebox{0.75pt}{ $\mathsf{L}\_{G}$ }}\,\middle|\,\mathrm{F}\right] \equiv \mathsf{n}\,\mathsf{L}\_{G}^{\tiny\raisebox{0.75pt}{ $\mathsf{L}\_{G}$ }}\,\mathsf{A}\qquad\left[\mathsf{L}\_{G,}\,\mathsf{F}\,\middle|\,\mathrm{F}\right] \equiv \mathsf{n}\,\mathsf{F}^{\mathop\rule{0.75pt}{ $\mathsf{L}\_{F}$ }}\,\mathsf{G}\,\middle|\,\mathrm{G}\right] \equiv \mathsf{n}\,\mathsf{L}\_{F}^{\mathop\rule{0.75pt}{ $\mathsf{L}\_{F}$ }}\,\mathsf{A}\qquad\left[\mathsf{L}\_{F},\,\mathsf{G}\,\middle|\,\mathrm{F}\right] \equiv \mathsf{n}\,\mathsf{G}^{\mathop\rule{0.75pt}{ $\mathsf{L}\_{F}$ }}\,\mathsf{A}\tag{31}$$

Based on the above equalities, we can get translation relationships for functions on *T \* Q*. We first note that, for a holomorphic function *u*(*x*)= ∑ *n*=0 *∞ unx <sup>n</sup>*,

$$\left[\mu\left(\mathsf{L}\_{\mathsf{G}}\right),\,\,F\right] = \left[\sum\_{n=0}^{\mathsf{m}}\mu\_{n}\mathsf{L}\_{\mathsf{G},\prime}^{\mathsf{m}}\,\,\_{\mathsf{F}}\right] = \sum\_{n=0}^{\mathsf{m}}\mu\mu\_{n}\mathsf{L}\_{\mathsf{G}}^{\mathsf{m}-1} = \mu^{\prime}\left(\mathsf{L}\_{\mathsf{G}}\right)\,\,. \tag{32}$$

In particular, we have that

$$\begin{bmatrix} \int e^{f\,\mathsf{L}\_{G}} \, . \, F \end{bmatrix} = f \, e^{f\,\mathsf{L}\_{G}} \, . \tag{33}$$

Then, *e <sup>f</sup>* <sup>L</sup>*G* is the eigenfunction of the commutator ∙, *F* with eigenvalue *f* .

From Eq. (32), we find that

$$
u(\mathsf{L}\_G) \mathcal{F} \text{ -- } \mathsf{F} \mathcal{u}(\mathsf{L}\_G) \coloneqq \mathsf{u}^\top(\mathsf{L}\_G).\tag{34}$$

But, if we multiply by *u* -1(L*G*) from the right, we arrive to

$$
\mu\left(\mathsf{L}\_{G}\right)\mathsf{F}\,\mu^{-1}\left(\mathsf{L}\_{G}\right) = \mathsf{F} + \mu^{\top}\left(\mathsf{L}\_{G}\right)\mu^{-1}\left(\mathsf{L}\_{G}\right).\tag{35}
$$

This is a generalized version of a shift of *F* , and the classical analogue of a generalization of the quantum Weyl relationship. A simple form of the above equality, a familiar form, is ob‐ tained with the exponential function, i.e.

$$\text{If } e^{f\perp\_{\mathcal{C}}} \mathcal{F} e^{-f\perp\_{\mathcal{C}}} = \mathcal{F} + f \text{ .} \tag{36}$$

This is a relationship that indicates how to translate the function *F* (*z*) as an operator. When this equality is acting on the number one, we arrive at the translation property for *F* as a function

$$F\left(z;f\right) = \left(e^{f\left\lfloor\operatorname{L}\_{\mathbb{C}}\right\rfloor}F\left(z\right)\right) = F\left(z\right)\left(e^{f\left\lfloor\operatorname{L}\_{\mathbb{C}}\right\rfloor}1 + f\right)e^{f\left\lfloor\operatorname{L}\_{\mathbb{C}}\right\rfloor}1 = F\left(z\right) + f\left.\right.\tag{37}$$

This implies that

*u*(*z*; *g*)=*e*-*g*L*<sup>F</sup> u*(*z*) , and *u*(*z*; *f* )=*e*- *<sup>f</sup>* <sup>L</sup>*Gu*(*z*). (30)

points of their support move according to the dynamical systems Eqs. (15) and the total

As in quantum theory, we have found commutators and there are many properties based on

Since the commutator is a derivation, for conjugate variables *F* (*z*) and *G*(*z*) we have that, for

*<sup>n</sup>* , *<sup>G</sup>* <sup>=</sup>*<sup>n</sup>* <sup>L</sup>*<sup>F</sup>*

*n*=0 *∞ unx <sup>n</sup>*,

, L*<sup>F</sup>*

Based on the above equalities, we can get translation relationships for functions on *T \**

*<sup>n</sup>* , *F* = ∑ *n*=0 ∞ *nun*L*<sup>G</sup> n*-1 =*u* ' *Q* in such a way that the

*<sup>n</sup>*-1 , <sup>L</sup>*<sup>F</sup>* , *<sup>G</sup> <sup>n</sup>* <sup>=</sup>*<sup>n</sup> <sup>G</sup> <sup>n</sup>*-1 . (31)

(L*G*) . (32)

(L*G*). (34)

(L*G*)*u* -1(L*G*). (35)

*e <sup>f</sup>* <sup>L</sup>*<sup>G</sup> F e*- *<sup>f</sup>* <sup>L</sup>*<sup>G</sup>* = *F* + *f* . (36)

*e <sup>f</sup>* <sup>L</sup>*G*, *F* = *f e <sup>f</sup>* <sup>L</sup>*<sup>G</sup>* . (33)

*Q*. We

With these equations, we can now move a function *u*(*z*) on *T \**

them, taking advantage of the fact that a commutator is a derivation.

**1.6. The commutator as a derivation and its consequences**

*<sup>n</sup>*-1 , <sup>L</sup>*G*, *<sup>F</sup> <sup>n</sup>* <sup>=</sup>*<sup>n</sup> <sup>F</sup> <sup>n</sup>*-1

*u*(L*G*), *F* = ∑

But, if we multiply by *u* -1(L*G*) from the right, we arrive to

tained with the exponential function, i.e.

*n*=0 ∞ *un*L*<sup>G</sup>*

Then, *e <sup>f</sup>* <sup>L</sup>*G* is the eigenfunction of the commutator ∙, *F* with eigenvalue *f* .

*u*(L*G*)*F* - *Fu*(L*G*) =*u* '

This is a generalized version of a shift of *F* , and the classical analogue of a generalization of the quantum Weyl relationship. A simple form of the above equality, a familiar form, is ob‐

*u*(L*G*)*F u* -1(L*G*) = *F* + *u* '

first note that, for a holomorphic function *u*(*x*)= ∑

amount of *u* is conserved.

10 Advances in Quantum Mechanics

integer *n*,

L*<sup>G</sup>*

*<sup>n</sup>* , *<sup>F</sup>* <sup>=</sup>*<sup>n</sup>* <sup>L</sup>*<sup>G</sup>*

In particular, we have that

From Eq. (32), we find that

$$\mathbf{1} \frac{d}{d\mathbf{f}} \mathbf{F}\left(z; f\right) = \mathbf{1} \,, \tag{38}$$

i.e., up to an additive constant, *f* is the value of *F* (*z*) itself, one can be replaced by the other and actually they are the same object, with *f* the classical analogue of the spectrum of a quantum operator.

Continuing in a similar way, we can obtain the relationships shown in the following diagram

**Diagram 1.**

where the constant *s* has units of action, length times momentum, the same units as the quantum constant ℏ.

The eigenvectors of the position, momentum and energy operators have been used to pro‐ vide a representation of wave functions and of operators. So, in general, the eigenvectors

^ provide with a set of vectors for a represen‐

. (40)

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13

^ ) , (41)

^ . (42)

^ ). (43)

^ ). (44)

. The usual shift

^ <sup>+</sup> *<sup>f</sup>* . (45)

^ *<sup>n</sup>*-1

Classical and Quantum Conjugate Dynamics – The Interplay Between Conjugate Variables

*unz <sup>n</sup>* we have that

^ and *G*

With the help of the properties of commutators between operators, we can see that

, *F* ^ , *G* ^ *<sup>n</sup>* <sup>=</sup>*i*ℏ*<sup>G</sup>*

*n*=0 *∞*

^) , *<sup>F</sup>*

^ , <sup>1</sup>

A set of equalities is obtained from Eq. (43) by first writing them in expanded form as

^), and *<sup>F</sup>*

^), and *<sup>u</sup>*

^ <sup>+</sup> *<sup>g</sup>*, and *<sup>F</sup>*

These are a set of generalized shift relationships for the operators *G*

relationships are obtained when *u*(*x*) is the exponential function, i.e.

^ , *u* ^(*<sup>G</sup>* ^ ) <sup>=</sup>*i*ℏ*<sup>u</sup>* ^' (*G*

i.e., the commutators behave as derivations with respect to operators. In an abuse of nota‐

*<sup>i</sup>*<sup>ℏ</sup> *F* ^

> ^ *u* ^(*<sup>G</sup>* ^ ) - *<sup>u</sup>* ^(*<sup>G</sup>* ^ )*F* ^ <sup>=</sup>*i*ℏ*<sup>u</sup>* ^' (*G*

^-1(*<sup>G</sup>* ^ )*F* ^ *u* ^(*<sup>G</sup>* ^ )= *<sup>F</sup>*

^( *<sup>f</sup>* ): <sup>=</sup>*eif <sup>G</sup>*

^ /ℏ*F* ^ *e*-*if <sup>G</sup>* ^ /<sup>ℏ</sup> = *F*

^ <sup>+</sup> *<sup>i</sup>*ℏ*<sup>u</sup>*

^-1(*<sup>G</sup>* ^ )*u* ^' (*G*

^ and *<sup>F</sup>* ^

Next, we multiply these equalities by the inverse operator to the right or to the left in order

We can take advantage of this fact and derive the quantum versions of the equalities found

, <sup>∙</sup> <sup>=</sup> *<sup>d</sup>* <sup>∙</sup> *dG*

tation of dynamical quantities like the wave functions *f* | *ψ* and *g* | *ψ* .

^ <sup>=</sup>*i*ℏ*<sup>F</sup>* ^ *<sup>n</sup>*-1

^ <sup>=</sup>*i*ℏ*<sup>u</sup>* ^' (*F*

> ^ <sup>=</sup> *<sup>d</sup>* <sup>∙</sup> *d F*


Hence, for a holomorphic function *u*(*z*)= ∑

*u* ^(*<sup>F</sup>* ^), *<sup>G</sup>*

tion, we have that

in the classical realm.

to obtain

*u* ^(*<sup>F</sup>* ^)*<sup>G</sup>* ^ *u* ^-1(*<sup>F</sup>* ^)=*<sup>G</sup>*

*G*

^ (*g*): <sup>=</sup>*e*-*ig <sup>F</sup>*

^ /ℏ*G* ^ *<sup>e</sup>ig <sup>F</sup>* ^ /<sup>ℏ</sup> =*G*

*u* ^(*<sup>F</sup>* ^)*<sup>G</sup>* ^ - *<sup>G</sup>* ^ *u* ^(*<sup>F</sup>* ^)=*<sup>i</sup>*ℏ*<sup>u</sup>* ^' (*F*

*F* ^ *<sup>n</sup>*, *<sup>G</sup>*

1 *<sup>i</sup>*<sup>ℏ</sup> ∙, *G*

^ <sup>+</sup> *<sup>i</sup>*ℏ*<sup>u</sup>* ^' (*F* ^)*u* ^-1(*<sup>F</sup>*

Some of the things to note are:

The operator *e <sup>g</sup>*L*F* is the eigenoperator of the commutator ∙, *G* and can be used to gener‐ ate translations of *G*(*z*) as an operator or as a function. This operator is also the propagator for the evolution of functions along the *g* direction. The variable *g* is more than just a shift parameter; it actually labels the values that *G*(*z*) takes, the classical analogue of the spec‐ trum of a quantum operator.

The operators L*F* and *G*(*z*) are also a pair of conjugate operators, as well as the pair L*<sup>G</sup>* and *F* (*z*).

But L*<sup>F</sup>* commutes with *F* (*z*) and then it cannot be used to translate functions of *F* (*z*), *F* (*z*) is a conserved quantity when motion occurs along the *G*(*z*) direction.

The eigenfunction of L*<sup>F</sup>* , <sup>∙</sup> and of *s*L*F* is *<sup>e</sup> fG*(*z*)/*<sup>s</sup>* and this function can be used to shift L*F* as an operator or as a function.

The variable *f* is more than just a parameter in the shift of *s*L*<sup>F</sup>* , it actually is the value that *s*L*F* can take, the classical analogue of the spectra of a quantum operator.

The steady state of L*F* is a function of *F* (*z*), but *e gF* (*z*)/*<sup>s</sup>* is an eigenfunction of L*G* and of L*G*, ∙ and it can be used to translate L*G*.

These comments involve the left hand side of the above diagram. There are similar conclu‐ sions that can be drawn by considering the right hand side of the diagram.

Remember that the above are results valid for classical systems. Below we derive the corre‐ sponding results for quantum systems.

### **2. Quantum systems**

We now derive the quantum analogues of the relationships found in previous section. We start with a Hilbert space H of wave functions and two conjugate operators *F* ^ and *G* ^ acting on vectors in H, and with a constant commutator between them

$$\begin{aligned} \text{[}\stackrel{\wedge}{F}, \stackrel{\wedge}{G}] &= i\hbar \, , \end{aligned} \tag{39}$$

together with the domain D=D(*F* ^ *G* ^ )∩D(*<sup>G</sup>* ^ *F* ^) in which the commutator holds. Examples of these operators are coordinate *Q* ^ and momentum *<sup>P</sup>* ^ operators, energy *H* ^ and time *<sup>T</sup>* ^ opera‐ tors, creation *a* ^† and annihilation *a* ^ operators.

The eigenvectors of the position, momentum and energy operators have been used to pro‐ vide a representation of wave functions and of operators. So, in general, the eigenvectors | *f* and | *g* of the conjugate operators *F* ^ and *G* ^ provide with a set of vectors for a represen‐ tation of dynamical quantities like the wave functions *f* | *ψ* and *g* | *ψ* .

With the help of the properties of commutators between operators, we can see that

$$[\stackrel{\frown}{F}^n, \stackrel{\frown}{G}] = i\hbar \hat{F}^{n-1}, \qquad [\stackrel{\frown}{F}\_{\prime}, \stackrel{\frown}{G}^n] = i\hbar \hat{G}^{n-1}.\tag{40}$$

Hence, for a holomorphic function *u*(*z*)= ∑ *n*=0 *∞ unz <sup>n</sup>* we have that

where the constant *s* has units of action, length times momentum, the same units as the

The operator *e <sup>g</sup>*L*F* is the eigenoperator of the commutator ∙, *G* and can be used to gener‐ ate translations of *G*(*z*) as an operator or as a function. This operator is also the propagator for the evolution of functions along the *g* direction. The variable *g* is more than just a shift parameter; it actually labels the values that *G*(*z*) takes, the classical analogue of the spec‐

The operators L*F* and *G*(*z*) are also a pair of conjugate operators, as well as the pair L*<sup>G</sup>*

But L*<sup>F</sup>* commutes with *F* (*z*) and then it cannot be used to translate functions of *F* (*z*), *F* (*z*) is

The eigenfunction of L*<sup>F</sup>* , <sup>∙</sup> and of *s*L*F* is *<sup>e</sup> fG*(*z*)/*<sup>s</sup>* and this function can be used to shift L*F* as

The variable *f* is more than just a parameter in the shift of *s*L*<sup>F</sup>* , it actually is the value that

These comments involve the left hand side of the above diagram. There are similar conclu‐

Remember that the above are results valid for classical systems. Below we derive the corre‐

We now derive the quantum analogues of the relationships found in previous section. We

^

start with a Hilbert space H of wave functions and two conjugate operators *F*

*F* ^ , *G*

^ operators.

^ and momentum *<sup>P</sup>*

on vectors in H, and with a constant commutator between them

^ *G* ^ )∩D(*<sup>G</sup>* ^ *F* is an eigenfunction of L*G* and of

^ and *G*

^ and time *<sup>T</sup>*

^ <sup>=</sup>*i*<sup>ℏ</sup> , (39)

^) in which the commutator holds. Examples of

operators, energy *H*

^ acting

^ opera‐

a conserved quantity when motion occurs along the *G*(*z*) direction.

*s*L*F* can take, the classical analogue of the spectra of a quantum operator.

sions that can be drawn by considering the right hand side of the diagram.

The steady state of L*F* is a function of *F* (*z*), but *e gF* (*z*)/*<sup>s</sup>*

L*G*, ∙ and it can be used to translate L*G*.

sponding results for quantum systems.

**2. Quantum systems**

together with the domain D=D(*F*

and annihilation *a*

these operators are coordinate *Q*

^†

tors, creation *a*

quantum constant ℏ.

12 Advances in Quantum Mechanics

Some of the things to note are:

trum of a quantum operator.

an operator or as a function.

and *F* (*z*).

$$\begin{array}{ll} \tiny \begin{array}{l} \widehat{\ulcorner} \widehat{\urcorner} \end{array} \widehat{\ulcorner} \end{array} \begin{array}{l} \widehat{\ulcorner} \end{array} \begin{array}{l} \widehat{\urcorner} \end{array} \begin{array}{l} \widehat{\urcorner} \end{array} \begin{array}{l} \widehat{\urcorner} \end{array} \begin{array}{l} \widehat{\urcorner} \end{array} \begin{array}{l} \widehat{\urcorner} \end{array} \begin{array}{l} \widehat{\urcorner} \end{array} \begin{array}{l} \widehat{\urcorner} \end{array} \begin{array}{l} \widehat{\urcorner} \end{array} \begin{array}{l} \widehat{\urcorner} \end{array} \begin{array}{l} \widehat{\urcorner} \end{array} \end{array} \tag{41}$$

i.e., the commutators behave as derivations with respect to operators. In an abuse of nota‐ tion, we have that

$$\begin{array}{ll}\frac{1}{\sqrt{\hbar}}\mathsf{f}\bullet\ \bigwedge^{\wedge}\mathsf{G}\begin{array}{c}\mathsf{I}\end{array}\mathsf{J}=\frac{d\,\mathsf{A}}{d\,\mathsf{F}}\end{array},\qquad\qquad\qquad\qquad\qquad\qquad\frac{1}{\sqrt{\hbar}}\mathsf{f}\begin{array}{c}\bigwedge^{\wedge}\mathsf{F}\ \rightarrow\mathsf{J}=\frac{d\,\mathsf{A}}{d\,\mathsf{G}}\end{array},\tag{42}$$

We can take advantage of this fact and derive the quantum versions of the equalities found in the classical realm.

A set of equalities is obtained from Eq. (43) by first writing them in expanded form as

$$
\hat{\mathfrak{u}}(\stackrel{\wedge}{F})\hat{\mathbf{G}} \cdot \stackrel{\wedge}{G}\hat{\mathfrak{u}}(\stackrel{\wedge}{F}) = i\hbar \hat{\mathfrak{u}}^{\wedge}(\stackrel{\wedge}{F}), \qquad \text{and} \qquad \hat{\mathcal{F}}\hat{\mathfrak{u}}(\stackrel{\wedge}{G}) \cdot \stackrel{\wedge}{\hat{\mathfrak{u}}}(\stackrel{\wedge}{G})\hat{\mathbf{F}} = i\hbar \hat{\mathfrak{u}}^{\wedge}(\stackrel{\wedge}{G}).\tag{43}
$$

Next, we multiply these equalities by the inverse operator to the right or to the left in order to obtain

$$
\stackrel{\scriptstyle\wedge}{u}(\stackrel{\scriptstyle\wedge}{F})\stackrel{\scriptstyle\wedge}{G}\stackrel{\scriptstyle\wedge}{u}^{-1}(\stackrel{\scriptstyle\wedge}{F}) = \stackrel{\scriptstyle\wedge}{G} + i\hbar\stackrel{\scriptstyle\wedge}{u}(\stackrel{\scriptstyle\wedge}{F})\stackrel{\scriptstyle\wedge}{u}^{-1}(\stackrel{\scriptstyle\wedge}{F}), \quad \text{and} \quad \stackrel{\scriptstyle\wedge}{u}^{-1}(\stackrel{\scriptstyle\wedge}{G})\stackrel{\scriptstyle\wedge}{F}\stackrel{\scriptstyle\wedge}{u}(\stackrel{\scriptstyle\wedge}{G}) = \stackrel{\scriptstyle\wedge}{F} + i\hbar\stackrel{\scriptstyle\wedge}{u}^{-1}(\stackrel{\scriptstyle\wedge}{G})\stackrel{\scriptstyle\wedge}{u}^{\wedge}(\stackrel{\scriptstyle\wedge}{G}).\tag{44}
$$

These are a set of generalized shift relationships for the operators *G* ^ and *<sup>F</sup>* ^ . The usual shift relationships are obtained when *u*(*x*) is the exponential function, i.e.

$$\stackrel{\wedge}{G}(\mathcal{G}) \colon= e^{-i\boldsymbol{\varrho}^{\hat{\sf F}}/\hbar} \stackrel{\wedge}{G} e^{i\boldsymbol{\varrho}^{\hat{\sf F}}/\hbar} = \stackrel{\wedge}{G} + \stackrel{\wedge}{g}, \qquad \text{and} \quad \stackrel{\wedge}{F}(f) \colon= e^{i\boldsymbol{f}^{\hat{\sf G}}/\hbar} \stackrel{\wedge}{F} e^{-i\boldsymbol{f}^{\hat{\sf G}}/\hbar} = \stackrel{\wedge}{F} + f \,. \tag{45}$$

Now, as in Classical Mechanics, the commutator between two operators can be seen as two different derivatives introducing quantum dynamical system as

$$\frac{\stackrel{\bigwedge}{d\mathbb{P}}(f)}{\stackrel{\bigwedge}{d\mathbb{P}}} = \text{ } \frac{\stackrel{\bigwedge}{\otimes}\stackrel{\bigwedge}{\mathbb{Q}}\stackrel{\bigwedge}{\mathbb{P}}}{\stackrel{\bigwedge}{\mathbb{Q}}\!\!} = \frac{1}{\stackrel{\bigwedge}{\mathbb{H}}} \stackrel{\bigwedge}{\mathbb{P}} \begin{pmatrix} f \\ f \end{pmatrix} \stackrel{\bigwedge}{\mathbb{G}} \begin{pmatrix} \stackrel{\bigwedge}{\mathbb{Q}}\stackrel{\bigwedge}{\mathbb{P}}\!\!}{\stackrel{\bigwedge}{\mathbb{Q}}\!\!} \end{pmatrix} \qquad \frac{\stackrel{\bigwedge}{d\mathbb{Q}}(f)}{\stackrel{\bigwedge}{\mathbb{P}}} = \frac{\stackrel{\bigwedge}{\mathbb{G}}\stackrel{\bigwedge}{\mathbb{Q}}\stackrel{\bigwedge}{\mathbb{P}}\stackrel{\bigwedge}{\mathbb{P}}}{\stackrel{\bigwedge}{\mathbb{P}}\mathbb{P}} = \frac{1}{\stackrel{\bigwedge}{\mathbb{R}}} \!\!} \stackrel{\bigwedge}{\mathbb{Q}}(f) \stackrel{\bigwedge}{\mathbb{Q}} \begin{pmatrix} \stackrel{\bigwedge}{\mathbb{Q}}\stackrel{\bigwedge}{\mathbb{P}}\stackrel{\bigwedge}{\mathbb{P}}\mathbb{P}}{\stackrel{\bigwedge}{\mathbb{Q}}\!} \end{pmatrix} \tag{46}$$

$$\frac{\stackrel{\frown}{AD}(\mathcal{G})}{\stackrel{\frown}{dg}} = \frac{\stackrel{\frown}{\partial}\stackrel{\frown}{F}(\stackrel{\frown}{Q},\stackrel{\frown}{P})}{\stackrel{\frown}{\partial}Q} = \frac{1}{\stackrel{\frown}{\hbar}\!\!\!} \stackrel{\frown}{F}(\stackrel{\frown}{Q},\stackrel{\frown}{P}),\ \stackrel{\frown}{P}(\mathcal{g})\mathbf{I},\qquad\text{and}\qquad\frac{\stackrel{\frown}{dQ}(\mathcal{g})}{\stackrel{\frown}{dg}} = -\frac{\stackrel{\frown}{\partial}\stackrel{\frown}{F}(\stackrel{\frown}{Q},\stackrel{\frown}{P})}{\stackrel{\frown}{\partial}P} = \frac{1}{\stackrel{\frown}{\hbar}\!\!} \!\!\!\!\!\!\!\!\!\!\!\!\!\begin{B}\stackrel{\frown}{Q}(\stackrel{\frown}{Q},\stackrel{\frown}{Q}),\ \stackrel{\frown}{Q}(\mathcal{g})\mathbf{I},\tag{47}$$

where

$$
\stackrel{\scriptstyle \hat{\Lambda}}{P}(f) = e^{\circ \stackrel{\scriptstyle \hat{G}}{} \stackrel{\scriptstyle \hat{G}}{/\hbar} \stackrel{\scriptstyle \hat{\Lambda}}{P} e^{\circ \stackrel{\scriptstyle \hat{G}}{} \stackrel{\scriptstyle \hat{G}}{/\hbar} \stackrel{\scriptstyle \hat{\Lambda}}{}{/\hbar}}, \qquad \stackrel{\scriptstyle \hat{\Lambda}}{Q}(f) = e^{\circ \stackrel{\scriptstyle \hat{G}}{} \stackrel{\scriptstyle \hat{G}}{} / \hbar} \stackrel{\scriptstyle \hat{\Lambda}}{Q} e^{\circ \stackrel{\scriptstyle \hat{G}}{} \stackrel{\scriptstyle \hat{G}}{} / \hbar}, \tag{48}
$$

There are many equalities that can be obtained as in the classical case. The following dia‐

Classical and Quantum Conjugate Dynamics – The Interplay Between Conjugate Variables

http://dx.doi.org/10.5772/53598

15

gram shows some of them:

**Diagram 2.**

$$
\stackrel{\frown}{P}(\mathbf{g}) = e^{-i\mathbf{g}\cdot\hat{\mathbf{F}}/\hbar} \stackrel{\frown}{P} e^{i\mathbf{f}\cdot\hat{\mathbf{F}}/\hbar}, \qquad \text{and} \qquad \stackrel{\frown}{Q}(\mathbf{g}) = e^{-i\mathbf{g}\cdot\hat{\mathbf{F}}/\hbar} \stackrel{\frown}{Q} e^{i\mathbf{g}\cdot\hat{\mathbf{F}}/\hbar}. \tag{49}
$$

These equations can be written in the form of a set of quantum dynamical systems

$$\stackrel{\hat{d}\cdot\hat{z}}{\frac{d\hat{\mathcal{L}}}{d\hat{f}}} = \stackrel{\bigwedge}{\mathcal{X}}\_{G\text{-}f}, \qquad \stackrel{\bigwedge}{\mathcal{X}}\_{G} = \begin{pmatrix} \stackrel{\wedge}{\partial\!\hat{Q}}\_{\mathcal{P}} & \stackrel{\wedge}{\partial\!\hat{Q}}\_{\mathcal{P}} \end{pmatrix}, \qquad \qquad \stackrel{\hat{d}\cdot\hat{z}}{\frac{d\hat{z}}{d\mathbf{g}}} = \stackrel{\wedge}{\hat{\mathcal{X}}}\_{F\text{-}f}, \qquad \qquad \stackrel{\bigwedge}{\hat{\mathcal{X}}}\_{F} = \begin{pmatrix} \stackrel{\wedge}{\partial\!\hat{F}}\_{\overline{\mathcal{Q}}} & \stackrel{\wedge}{\partial\!\hat{Q}}\_{\overline{\mathcal{Q}}} \end{pmatrix},\tag{50}$$

where *z* ^ =(*<sup>Q</sup>* ^ , *<sup>P</sup>* ^).

The inner product between the operator vector fields is

$$\stackrel{\frown}{X}\stackrel{+}{F}\_{F}\bullet\stackrel{\frown}{X}\_{F} = \begin{pmatrix} \stackrel{\frown}{\partial\,\tilde{\mathbb{E}}} & \stackrel{\frown}{\partial\,\tilde{\mathbb{E}}} \\ \stackrel{\frown}{\partial P}\prime & \stackrel{\frown}{\partial Q} \end{pmatrix} \stackrel{\sharp}{\bullet} \bullet \begin{pmatrix} \stackrel{\frown}{\partial\,\tilde{\mathbb{E}}} & \stackrel{\frown}{\partial\,\tilde{\mathbb{E}}} \\ \stackrel{\frown}{\partial P}\prime & \stackrel{\frown}{\partial Q} \end{pmatrix} \equiv \begin{pmatrix} \stackrel{\frown}{d\,\tilde{Q}} \\ \stackrel{\frown}{\partial\,\tilde{\mathbb{E}}} \end{pmatrix} + \begin{pmatrix} \stackrel{\frown}{d\,\tilde{P}} \\ \stackrel{\frown}{\partial\,\tilde{\mathbb{E}}} \end{pmatrix} \tag{51}$$

where (*d l* ^ *<sup>F</sup>* )2≔(*dQ* ^ )<sup>2</sup> <sup>+</sup> (*dP* ^)2 , evaluated along the *g* direction, is the quantum analogue of the square of the line element (*dlF* )<sup>2</sup> =(*dq*)<sup>2</sup> <sup>+</sup> (*dp*)<sup>2</sup> .

We can define many of the classical quantities but now in the quantum realm. Liouville type operators are

$$\stackrel{\scriptstyle \mathsf{A}}{\mathsf{L}}\_{F} \coloneqq \frac{1}{i\hbar} \stackrel{\scriptstyle \mathsf{f}}{\mathsf{f}}\_{F} \bullet \mathsf{J}\_{\prime} \qquad \text{and} \quad \stackrel{\scriptstyle \mathsf{A}}{\mathsf{L}}\_{G} \coloneqq \frac{1}{i\hbar} \mathsf{I} \bullet \stackrel{\scriptstyle \mathsf{A}}{\mathsf{G}} \mathsf{J} \,. \tag{52}$$

These operators will move functions of operators along the conjugate directions *G* ^ or *<sup>F</sup>* ^ , re‐ spectively. This is the case when *G* ^ is the Hamiltonian *<sup>H</sup>* ^ of a physical system, a case in which we get the usual time evolution of operator.

There are many equalities that can be obtained as in the classical case. The following dia‐ gram shows some of them:

**Diagram 2.**

Now, as in Classical Mechanics, the commutator between two operators can be seen as two

^ ( *<sup>f</sup>* ) *df* <sup>=</sup> <sup>∂</sup> *<sup>G</sup>*

^ (*g*) *dg* <sup>=</sup> - <sup>∂</sup> *<sup>F</sup>*

^ ( *<sup>f</sup>* )=*eif <sup>G</sup>*

^ (*g*)=*e*-*ig <sup>F</sup>*

^ /ℏ*Q* ^ *<sup>e</sup>ig <sup>F</sup>* ^

*<sup>F</sup>* , *X*

<sup>+</sup> ( *<sup>d</sup> <sup>P</sup>* ^ *dg* )<sup>2</sup>

*<sup>i</sup>*<sup>ℏ</sup> ∙, *G*

, evaluated along the *g* direction, is the quantum analogue of

^ *<sup>F</sup>* =(- ∂ *F* ^ ∂ *P* ^ , <sup>∂</sup> *<sup>F</sup>* ^ ∂ *Q*

<sup>≔</sup>( *<sup>d</sup> <sup>l</sup>* ^ *F dg* )<sup>2</sup>

^ /ℏ*Q* ^ *<sup>e</sup>*-*if <sup>G</sup>* ^

^ (*Q* ^ , *P* ^ ) ∂ *P* ^ <sup>=</sup> <sup>1</sup>

> ^ (*Q* ^ , *P* ^ ) ∂ *P* ^ <sup>=</sup> <sup>1</sup> *ih F* ^(*<sup>Q</sup>* ^ , *<sup>P</sup>* ^), *<sup>Q</sup>*

*<sup>i</sup>*<sup>ℏ</sup> *Q*

^ ( *<sup>f</sup>* ), *<sup>G</sup>* ^ (*<sup>Q</sup>* ^ , *<sup>P</sup>*

^) , (46)

^ (*g*) , (47)

/<sup>ℏ</sup> , (48)

/<sup>ℏ</sup> . (49)

^ ) , (50)

, (51)

^ . (52)

^ of a physical system, a case in

^ or *<sup>F</sup>* ^ , re‐

^) , *dQ*

^(*g*) , and *dQ*

/<sup>ℏ</sup> , *Q*

/<sup>ℏ</sup> , and *Q*

These equations can be written in the form of a set of quantum dynamical systems

^ ) , *<sup>d</sup> <sup>z</sup>*

^ *dg* = *X* ^

.

We can define many of the classical quantities but now in the quantum realm. Liouville type

^ ^ *<sup>G</sup>* <sup>≔</sup> <sup>1</sup>

^ is the Hamiltonian *<sup>H</sup>*

, ∙ , and L

These operators will move functions of operators along the conjugate directions *G*

different derivatives introducing quantum dynamical system as

^( *<sup>f</sup>* ), *<sup>G</sup>* ^ (*<sup>Q</sup>* ^ , *<sup>P</sup>*

> ^ /ℏ*P* ^ *eif <sup>F</sup>* ^

The inner product between the operator vector fields is

*d P* ^ ( *f* ) *df* <sup>=</sup> - <sup>∂</sup> *<sup>G</sup>*

*d P* ^ (*g*) *dg* <sup>=</sup> <sup>∂</sup> *<sup>F</sup>* ^ (*Q* ^ , *P* ^ ) ∂ *Q* ^ <sup>=</sup> <sup>1</sup> *<sup>i</sup>*<sup>ℏ</sup> *F* ^(*<sup>Q</sup>* ^ , *<sup>P</sup>* ^), *<sup>P</sup>*

where

*d z* ^ *df* = *X* ^

^ =(*<sup>Q</sup>* ^ , *<sup>P</sup>* ^).

^

*X* ^ *F* † ∙ *X* ^ *<sup>F</sup>* =(- ∂ *F* ^ ∂ *P* ^ , <sup>∂</sup> *<sup>F</sup>* ^ ∂ *Q* ^ )† ∙(- ∂ *F* ^ ∂ *P* ^ , <sup>∂</sup> *<sup>F</sup>* ^ ∂ *Q* ^ ) =( *dQ* ^ *dg* )<sup>2</sup>

*<sup>F</sup>* )2≔(*dQ*

where *z*

where (*d l*

operators are

^ (*Q* ^ , *P* ^ ) ∂ *Q* ^ <sup>=</sup> <sup>1</sup> *ih P*

14 Advances in Quantum Mechanics

*P* ^( *<sup>f</sup>* )=*eif <sup>G</sup>* ^ /ℏ*P* ^ *e*-*if <sup>G</sup>* ^

*P* ^(*g*)=*e*-*ig <sup>F</sup>*

*<sup>G</sup>* , *X*

^ *<sup>G</sup>* =( <sup>∂</sup> *<sup>G</sup>* ^ ∂ *P* ^ , - <sup>∂</sup> *<sup>G</sup>* ^ ∂ *Q*

^ )<sup>2</sup> <sup>+</sup> (*dP* ^)2

the square of the line element (*dlF* )<sup>2</sup> =(*dq*)<sup>2</sup> <sup>+</sup> (*dp*)<sup>2</sup>

L ^ ^ *<sup>F</sup>* <sup>≔</sup> <sup>1</sup> *<sup>i</sup>*<sup>ℏ</sup> *F* ^

which we get the usual time evolution of operator.

spectively. This is the case when *G*

Note that the conclusions mentioned at the end of the previous section for classical systems also hold in the quantum realm.

Next, we illustrate the use of these ideas with a simple system.

#### **3. Time evolution using energy and time eigenstates**

As a brief application of the abovee ideas, we show how to use the energy-time coordinates and eigenfunctions in the reversible evolution of probability densities.

Earlier, there was an interest on the classical and semi classical analysis of energy transfer in molecules. Those studies were based on the quantum procedure of expanding wave func‐ tions in terms of energy eigenstates, after the fact that the evolution of energy eigenstates is quite simple in Quantum Mechanics because the evolution equation for a wave function *<sup>i</sup>*<sup>ℏ</sup> <sup>∂</sup> ∂ *t* |*ψ*> =*H* ^ <sup>|</sup>*ψ*> is linear and contains the Hamiltonian operator. In those earlier calcula‐ tions, an attempt to use the eigenfunctions of a complex classical Liouville operator was made [5-8]. The results in this chapter show that the eigenfunction of the Liouville operator L*H* is *e gT* (*z*) and that it do not seems to be a good set of functions in terms of which any other function can be written, as is the case for the eigenfunctions of the Hamiltonian operator in Quantum Mechanics. In this section, we use the time eigenstates instead.

With energy-time eigenstates the propagation of classical densities is quite simple. In order to illustrate our procedure, we will apply it to the harmonic oscillator with Hamiltonian giv‐ en by (we will use dimensionless units)

$$H(z) = \frac{p^2}{2} + \frac{q^2}{2} \,. \tag{53}$$

*<sup>p</sup>* <sup>=</sup>*<sup>q</sup>* tan (*<sup>t</sup>* <sup>+</sup> *<sup>π</sup>*

is a periodic system, so we will only consider one period in time.

<sup>2</sup> ), or *<sup>q</sup>* <sup>=</sup> *<sup>p</sup>* cot (*<sup>t</sup>* <sup>+</sup> *<sup>π</sup>*

Classical and Quantum Conjugate Dynamics – The Interplay Between Conjugate Variables

These are just straight lines passing through the origin, equivalent to the polar coordinates. The value of time on these points is *t*, precisely. In Fig. 1, we show both coordinate systems, the phase space coordinates (*q*, *p*), and the energy time coordinates (*E*, *t*) on the plane. This

**Figure 1.** Two conjugate coordinate systems for the classical harmonic oscillator in dimensionless units. Blue and black

At this point, there are two options for time curves. Both options will cover the plane and we can distinguish between the regions of phase space with negative or positive momen‐ tum. One is to use half lines and *t* in the range from -*π* to *π*, with the curve *t* =0 coinciding with the positive *p* axes. The other option is to use the complete curve including positive

lines correspond to the (*q*, *p*) coordinates and the red and green curves to the (*E*, *t*) coordinates.

<sup>2</sup> ) . (56)

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17

Given and energy scaling parameter *Es* and the frequency *ω* of the harmonic oscillator, the remaining scaling parameters are

$$p\_s = \sqrt{mE\_s} \quad , \qquad q\_s = \sqrt{\frac{E\_s}{m\omega^2}} \quad , \qquad t\_s = \frac{1}{\omega} \quad . \tag{54}$$

We need to define time eigensurfaces for our calculations. The procedure to obtain them is to take the curve *q* =0 as the zero time curve. The forward and backward propagation of the zero time curve generates the time coordinate system in phase space. The trajectory generat‐ ed with the harmonic oscillator Hamiltonian is

$$q(t) = \sqrt{2E} \cos\left(t + \frac{\pi}{2}\right), \qquad p(t) = \sqrt{2E} \sin\left(t + \frac{\pi}{2}\right). \tag{55}$$

With the choice of phase we have made, *q* =0 when *t* =0, which is the requirement for an ini‐ tial time curve. Then, the equation for the time curve is

Classical and Quantum Conjugate Dynamics – The Interplay Between Conjugate Variables http://dx.doi.org/10.5772/53598 17

$$p = q \tan\left(t + \frac{\pi}{2}\right), \quad \text{or} \quad q = p \cot\left(t + \frac{\pi}{2}\right). \tag{56}$$

These are just straight lines passing through the origin, equivalent to the polar coordinates. The value of time on these points is *t*, precisely. In Fig. 1, we show both coordinate systems, the phase space coordinates (*q*, *p*), and the energy time coordinates (*E*, *t*) on the plane. This is a periodic system, so we will only consider one period in time.

Note that the conclusions mentioned at the end of the previous section for classical systems

As a brief application of the abovee ideas, we show how to use the energy-time coordinates

Earlier, there was an interest on the classical and semi classical analysis of energy transfer in molecules. Those studies were based on the quantum procedure of expanding wave func‐ tions in terms of energy eigenstates, after the fact that the evolution of energy eigenstates is quite simple in Quantum Mechanics because the evolution equation for a wave function

tions, an attempt to use the eigenfunctions of a complex classical Liouville operator was made [5-8]. The results in this chapter show that the eigenfunction of the Liouville operator

function can be written, as is the case for the eigenfunctions of the Hamiltonian operator in

With energy-time eigenstates the propagation of classical densities is quite simple. In order to illustrate our procedure, we will apply it to the harmonic oscillator with Hamiltonian giv‐

<sup>2</sup> <sup>+</sup> *<sup>q</sup>* <sup>2</sup>

Given and energy scaling parameter *Es* and the frequency *ω* of the harmonic oscillator, the

We need to define time eigensurfaces for our calculations. The procedure to obtain them is to take the curve *q* =0 as the zero time curve. The forward and backward propagation of the zero time curve generates the time coordinate system in phase space. The trajectory generat‐

With the choice of phase we have made, *q* =0 when *t* =0, which is the requirement for an ini‐

*<sup>m</sup><sup>ω</sup>* <sup>2</sup> , *ts* <sup>=</sup> <sup>1</sup>

<sup>2</sup> ) , *<sup>p</sup>*(*t*)= <sup>2</sup>*E*sin (*<sup>t</sup>* <sup>+</sup> *<sup>π</sup>*

^ <sup>|</sup>*ψ*> is linear and contains the Hamiltonian operator. In those earlier calcula‐

and that it do not seems to be a good set of functions in terms of which any other

<sup>2</sup> . (53)

*<sup>ω</sup>* . (54)

<sup>2</sup> ) . (55)

also hold in the quantum realm.

16 Advances in Quantum Mechanics

*<sup>i</sup>*<sup>ℏ</sup> <sup>∂</sup> ∂ *t*

L*H* is *e gT* (*z*)


en by (we will use dimensionless units)

remaining scaling parameters are

ed with the harmonic oscillator Hamiltonian is

*<sup>q</sup>*(*t*)= <sup>2</sup>*E*cos (*<sup>t</sup>* <sup>+</sup> *<sup>π</sup>*

tial time curve. Then, the equation for the time curve is

Next, we illustrate the use of these ideas with a simple system.

**3. Time evolution using energy and time eigenstates**

and eigenfunctions in the reversible evolution of probability densities.

Quantum Mechanics. In this section, we use the time eigenstates instead.

*<sup>H</sup>* (*z*)= *<sup>p</sup>* <sup>2</sup>

*ps* <sup>=</sup> *mEs* , *qs* <sup>=</sup> *Es*

**Figure 1.** Two conjugate coordinate systems for the classical harmonic oscillator in dimensionless units. Blue and black lines correspond to the (*q*, *p*) coordinates and the red and green curves to the (*E*, *t*) coordinates.

At this point, there are two options for time curves. Both options will cover the plane and we can distinguish between the regions of phase space with negative or positive momen‐ tum. One is to use half lines and *t* in the range from -*π* to *π*, with the curve *t* =0 coinciding with the positive *p* axes. The other option is to use the complete curve including positive and negative momentum values and with *t* ∈( - *π* / 2, *π* / 2). In the first option, the positive momentum part of a probability density will correspond to the range *t* ∈( - *π* / 2, *π* / 2), and the negative values will correspond to *<sup>t</sup>* <sup>∈</sup>(-*π*, - *<sup>π</sup>* <sup>2</sup> ) ∪(*π* / 2, *π*). We take this option.

Now, based on the equalities derived in this chapter, we find the following relationship for a marginal density dependent only upon *H* (*z*), assuming that the function *ρ*(*H* ) can be writ‐ ten as a power series of *H* , *ρ*(*H* )=∑ *ρi H <sup>i</sup>* ,

*i*

$$\operatorname{e}^{-\tau \mathsf{L}\_H} \rho(H) = \sum\_{n} \frac{(\cdot \cdot \tau)^n}{n!} \mathsf{L}\_H^n \sum\_{i} \rho\_i H^i \stackrel{i}{=} \sum\_{i} \rho\_i H^i = \rho(H) \ , \tag{57}$$

where we have made use of the equality L*<sup>H</sup> H* =0. Then, a function of *H* does not evolve in time, it is a steady state. For a marginal function dependent upon *t*, we also have that

$$e^{-\tau \mathcal{L}\_H} \rho(t) = e^{\tau d/dt} \rho(t) = \rho(t+\tau) \,. \tag{58}$$

**Figure 2.** Contour plots of the time evolution of a probability density on phase-space and on energy-time space. Initial densities (a) in phase space, and (b) in energy-time space. (d) Evolution in energy-time space is accomplished by a shift

Classical and Quantum Conjugate Dynamics – The Interplay Between Conjugate Variables

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19

This behaviour is also observed in quantum systems. Time eigenfunctions can be defined in a similar way as for classical systems. We start with a coordinate eigenfunction |*q* > for the

<sup>ℏ</sup> |*q* =0> . (61)

<sup>ℏ</sup> |ψ> =*ψ*(q=0;t) , (62)

<sup>ℏ</sup> |ψ> = <*t* + *τ*|ψ> . (63)

along the *t*axes. (c) In phase space, the density is also translated to the corresponding time eigensurfaces.

*it H* ^

> *it H* ^

*it H* ^ <sup>ℏ</sup> *eiτ H* ^

Which is the time dependent wave function, in the coordinate representation, and evaluated

Then, time evolution is the translation in time representation, without a change in shape. Note that the variable *τ* is the time variable that appears in the Schrödinger equation for the

eigenvalue *q* =0 and propagate it in time. This will be our time eigenstate


<*t*|*ψ*> = <*q* =0|*e*-

at *q* =0. This function is the time component of the wave function.

The time component of a propagated wave function for a time *τ* is

<*t*|*ψ*(*τ*)> = <*q* =0|*e*-

wave function.

The projection of a wave function onto this vector is

where we have made use of the result that *<sup>d</sup> dt* = - L*<sup>H</sup>* . Therefore, a function of *t* is only shifted in time without changing its shape.

For a function of *H* and *t* we find that

$$e^{-\tau \mathcal{L}\_H} \rho(H, t) = e^{-\tau d/dt} \rho(H, t) = \rho(H, t + \tau) \,. \tag{59}$$

This means that evolution in energy-time space also is quite simple, it is only a shift of the function along the *t* axes without a change of shape.

So, let us take a concrete probability density and let us evolve it in time. The probability density, in phase space, that we will consider is

$$\rho(z) = H(z)e^{-\left( (q \cdot q\_0)^2 + (p \cdot p\_0)^2 \right) / 2\sigma^2} \,\, \, \, \, \tag{60}$$

with (*q*0, *p*0) =(1,2) and *σ* =1. A contour plot of this density in phase-space is shown in (a) of Fig. 2. The energy-time components of this density are shown in (b) of the same figure. Time evolution by an amount *τ* correspond to a translation along the *t* axes, from *t* to *t* + *τ*, with‐ out changing the energy values. This translation is illustrated in (d) of Fig. 2 in energy-time space and in (c) of the same figure in phase-space.

Recall that the whole function *ρ*(*z*) is translated in time with the propagator *e*-*τ*L*<sup>H</sup>* . Then, there are two times involved here, the variable *t* as a coordinate and the shift in time *τ*. The latter is the time variable that appears in the Liouville equation of motion *dρ*(*z*; *τ*) *<sup>d</sup><sup>τ</sup>* = - L*<sup>H</sup> ρ*(*z*;*τ*).

Classical and Quantum Conjugate Dynamics – The Interplay Between Conjugate Variables http://dx.doi.org/10.5772/53598 19

**Figure 2.** Contour plots of the time evolution of a probability density on phase-space and on energy-time space. Initial densities (a) in phase space, and (b) in energy-time space. (d) Evolution in energy-time space is accomplished by a shift along the *t*axes. (c) In phase space, the density is also translated to the corresponding time eigensurfaces.

This behaviour is also observed in quantum systems. Time eigenfunctions can be defined in a similar way as for classical systems. We start with a coordinate eigenfunction |*q* > for the eigenvalue *q* =0 and propagate it in time. This will be our time eigenstate

$$\left. \right|\_{t>} = e^{\frac{\hat{\imath}}{\pi}} \left. \right|\_{q=0>} \tag{61}$$

The projection of a wave function onto this vector is

and negative momentum values and with *t* ∈( - *π* / 2, *π* / 2). In the first option, the positive momentum part of a probability density will correspond to the range *t* ∈( - *π* / 2, *π* / 2), and

Now, based on the equalities derived in this chapter, we find the following relationship for a marginal density dependent only upon *H* (*z*), assuming that the function *ρ*(*H* ) can be writ‐

where we have made use of the equality L*<sup>H</sup> H* =0. Then, a function of *H* does not evolve in

This means that evolution in energy-time space also is quite simple, it is only a shift of the

So, let us take a concrete probability density and let us evolve it in time. The probability


with (*q*0, *p*0) =(1,2) and *σ* =1. A contour plot of this density in phase-space is shown in (a) of Fig. 2. The energy-time components of this density are shown in (b) of the same figure. Time evolution by an amount *τ* correspond to a translation along the *t* axes, from *t* to *t* + *τ*, with‐ out changing the energy values. This translation is illustrated in (d) of Fig. 2 in energy-time

Recall that the whole function *ρ*(*z*) is translated in time with the propagator *e*-*τ*L*<sup>H</sup>* . Then, there are two times involved here, the variable *t* as a coordinate and the shift in time *τ*. The latter is the time variable that appears in the Liouville equation of motion

time, it is a steady state. For a marginal function dependent upon *t*, we also have that

<sup>2</sup> ) ∪(*π* / 2, *π*). We take this option.

*ρ*(*t*)=*ρ*(*t* + *τ*) . (58)

*dt* = - L*<sup>H</sup>* . Therefore, a function of *t* is only shifted

*ρ*(*H* , *t*)=*ρ*(*H* , *t* + *τ*) . (59)

, (60)

=*ρ*(*H* ) , (57)

the negative values will correspond to *<sup>t</sup>* <sup>∈</sup>(-*π*, - *<sup>π</sup>*

*e*-*τ*L*<sup>H</sup> ρ*(*H* )=∑

where we have made use of the result that *<sup>d</sup>*

in time without changing its shape.

For a function of *H* and *t* we find that

*i ρi H <sup>i</sup>* ,

( - *τ*)*<sup>n</sup> <sup>n</sup>* ! L*<sup>H</sup> <sup>n</sup>* ∑ *i ρi H <sup>i</sup>* =∑ *i ρi H <sup>i</sup>*

*e*-*τ*L*<sup>H</sup> ρ*(*t*)=*e <sup>τ</sup><sup>d</sup>* /*dt*

*n*

*e*-*τ*L*<sup>H</sup> ρ*(*H* , *t*)=*e*-*τ<sup>d</sup>* /*dt*

*ρ*(*z*)=*H* (*z*)*e*

function along the *t* axes without a change of shape.

density, in phase space, that we will consider is

space and in (c) of the same figure in phase-space.

*dρ*(*z*; *τ*)

*<sup>d</sup><sup>τ</sup>* = - L*<sup>H</sup> ρ*(*z*;*τ*).

ten as a power series of *H* , *ρ*(*H* )=∑

18 Advances in Quantum Mechanics

$$<\psi \mid \psi> =  = \psi \{ \mathbf{q}=0; \mathbf{t} \} \tag{62}$$

Which is the time dependent wave function, in the coordinate representation, and evaluated at *q* =0. This function is the time component of the wave function.

The time component of a propagated wave function for a time *τ* is

$$<\langle t \mid \psi(\tau) > =  =  . \tag{63}$$

Then, time evolution is the translation in time representation, without a change in shape. Note that the variable *τ* is the time variable that appears in the Schrödinger equation for the wave function.

Now, assuming a discrete energy spectrum with energy eigenvalue *En* and corresponding eigenfunction |*n* >, in the energy representation we have that

$$<\mu \mid \psi(\tau) > = <\nu \mid e^{\cdot \frac{\hat{\imath}H}{\hbar}} \Big| \psi > = e^{\cdot \cdot \frac{\hat{\imath}H\_n}{\hbar}} <\nu \Big| \psi > \Big. \tag{64}$$

[5] Jaffé C. Classical Liouville mechanics and intramolecular relaxation dynamics. The

Classical and Quantum Conjugate Dynamics – The Interplay Between Conjugate Variables

http://dx.doi.org/10.5772/53598

21

[6] Jaffé C and Brumer C. Classical-quantum correspondence in the distribution dynam‐

[7] Jaffé C. Semiclassical quantization of the Liouville formulation of classical mechanics.

[8] Jaffé C. Sheldon Kanfer and Paul Brumer, Classical analog of pure-state quantum dy‐

ics of integrable systems. Journal of Chemical Physics 1985; 82 2330.

Journal of Physical Chemistry 1984; 88 4829.

Journal of Chemical Physics 1988; 88 7603.

namics. Physical Review Letters 1985; 54 8.

[10] Galapon EA, Proc R Soc Lond A 2002; 458 451

[11] Galapon EA, Proc R Soc Lond A 2002; 458 2671

[14] Garrison JC and Wong J, J Math Phys 1970; 11 2242

[15] Torres-Vega G, J Phys A: Math Theor 45, 215302 (2012)

[12] Galapon EA, quant-ph/0303106

[13] Galindo A, Lett Math Phys 1984; 8 495

[9] Pauli W. Handbuch der Physics. Berlin: Springer-Verlag; 1926

i.e. the wave function in energy space only changes its phase after evolution for a time *τ*.

### **4. Concluding remarks**

Once that we have made use of the same concepts in both classical and quantum mechanics, it is more easy to understand quantum theory since many objects then are present in both theories.

Actually, there are many things in common for both classical and quantum systems, as is the case of the eigensurfaces and the eigenfunctions of conjugate variables, which can be used as coordinates for representing dynamical quantities.

Another benefit of knowing the influence of conjugate dynamical variables on themselves and of using the same language for both theories lies in that some puzzling things that are found in one of the theories can be analysed in the other and this helps in the understanding of the original puzzle. This is the case of the Pauli theorem [9-14] that prevents the existence of a hermitian time operator in Quantum Mechanics. The classical analogue of this puzzle is found in Reference [15].

These were some of the properties and their consequences in which both conjugate variables participate, influencing each other.

### **Author details**

Gabino Torres-Vega

Physics Department, Cinvestav, México

### **References**


Now, assuming a discrete energy spectrum with energy eigenvalue *En* and corresponding

<sup>ℏ</sup> <sup>|</sup>ψ<sup>&</sup>gt; <sup>=</sup>*e*-

Once that we have made use of the same concepts in both classical and quantum mechanics, it is more easy to understand quantum theory since many objects then are present in both

Actually, there are many things in common for both classical and quantum systems, as is the case of the eigensurfaces and the eigenfunctions of conjugate variables, which can be used as

Another benefit of knowing the influence of conjugate dynamical variables on themselves and of using the same language for both theories lies in that some puzzling things that are found in one of the theories can be analysed in the other and this helps in the understanding of the original puzzle. This is the case of the Pauli theorem [9-14] that prevents the existence of a hermitian time operator in Quantum Mechanics. The classical analogue of this puzzle is

These were some of the properties and their consequences in which both conjugate variables

[1] Woodhouse NMJ. Geometric Quantization. Oxford: Osford University Press; 1991.

[4] Torres-Vega G, Theoretical concepts of quantum mechanics. Rijeka: InTech; 2012.

i.e. the wave function in energy space only changes its phase after evolution for a time *τ*.

*iτ*En

<sup>ℏ</sup> <sup>&</sup>lt;*n*|ψ<sup>&</sup>gt; , (64)

*iτ H* ^

eigenfunction |*n* >, in the energy representation we have that

<*n*|*ψ*(*τ*)> = <*n*|*e*-

coordinates for representing dynamical quantities.

**4. Concluding remarks**

20 Advances in Quantum Mechanics

found in Reference [15].

**Author details**

Gabino Torres-Vega

**References**

participate, influencing each other.

Physics Department, Cinvestav, México

[2] Wigner E. Phys Rev A 1932; 40 749

[3] Husimi K. Proc Phys Math Soc Jpn 1940; 22 264

theories.


**Chapter 2**

**Provisional chapter**

**Classical and Quantum Correspondence in Anisotropic**

If the classical behavior of a given quantum system is chaotic, how is it reflected in the quantum properties of the system? To elucidate this correspondence is the main theme of the quantum chaos study. With the advent of nanophysics techniques, this has become also of experimental importance. With the advent of new technology, various quantum systems are now challenging us. These include nano-scale devices, laser trapping of atoms, the Bose-Einstein condensate, Rydberg atoms, and even web of chaos is observed in superlattices. In this note we devote ourselves to the investigation of the quantum scars which occurs in the Anisotropic Kepler Problem (AKP) – the classical and quantum physics of an electron trapped around a proton in semiconductors. The merit of AKP is that its chaotic property can be controlled by changing the anisotropy from integrable Kepler limit down ergodic limit where the tori are completely collapsed and isolated unstable periodic orbits occupy the classical phase space. Thus in AKP we are able to investigate the classical quantum correspondence at varying chaoticity. Furthermore each unstable periodic orbit (PO) can be coded in a Bernoulli code which is a large merit in the formulation of quantum chaos in term

The AKP is an old home ground of the quantum chaos study. Its low energy levels were used as a test of the periodic orbit theory in the seminal work of Gutzwiller [3-7]. Then an efficient matrix diagonalization scheme was devised by Wintgen et al. (WMB method) [8]. With this method, the statistics of up to nearly 8000 AKP quantum levels were examined and it was found that the quantum level statistics of AKP change from Poisson to Wigner distribution with the increase of mass anisotropy [9]. Furthermore, an intriguing classical Poincaré surface of section (POS) was found at medium anisotropy (*γ* ≡ *m*(light)/*m*(heavy) = 0.8),

> ©2012 Shimada et al., 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 Sumiya et al.; 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,

© 2012 Sumiya et al., 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.

**Classical and Quantum Correspondence in**

**Kepler Problem**

http://dx.doi.org/10.5772/55208

10.5772/55208

**1. Introduction**

Keita Sumiya, Hisakazu Uchiyama, Kazuhiro Kubo and Tokuzo Shimada

of the periodic orbit theory (POT) [1, 2, 6].

Keita Sumiya, Hisakazu Uchiyama, Kazuhiro Kubo and Tokuzo Shimada

**Anisotropic Kepler Problem**

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

**Provisional chapter**

### **Classical and Quantum Correspondence in Anisotropic Kepler Problem Classical and Quantum Correspondence in Anisotropic Kepler Problem**

Keita Sumiya, Hisakazu Uchiyama, Kazuhiro Kubo and Tokuzo Shimada Keita Sumiya, Hisakazu Uchiyama, Kazuhiro Kubo and Tokuzo Shimada

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/55208 10.5772/55208

### **1. Introduction**

If the classical behavior of a given quantum system is chaotic, how is it reflected in the quantum properties of the system? To elucidate this correspondence is the main theme of the quantum chaos study. With the advent of nanophysics techniques, this has become also of experimental importance. With the advent of new technology, various quantum systems are now challenging us. These include nano-scale devices, laser trapping of atoms, the Bose-Einstein condensate, Rydberg atoms, and even web of chaos is observed in superlattices.

In this note we devote ourselves to the investigation of the quantum scars which occurs in the Anisotropic Kepler Problem (AKP) – the classical and quantum physics of an electron trapped around a proton in semiconductors. The merit of AKP is that its chaotic property can be controlled by changing the anisotropy from integrable Kepler limit down ergodic limit where the tori are completely collapsed and isolated unstable periodic orbits occupy the classical phase space. Thus in AKP we are able to investigate the classical quantum correspondence at varying chaoticity. Furthermore each unstable periodic orbit (PO) can be coded in a Bernoulli code which is a large merit in the formulation of quantum chaos in term of the periodic orbit theory (POT) [1, 2, 6].

The AKP is an old home ground of the quantum chaos study. Its low energy levels were used as a test of the periodic orbit theory in the seminal work of Gutzwiller [3-7]. Then an efficient matrix diagonalization scheme was devised by Wintgen et al. (WMB method) [8]. With this method, the statistics of up to nearly 8000 AKP quantum levels were examined and it was found that the quantum level statistics of AKP change from Poisson to Wigner distribution with the increase of mass anisotropy [9]. Furthermore, an intriguing classical Poincaré surface of section (POS) was found at medium anisotropy (*γ* ≡ *m*(light)/*m*(heavy) = 0.8),

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 Sumiya et al.; 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. © 2012 Sumiya et al., 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 Shimada et al., licensee InTech. This is an open access chapter distributed under the terms of the

which indicates remnants of tori (cantori) in the classical phase space [9]. Thus, over two decades from the early 70th, AKP was a good testing ground of theories (along with billiards) as well as a constant source of important information to quantum chaos studies. However, there has not been much recent theory investigation on AKP. Especially, to our knowledge, the quantum scar of the classical periodic orbits in AKP has not been directly examined, even though intriguing phenomena was discovered by Heller [10] in 1984. On the other hand, for an analogous system – the hydrogen under a magnetic field (diamagnetic Kepler problem (DKP)), the scars of periodic orbits were extensively studied using highly efficient tool called as scar strength functions [11]. We note that AKP is by far simpler; for DKP it is necessary to code the POs by a sequence of symbols consisting of three letters.

10.5772/55208

25

http://dx.doi.org/10.5772/55208

seemingly contradicting features intriguingly compromise. The localization patterns in the wave functions or Husimi functions are swapped between two eigenstates of energy at every avoiding crossing. Repeating successively this swap process characteristic scarring patterns follow the POs responsible to them. In this sense the quantum scarring phenomena are

We first explain how we have prepared the energy levels and wave functions. Then we introduce the indispensable ingredients to study the scars in AKP. After briefly explaining Husimi functions, we explain periodic orbit theory. The quantum scars will be observed

robust. We conclude in section 4.

**2.1. Matrix diagonalization**

*2.1.1. AKP Hamiltonian*

with *γ* = *ν*/*µ* = 1/*µ*<sup>2</sup> or

with literature.

*2.1.2. Matrix diagonalization in Sturmian basis*

Firstly, in the Sturmian basis

**2. Manifestation of Scars in AKP**

along the classical unstable periodic orbits.

The AKP Hamiltonian in the dimensionless form is given by

*HG* <sup>=</sup> <sup>1</sup>

or equivalently it may be also written as [9,11] (Harmonic basis)

*H*<sup>88</sup> *<sup>W</sup>* <sup>=</sup> <sup>1</sup> 2 (*p*<sup>2</sup> *<sup>x</sup>* + *<sup>p</sup>*<sup>2</sup>

> *H*<sup>87</sup> *<sup>W</sup>* <sup>=</sup> *<sup>p</sup>*<sup>2</sup>

We here summarize WMB method for efficient matrix diagonalization.

<sup>2</sup>*<sup>µ</sup> <sup>p</sup>*<sup>2</sup> *<sup>x</sup>* + 1 2*ν* (*p*<sup>2</sup> *<sup>y</sup>* + *<sup>p</sup>*<sup>2</sup>

where *r* = *x*<sup>2</sup> + *y*<sup>2</sup> + *z*<sup>2</sup> and *µ* > *ν*[1,4] with which POT was formulated in the history,

*<sup>x</sup>* + *<sup>p</sup>*<sup>2</sup>

as used in WMB (Sturmian basis) [8]. We recapitulate POT predictions in terms of (1), our formula for AKP eigenvalue calculation in tensored harmonic wave function basis in terms of (2), and we discuss quantum scars using energy values in (3) in order to facilitate comparison

*<sup>y</sup>*) + *<sup>γ</sup>* 2 *p*2 *<sup>z</sup>* <sup>−</sup> <sup>1</sup>

*<sup>y</sup>* + *<sup>γ</sup>p*<sup>2</sup>

*<sup>z</sup>* <sup>−</sup> <sup>2</sup>

*<sup>z</sup>* ) <sup>−</sup> <sup>1</sup>

Classical and Quantum Correspondence in Anisotropic Kepler Problem

*<sup>r</sup>* (1)

*<sup>r</sup>* (2)

*<sup>r</sup>* (3)

Recently the level statistics of AKP was examined from the random matrix theory view [12]. It was considered that the AKP level statistic in the transitive region from Poisson to Wigner distribution correspond to the critical level statistics of an extended GOE random matrix theory and it was conjectured that the wave functions should exhibit characteristic multifractality. This aspect has been further developed in [13, 14]; it is considered that *Anderson transition* occurs in the quantum physics of a class of physical systems such as AKP and periodically driven kicked rotator in their critical parameter regions. Further very recently a well devised new solid state experiment has been conducted for AKP and ADKP [15, 16]. We also refer [17] for a recent overview including this interesting conjecture.

Such is the case we have recently conducted AKP high accuracy matrix diagonalization based on the WMB method. This is not a perturbation calculation; the anisotropy term is not regarded as a perturbation and the full Hamiltonian matrix is diagonalized. Thus the approximation comes only from the size of the matrix. But, as a trade-off, a scaling parameter is unavoidably included; it is crucial to choose a correct parameter value at every anisotropy parameter. We have derived a simple rule of thumb to choose a suitable value [17]. After comparing with original WMB result in Sturmian basis, we have also worked with tensored-harmonic-wavefunction basis (THWFB) [11], which is more suitable for the Husimi function calculation to investigate the quantum scars. Our contribution here is the calculation of anisotropy term in the AKP Hamiltonian in THWFB [17], which is harder than the diamagnetic case. Comparing the results from two independent bases we have verified that both results agree completely thus the choices of scaling parameters (in both bases) are validated.

Aimed by these numerical data, we report in section 2 salient evidences of quantum scars in AKP for the first time. We compare the features of various known observables; thus this section will serve as a comparative test of methods and fulfills the gap in the literature pointed out above. Most interesting is the test using the scar strength function. We show that even in the ergodic regime (*γ* = 0.2), we can quantitatively observe that prominent periodic orbits systematically contribute to the quantum theory endowed with random energy spectrum.

In section 3 we investigate that how the scaring phenomena are affected by the variation of the anisotropy parameter. It is well known that the energy levels show successive avoiding crossings. On the other hand, in the periodic orbit formula, each term in the series for the density of states (DOS) consists of a contribution of an unstable PO with a pole (with an imaginary part given by the Lyapunov exponent of the PO) at the Bohr-Sommerfeld-type energy; thus each term smoothly varies with the anisotropy. We show that how these two seemingly contradicting features intriguingly compromise. The localization patterns in the wave functions or Husimi functions are swapped between two eigenstates of energy at every avoiding crossing. Repeating successively this swap process characteristic scarring patterns follow the POs responsible to them. In this sense the quantum scarring phenomena are robust. We conclude in section 4.

### **2. Manifestation of Scars in AKP**

We first explain how we have prepared the energy levels and wave functions. Then we introduce the indispensable ingredients to study the scars in AKP. After briefly explaining Husimi functions, we explain periodic orbit theory. The quantum scars will be observed along the classical unstable periodic orbits.

#### **2.1. Matrix diagonalization**

#### *2.1.1. AKP Hamiltonian*

2 Advances in Quantum Mechanics

validated.

energy spectrum.

which indicates remnants of tori (cantori) in the classical phase space [9]. Thus, over two decades from the early 70th, AKP was a good testing ground of theories (along with billiards) as well as a constant source of important information to quantum chaos studies. However, there has not been much recent theory investigation on AKP. Especially, to our knowledge, the quantum scar of the classical periodic orbits in AKP has not been directly examined, even though intriguing phenomena was discovered by Heller [10] in 1984. On the other hand, for an analogous system – the hydrogen under a magnetic field (diamagnetic Kepler problem (DKP)), the scars of periodic orbits were extensively studied using highly efficient tool called as scar strength functions [11]. We note that AKP is by far simpler; for DKP it is necessary to

Recently the level statistics of AKP was examined from the random matrix theory view [12]. It was considered that the AKP level statistic in the transitive region from Poisson to Wigner distribution correspond to the critical level statistics of an extended GOE random matrix theory and it was conjectured that the wave functions should exhibit characteristic multifractality. This aspect has been further developed in [13, 14]; it is considered that *Anderson transition* occurs in the quantum physics of a class of physical systems such as AKP and periodically driven kicked rotator in their critical parameter regions. Further very recently a well devised new solid state experiment has been conducted for AKP and ADKP [15, 16]. We also refer [17] for a recent overview including this interesting conjecture.

Such is the case we have recently conducted AKP high accuracy matrix diagonalization based on the WMB method. This is not a perturbation calculation; the anisotropy term is not regarded as a perturbation and the full Hamiltonian matrix is diagonalized. Thus the approximation comes only from the size of the matrix. But, as a trade-off, a scaling parameter is unavoidably included; it is crucial to choose a correct parameter value at every anisotropy parameter. We have derived a simple rule of thumb to choose a suitable value [17]. After comparing with original WMB result in Sturmian basis, we have also worked with tensored-harmonic-wavefunction basis (THWFB) [11], which is more suitable for the Husimi function calculation to investigate the quantum scars. Our contribution here is the calculation of anisotropy term in the AKP Hamiltonian in THWFB [17], which is harder than the diamagnetic case. Comparing the results from two independent bases we have verified that both results agree completely thus the choices of scaling parameters (in both bases) are

Aimed by these numerical data, we report in section 2 salient evidences of quantum scars in AKP for the first time. We compare the features of various known observables; thus this section will serve as a comparative test of methods and fulfills the gap in the literature pointed out above. Most interesting is the test using the scar strength function. We show that even in the ergodic regime (*γ* = 0.2), we can quantitatively observe that prominent periodic orbits systematically contribute to the quantum theory endowed with random

In section 3 we investigate that how the scaring phenomena are affected by the variation of the anisotropy parameter. It is well known that the energy levels show successive avoiding crossings. On the other hand, in the periodic orbit formula, each term in the series for the density of states (DOS) consists of a contribution of an unstable PO with a pole (with an imaginary part given by the Lyapunov exponent of the PO) at the Bohr-Sommerfeld-type energy; thus each term smoothly varies with the anisotropy. We show that how these two

code the POs by a sequence of symbols consisting of three letters.

The AKP Hamiltonian in the dimensionless form is given by

$$H\_G = \frac{1}{2\mu}p\_x^2 + \frac{1}{2\nu}(p\_y^2 + p\_z^2) - \frac{1}{r} \tag{1}$$

where *r* = *x*<sup>2</sup> + *y*<sup>2</sup> + *z*<sup>2</sup> and *µ* > *ν*[1,4] with which POT was formulated in the history, or equivalently it may be also written as [9,11] (Harmonic basis)

$$H\_W^{88} = \frac{1}{2}(p\_x^2 + p\_y^2) + \frac{\gamma}{2}p\_z^2 - \frac{1}{r} \tag{2}$$

with *γ* = *ν*/*µ* = 1/*µ*<sup>2</sup> or

$$H\_W^{87} = p\_x^2 + p\_y^2 + \gamma p\_z^2 - \frac{2}{r} \tag{3}$$

as used in WMB (Sturmian basis) [8]. We recapitulate POT predictions in terms of (1), our formula for AKP eigenvalue calculation in tensored harmonic wave function basis in terms of (2), and we discuss quantum scars using energy values in (3) in order to facilitate comparison with literature.

#### *2.1.2. Matrix diagonalization in Sturmian basis*

We here summarize WMB method for efficient matrix diagonalization.

Firstly, in the Sturmian basis

$$
\langle \overrightarrow{r} \vert n\ell m \rangle = \frac{1}{r} \sqrt{\frac{n!}{(2\ell+n+1)!}} e^{-\frac{\lambda r}{2}} (\lambda r)^{\ell+1} L\_n^{2\ell+1}(\lambda r) Y\_{\ell m}(\theta, \varphi) \tag{4}
$$

27

<sup>|</sup>**Ψ**� <sup>=</sup> *<sup>E</sup>* <sup>|</sup>**Ψ**�. (11)

http://dx.doi.org/10.5772/55208


and the AKP Schrödinger equation becomes

 ∆(2) *<sup>µ</sup>* <sup>+</sup> <sup>∆</sup>(2) *ν* + 1 − *γ* 2

�*µ*, *<sup>ν</sup>* <sup>|</sup> *<sup>i</sup>*, *<sup>j</sup>*, *<sup>κ</sup>*� <sup>=</sup> *<sup>κ</sup>*

*π*

with a scaling parameter *κ* and, corresponding to *ε* in (6), we introduce a parameter

*En* <sup>=</sup> <sup>−</sup>*κ*<sup>2</sup> *n* <sup>2</sup> *<sup>ε</sup>*˜ <sup>=</sup> <sup>−</sup> <sup>2</sup> Λ2 *n*

*ε*˜

We have found precise agreement between our calculations by the Sturmian basis and by the harmonic oscillator basis which in turn validates our choices of scaling parameter *ε* and *ε*˜. For the calculation of Husimi functions and scar strength function which uses Husimi functions, we use the THWFB since the projection of the basis functions to the Gaussian

*ε*˜ = 2

and we solve (12) after transforming it into the matrix equation of WMB form with eigenvalues Λ*<sup>n</sup>* = 2/*κn*. The matrix element calculation of the mass anisotropy term in (12) is somewhat involved and we refer to [17] for detail. Energy levels are then determined

*∂*2 *<sup>∂</sup>z*<sup>2</sup> <sup>−</sup> <sup>2</sup>

> *µ*<sup>2</sup> + *ν*<sup>2</sup>

Multiplying by *µ*<sup>2</sup> + *ν*2and swapping the Coulombic interaction term and the *E* term one

Thanks to the semi-parabolic coordinates, the Coulombic singularity has removed [19] for *γ* = 1. Corresponding to the Sturmian basis with a scaling parameter *λ* in (4), we introduce

*Li*(*κµ*2)*Lj*(*κν*2) exp


*µ*<sup>2</sup> + *ν*<sup>2</sup>

Classical and Quantum Correspondence in Anisotropic Kepler Problem

 *<sup>∂</sup>*<sup>2</sup> *∂z*<sup>2</sup> 

<sup>−</sup>*µ*<sup>2</sup> <sup>+</sup> *<sup>ν</sup>*<sup>2</sup> 2

*<sup>κ</sup>*<sup>2</sup> (13)

*ε*˜ (14)

<sup>∗</sup> <sup>≈</sup> *<sup>γ</sup>* (15)

 − 1 2 (*µ*<sup>2</sup> + *ν*2)

 −1 2 ∆(2) *<sup>µ</sup>* <sup>+</sup> <sup>∆</sup>(2) *ν* + |*E*| *µ*<sup>2</sup> + *ν*<sup>2</sup> + 1 − *γ* 2

the harmonic wave function basis

The best value of *ε*˜ is given by

packets are easy to calculate [17].

which is similar to (7).

obtains

by

with a scaling parameter *λ*, the Schrödinger equation of the AKP becomes a matrix equation:

$$\left[ -\lambda \overleftarrow{\Delta^{(3)}} + (1 - \gamma)\lambda \overleftarrow{\frac{\partial^2}{\partial z^2}} - 2\frac{\leftrightarrow}{r} \right] \mathbf{v} = \frac{E}{\lambda} \overleftarrow{\text{Id}} \,\mathbf{v} \tag{5}$$

Dividing the whole equation by *λ* and packing *E*/*λ*<sup>2</sup> into a parameter *ε*, one obtains

$$\overset{\textstyle \longleftrightarrow}{\widetilde{M}} \mathbf{Y} \equiv \left[ \overset{\textstyle \longleftrightarrow}{-\Delta^{\langle 3 \rangle}} + (1 - \gamma) \frac{\overset{\textstyle \longleftrightarrow}{\partial^2}}{\partial z^2} - \epsilon \overset{\textstyle \longleftrightarrow}{\operatorname{Id}} \right] \mathbf{Y} = \frac{2}{\lambda} \mathbf{Y}. \tag{6}$$

This *ǫ* is to be fixed at some constant value. In principle any value will do, but for finite size of Hamiltonian matrix, the best choice is given [17] approximately

$$
\varepsilon^\* \simeq -\frac{1}{4}\gamma.\tag{7}
$$

With this choice, we can get the largest number of reliable energy levels at a given matrix size. The ratio of reliable levels to the matrix size can be estimated as

$$R\_{eff} \simeq \sqrt{\gamma} \,. \tag{8}$$

After fixing *ε*, the diagonalization of (6) is performed for 2/*λ<sup>i</sup>* s and finally we obtain the energy eigenvalues by

$$E\_{\hat{l}} = \varepsilon \lambda\_{\hat{l}}^2. \tag{9}$$

#### *2.1.3. Matrix diagonalization in Sturmian basis*

For the (tensored) harmonic wave function basis (THWFB) [11] we convert the Hamiltonian of AKP into the Hamiltonian of two of two-dimensional harmonic oscillators.

For this purpose semi-parabolic coordinates are introduced

$$
\mu \nu = \rho = \sqrt{x^2 + y^2}, \quad \frac{1}{2}(\mu^2 - \nu^2) = z, \quad \phi = \tan^{-1}\left(\frac{y}{x}\right) \tag{10}
$$

and the AKP Schrödinger equation becomes

4 Advances in Quantum Mechanics

energy eigenvalues by

*2.1.3. Matrix diagonalization in Sturmian basis*

*µν* = *ρ* =

��*r*|*n*ℓ*m*� <sup>=</sup> <sup>1</sup>

 −*<sup>λ</sup>*

←→*<sup>M</sup>* **<sup>Ψ</sup>** <sup>≡</sup>

*r* �

←→

 −

←→

of Hamiltonian matrix, the best choice is given [17] approximately

size. The ratio of reliable levels to the matrix size can be estimated as

*n*! (2ℓ + *n* + 1)!

∆(3) + (1 − *γ*)*λ*

*e* − *<sup>λ</sup><sup>r</sup>*

with a scaling parameter *λ*, the Schrödinger equation of the AKP becomes a matrix equation:

←→ *∂*2 *<sup>∂</sup>z*<sup>2</sup> <sup>−</sup> <sup>2</sup>

> ←→ *∂*2 *<sup>∂</sup>z*<sup>2</sup> <sup>−</sup> *<sup>ǫ</sup>*

This *ǫ* is to be fixed at some constant value. In principle any value will do, but for finite size

With this choice, we can get the largest number of reliable energy levels at a given matrix

After fixing *ε*, the diagonalization of (6) is performed for 2/*λ<sup>i</sup>* s and finally we obtain the

*Ei* = *ελ*<sup>2</sup>

For the (tensored) harmonic wave function basis (THWFB) [11] we convert the Hamiltonian

(*µ*<sup>2</sup> <sup>−</sup> *<sup>ν</sup>*2) = *<sup>z</sup>*, *<sup>φ</sup>* <sup>=</sup> tan−<sup>1</sup>

of AKP into the Hamiltonian of two of two-dimensional harmonic oscillators.

*<sup>x</sup>*<sup>2</sup> <sup>+</sup> *<sup>y</sup>*2, <sup>1</sup>

2

For this purpose semi-parabolic coordinates are introduced

�

Dividing the whole equation by *λ* and packing *E*/*λ*<sup>2</sup> into a parameter *ε*, one obtains

∆(3) + (1 − *γ*)

*ε* <sup>∗</sup> ≃ −1 4 ←→1 *r*

 **<sup>Ψ</sup>** <sup>=</sup> *<sup>E</sup> λ*

←→Id **<sup>Ψ</sup>** <sup>=</sup> <sup>2</sup>

<sup>2</sup> (*λr*)ℓ+1*L*2ℓ+<sup>1</sup> *<sup>n</sup>* (*λr*)*Y*ℓ*m*(*θ*, *<sup>ϕ</sup>*) (4)

Id **Ψ** (5)

*<sup>λ</sup>* **<sup>Ψ</sup>**. (6)

*γ*. (7)

*<sup>i</sup>* . (9)

� *y x* �

(10)

*Reff* <sup>≃</sup> <sup>√</sup>*γ*. (8)

←→

$$\left[ -\frac{1}{2\left(\mu^2 + \nu^2\right)} \left( \Delta\_{\mu}^{(2)} + \Delta\_{\nu}^{(2)} \right) + \frac{1-\gamma}{2} \frac{\partial^2}{\partial z^2} - \frac{2}{\mu^2 + \nu^2} \right] |\mathbf{Y}\rangle = E |\mathbf{Y}\rangle\_{\cdot} \tag{11}$$

Multiplying by *µ*<sup>2</sup> + *ν*2and swapping the Coulombic interaction term and the *E* term one obtains

$$\left[-\frac{1}{2}\left(\Delta\_{\mu}^{(2)} + \Delta\_{\nu}^{(2)}\right) + |E|\left(\mu^2 + \nu^2\right) + \frac{1-\gamma}{2}\left(\mu^2 + \nu^2\right)\frac{\partial^2}{\partial z^2}\right]|\Psi\rangle = 2\left|\Psi\right\rangle. \tag{12}$$

Thanks to the semi-parabolic coordinates, the Coulombic singularity has removed [19] for *γ* = 1. Corresponding to the Sturmian basis with a scaling parameter *λ* in (4), we introduce the harmonic wave function basis

$$\langle \mu, \nu \mid i, j, \kappa \rangle = \frac{\kappa}{\pi} L\_i(\kappa \mu^2) L\_j(\kappa \nu^2) \exp\left(-\frac{\mu^2 + \nu^2}{2}\right).$$

with a scaling parameter *κ* and, corresponding to *ε* in (6), we introduce a parameter

$$
\tilde{\varepsilon} = 2 \frac{|E|}{\kappa^2} \tag{13}
$$

and we solve (12) after transforming it into the matrix equation of WMB form with eigenvalues Λ*<sup>n</sup>* = 2/*κn*. The matrix element calculation of the mass anisotropy term in (12) is somewhat involved and we refer to [17] for detail. Energy levels are then determined by

$$E\_n = -\frac{\kappa\_n^2}{2}\tilde{\varepsilon} = -\frac{2}{\Lambda\_n^2}\tilde{\varepsilon}\tag{14}$$

The best value of *ε*˜ is given by

$$
\mathfrak{E}^\* \approx \gamma \tag{15}
$$

which is similar to (7).

We have found precise agreement between our calculations by the Sturmian basis and by the harmonic oscillator basis which in turn validates our choices of scaling parameter *ε* and *ε*˜.

For the calculation of Husimi functions and scar strength function which uses Husimi functions, we use the THWFB since the projection of the basis functions to the Gaussian packets are easy to calculate [17].

#### **2.2. Husimi function**

Husimi function is defined via the scalar product of the wave function |*ψ*� with a coherent state(CHS) |*q*0, *<sup>p</sup>*0� of the system [11]:

$$\mathcal{W}^{Hus}\_{\psi} \left( q\_{0\prime} \ p\_0 \right) = \left| \langle \psi \mid q\_{0\prime} \ p\_0 \rangle \right|^2 \tag{16}$$

10.5772/55208

29

(21)

(24)

(20)

http://dx.doi.org/10.5772/55208

*<sup>ρ</sup>*(*E*) <sup>≡</sup> ∑

DOS

Because

DOS

(24) is given by

*<sup>ρ</sup>*(*E*) ≈ − <sup>1</sup>

*π* Im

the stationary phase condition gives

*n*

*dq*′ *<sup>G</sup>*˜(*q*′′, *<sup>q</sup>*′

, *E*) 

*<sup>p</sup>*′′ <sup>=</sup> *<sup>∂</sup>S*(*q*′′, *<sup>q</sup>*′

<sup>0</sup> <sup>=</sup> *<sup>∂</sup>S*(*q*′

*r*∈*PO*

and the matrix *M* stands for the monodromy matrix of the primary PO.

<sup>2</sup> [det(*M<sup>n</sup>* <sup>−</sup> <sup>1</sup>)]

*<sup>ρ</sup>*(*E*) <sup>≃</sup> *<sup>ρ</sup>*(*E*) + Im ∑

1

*<sup>δ</sup>*(*<sup>E</sup>* <sup>−</sup> *En*) = <sup>−</sup> <sup>1</sup>

*π* Im *Trn*

where in the second equality an identity 1/(*x* + *iε*) = *P*(1/*x*) − *iπδ*(*x*) is used and trace is taken over all energy eigenstates {|*n*�}. Trading this tracing with the tracing over the eigenstates of coordinate operator {|*q*�}, we obtain a semiclassical approximation for the

*<sup>q</sup>*′′=*q*′ <sup>=</sup> <sup>−</sup> <sup>1</sup>

)

, *q*′ )

which dictates the periodic orbits. We obtain finally the periodic orbit theory formula for the

*Tr <sup>π</sup>h*¯ <sup>∑</sup> *n*�=0

Here the first sum runs over all primitive POs and the *n* sum counts the repetitions of each peridic orbit; *Tr*, *Sr*, and *lr* denote the period, action, and Maslov index of the primary PO,

In AKP *m* = 0 sector, the motion is restricted in a fixed plane which includes the heavy axis, and the problem essentially reduces to two dimensional one. (Later on the three dimensional feature is recovered only by the proper choice of the Maslov index [4]). As for AKP unstable periodic orbits, *<sup>M</sup>* has two eigenvalues *<sup>e</sup><sup>u</sup>* and *<sup>e</sup>*−*<sup>u</sup>* (hyperbolic case) and the determinant in

1

exp *in Sr <sup>h</sup>*¯ <sup>−</sup> *<sup>π</sup>* 2 *lr* 

[det((*Mr*)*<sup>n</sup>* − 1)]

*π* Im

*<sup>∂</sup>q*′′ , *<sup>p</sup>*′ <sup>=</sup> <sup>−</sup>*∂S*(*q*′′, *<sup>q</sup>*′

The integration over *<sup>q</sup>*′ can be again approximated by a stationary phase approximation.

 1 *E* + *iε* − *H*ˆ

Classical and Quantum Correspondence in Anisotropic Kepler Problem

*dq*′ ∑ Γ

*<sup>A</sup>*<sup>Γ</sup> exp *<sup>i</sup>*

*<sup>∂</sup>q*′ <sup>=</sup> *<sup>p</sup>*′′ <sup>−</sup> *<sup>p</sup>*′ (23)

1 2

<sup>2</sup> = −*i* sinh (*nu*/2) (25)

)

*h*¯ *S*<sup>Γ</sup> − *i ν*Γ 2 .

*<sup>∂</sup>q*′ (22)

A detailed account is given in [17].

#### **2.3. Periodic orbit theory**

#### *2.3.1. Periodic orbit theory and the density of state*

Let us recapitulate Gutzwiller's periodic orbit theory [4,20]. The starting point is Feynman's path integral formula for the propagator of a particle from *<sup>q</sup>*′ to *<sup>q</sup>*′′ during the time interval 0 to *T*;

$$\mathbb{K}\left(q'',q',T\right) \equiv \left\langle q'' \middle| \exp\left(-i\frac{H}{\hbar}T\right) \left| q' \right\rangle = \int\_{q'}^{q''} D\left[q\right] e^{\frac{i}{\hbar}\int\_{0}^{T} L(q,q,t)dt} \tag{17}$$

The Green function (response function) is given by the Fourier transformation of the propagator

$$G\left(q^{\prime\prime}, q^{\prime}, E\right) \equiv -\frac{i}{\hbar} \int\_0^\infty dt e^{\frac{iEt}{\hbar}} K\left(q^{\prime\prime}, q^{\prime}, T\right).$$

where *E* has infinitesimally small imaginary part for convergence. Thus we have

$$\mathcal{G}\left(q'',q',E\right) = \left\langle q'' \middle| \begin{array}{c} 1 \\ E+i\varepsilon-\hat{H} \end{array} \middle| q' \right\rangle = -\frac{i}{\hbar} \int\_0^\infty dt e^{\frac{iEt}{\hbar}} \left[ \int\_{q'}^{q''} D\left[q\right] e^{\frac{i}{\hbar} \int\_0^T L(q,q,t)dt} \right] \tag{18}$$

By a stationary approximation we obtain a semiclassical formula for the Green function

$$\tilde{G}\left(q'',q',E\right) \simeq \sum\_{\Gamma} A\_{\Gamma} \exp\left(\frac{i}{\hbar}\mathcal{S}\_{\Gamma} - i\frac{\nu\_{\Gamma}}{2}\right) \tag{19}$$

where Γ denotes a classically arrowed orbit, *ν*<sup>Γ</sup> is the number of conjugate points on the orbit, and the amplitude *A*Γ accounts for the Van Vleck determinant. Note that the principal function in (18) is changed into the action *<sup>S</sup>* <sup>=</sup> *<sup>q</sup>*′′ *<sup>q</sup>*′ *pdq* and the phase *iπ*/4 from the stationary point approximation is shifted into *A*Γ.

Now the density of states is given by

$$\rho(E) \equiv \sum\_{\mathfrak{n}} \delta(E - E\_{\mathfrak{n}}) = -\frac{1}{\pi} \text{Im} \left( T\_{r\_{\mathfrak{n}}} \left( \frac{1}{E + i\varepsilon - \hat{H}} \right) \right) \tag{20}$$

where in the second equality an identity 1/(*x* + *iε*) = *P*(1/*x*) − *iπδ*(*x*) is used and trace is taken over all energy eigenstates {|*n*�}. Trading this tracing with the tracing over the eigenstates of coordinate operator {|*q*�}, we obtain a semiclassical approximation for the DOS

$$\rho(\mathbf{E}) \approx -\frac{1}{\pi} \text{Im} \int d\boldsymbol{q}' \,\,\tilde{\mathbf{G}}(\boldsymbol{q}'', \boldsymbol{q}', \mathbf{E})\big|\_{\boldsymbol{q}'' = \boldsymbol{q}'} = -\frac{1}{\pi} \text{Im} \int d\boldsymbol{q}' \sum\_{\Gamma} A\_{\Gamma} \,\exp\left(\frac{i}{\hbar} \mathbf{S}\_{\Gamma} - i\frac{\nu\_{\Gamma}}{2}\right) \tag{21}$$

The integration over *<sup>q</sup>*′ can be again approximated by a stationary phase approximation. Because

$$p^{\prime\prime} = \frac{\partial S(q^{\prime\prime}, q^{\prime})}{\partial q^{\prime\prime}}, \ p^{\prime} = -\frac{\partial S(q^{\prime\prime}, q^{\prime})}{\partial q^{\prime}} \tag{22}$$

the stationary phase condition gives

6 Advances in Quantum Mechanics

**2.2. Husimi function**

state(CHS) |*q*0, *<sup>p</sup>*0� of the system [11]:

A detailed account is given in [17].

*2.3.1. Periodic orbit theory and the density of state*

*G <sup>q</sup>*′′, *<sup>q</sup>*′ , *E* ≡ − *i h*¯ <sup>∞</sup> 0

1 *E* + *iε* − *H*ˆ

*G*˜ *<sup>q</sup>*′′, *<sup>q</sup>*′ , *E* ≃ ∑ Γ

function in (18) is changed into the action *<sup>S</sup>* <sup>=</sup> *<sup>q</sup>*′′

point approximation is shifted into *A*Γ. Now the density of states is given by

**2.3. Periodic orbit theory**

*K <sup>q</sup>*′′, *<sup>q</sup>*′ , *T* ≡ *q*′′ exp −*i H h*¯ *T q*′ = *<sup>q</sup>*′′

0 to *T*;

propagator

*G <sup>q</sup>*′′, *<sup>q</sup>*′ , *E* = *q*′′ 

Husimi function is defined via the scalar product of the wave function |*ψ*� with a coherent

*<sup>ψ</sup>* (*q*0, *<sup>p</sup>*0) <sup>=</sup> |�*<sup>ψ</sup>* <sup>|</sup> *<sup>q</sup>*0, *<sup>p</sup>*0�|<sup>2</sup>

Let us recapitulate Gutzwiller's periodic orbit theory [4,20]. The starting point is Feynman's path integral formula for the propagator of a particle from *<sup>q</sup>*′ to *<sup>q</sup>*′′ during the time interval

The Green function (response function) is given by the Fourier transformation of the

where *E* has infinitesimally small imaginary part for convergence. Thus we have

 *q*′ <sup>=</sup> <sup>−</sup> *<sup>i</sup> h*¯ <sup>∞</sup> 0

By a stationary approximation we obtain a semiclassical formula for the Green function

*A*Γ exp(

where Γ denotes a classically arrowed orbit, *ν*<sup>Γ</sup> is the number of conjugate points on the orbit, and the amplitude *A*Γ accounts for the Van Vleck determinant. Note that the principal

*i h*¯ *S*<sup>Γ</sup> − *i ν*Γ

*dteiEt <sup>h</sup>*¯ *K*

*<sup>q</sup>*′ *<sup>D</sup>* [*q*]*<sup>e</sup>*

*<sup>q</sup>*′′, *<sup>q</sup>*′ , *T* 

*dteiEt h*¯ *<sup>q</sup>*′′

*i h*¯ *T*

*<sup>q</sup>*′ *<sup>D</sup>* [*q*]*<sup>e</sup>*

*i h*¯ *T* <sup>0</sup> *L*(*q*, *q*˙, *t*)*dt*

*<sup>q</sup>*′ *pdq* and the phase *iπ*/4 from the stationary

<sup>2</sup> ) (19)

. (16)

<sup>0</sup> *<sup>L</sup>*(*q*, *<sup>q</sup>*˙, *<sup>t</sup>*)*dt* (17)

(18)

*WHus*

$$0 = \frac{\partial \mathcal{S}(q', q')}{\partial q'} = p'' - p' \tag{23}$$

which dictates the periodic orbits. We obtain finally the periodic orbit theory formula for the DOS

$$\rho(E) \simeq \overline{\rho(E)} + \operatorname{Im} \sum\_{r \in \operatorname{PO}} \frac{T\_r}{\pi \hbar} \sum\_{n \neq 0} \frac{\exp\left\{ i n \left[ \frac{\mathbf{S}\_r}{\hbar} - \frac{\pi}{2} l\_r \right] \right\}}{\left[ \det((M\_r)^n - 1) \right]^{\frac{1}{2}}} \tag{24}$$

Here the first sum runs over all primitive POs and the *n* sum counts the repetitions of each peridic orbit; *Tr*, *Sr*, and *lr* denote the period, action, and Maslov index of the primary PO, and the matrix *M* stands for the monodromy matrix of the primary PO.

In AKP *m* = 0 sector, the motion is restricted in a fixed plane which includes the heavy axis, and the problem essentially reduces to two dimensional one. (Later on the three dimensional feature is recovered only by the proper choice of the Maslov index [4]). As for AKP unstable periodic orbits, *<sup>M</sup>* has two eigenvalues *<sup>e</sup><sup>u</sup>* and *<sup>e</sup>*−*<sup>u</sup>* (hyperbolic case) and the determinant in (24) is given by

$$\frac{1}{2}\left[\det(M^n - 1)\right]^{\frac{1}{2}} = -i\sinh\left(nu/2\right) \tag{25}$$

#### *2.3.2. Naming of a periodic orbit*

In AKP every PO can be coded by the sign of the heavy axis coordinate when the heavy axis is crossed by it. Note that number of the crossings must be even (2*nc*) for the orbit to close.

In this note we shall denote the PO according to Gutzwiller's identification number along with the Bernoulli sequence of POS. (See Table 1 in ref. [3] which gives a complete list1 of POs up to *nc* = 5 for the anisotropy *γ* = 0.2. ) For instance, PO36(+ + − + + −) is the identification number 6 among *nc* = 3 POs.

#### *2.3.3. The contribution of a periodic orbit to the density of state*

The contribution of a single periodic orbit *r* to the DOS is estimated by a resummation of the sum over the repetition j (after the approximation sinh *x* ≈ *ex*/2),

$$\left. \rho(E) \right|\_{r} \approx T\_{r} \sum\_{m} \frac{u\_{r} \hbar/2}{\left( \mathcal{S}\_{r} - 2 \pi \hbar (m + 1/4) \right)^{2} + \left( u\_{r} \hbar/2 \right)^{2}}. \tag{26}$$

This gives Lorentzian peak at

$$S\_{\Gamma} = 2\pi\hbar(m + l/4) \tag{27}$$

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**2.4. Scars as observed in the probability distributions and Husimi functions**

periodic orbit FPO (+ −) which reduces the Kepler ellipse orbit in the limit *γ* = 1.

to use the ability of POT predictions (24) to locate the scaring levels.

Let us start exploring the scars in AKP first by investigating the case of the fundamental

In Figure 1, we show the wave function squared in the *µν* plane and the Husimi distribution in the *µ p<sup>µ</sup>* plane. The FPO is shown by red line and compared with the probability distributions in the in the *µν* plane. At high anisotropy the orbit is largely distorted. Still at chosen energy levels (upper row) we find clear localization around the FPO for both anisotropies. In the lower row we have displayed other energy eigenstates. For these energy levels we see also characteristic probability distribution patterns but not around FPO. Now let us look at the Husimi distributions. In case of energies in the upper diagrams we see very clearly that around the Poincaré section of the FPO (the fixed points) the Husimi functions show clear scars, while in the lower we see anti-scars, the Husimi density is very low at the fixed points. It is clear that Husimi functions are superior observables. In this demonstration of scars we have scanned thousands of energy eigenstates and picked examples. Next task is

**Figure 1.** Scar and anti-scar phenomena with respect to the fundamental periodic orbit. The left set is for the anisotropy *γ* = 0.2 and right for *γ* = 0.6. In each set, the upper and lower row display prominent scar and anti-scar respectively, while the left and right columns exhibit the probability distribution on the *µν* plane and Husimi distribution on the *µ p<sup>µ</sup>* plane (*ν* = 0). The fundamental orbit is drawn by a red line on the *µν* plane and its Poincaré section on the *µ p<sup>µ</sup>* plane by red points. The respective eigenvalues are *E*786, *E*787, *E*438, *E*<sup>459</sup> in the m = 0, ℓ =even sector. Classical kinematical boundaries are shown by

yellow circles.

*2.4.1. The manifestation of the fundamental FPO*(+ −)

similar to the Bohr-Sommerfeld formula. In AKP the action *S* is given as

$$S\_{\mathcal{I}}(E) = \frac{T\_r}{\sqrt{-2E}}.\tag{28}$$

Hence the peak position of the Lorentzian form in the energy is given by

$$E\_{r,m} = -\frac{1}{2} \left( \frac{T\_r}{2\pi\hbar(m+1)} \right)^2,\tag{29}$$

where Maslov index *l* = 4 for three dimensions is taken.

We are aware that it is meant by (24) that the exact DOS with sharp delta function peaks on the energy axis corresponds to the sum of all PO contributions [20] (assuming convergence).

It is the collective addition of all POs that gives the dos. But, still, it is amusing to observe that the localization of wave functions occurs around the classical periodic orbits as we will see below.

<sup>1</sup> In [3, 5] an amazing approximation formula that gives a good estimate of the action of each periodic orbit from its symbolic code is presented. The trace formula has a difficulty coming from the proliferation of POs of long length. This approximation gives a nice way of estimating the sum. The table is created to fix the two parameters involved in the approximation. We thank Professor Gutzwiller for informing us of this formula a few years ago.

### **2.4. Scars as observed in the probability distributions and Husimi functions**

#### *2.4.1. The manifestation of the fundamental FPO*(+ −)

8 Advances in Quantum Mechanics

*2.3.2. Naming of a periodic orbit*

This gives Lorentzian peak at

see below.

identification number 6 among *nc* = 3 POs.

*2.3.3. The contribution of a periodic orbit to the density of state*

*<sup>ρ</sup>*(*E*)|*<sup>r</sup>* <sup>≈</sup> *Tr* ∑

sum over the repetition j (after the approximation sinh *x* ≈ *ex*/2),

*m*

similar to the Bohr-Sommerfeld formula. In AKP the action *S* is given as

Hence the peak position of the Lorentzian form in the energy is given by

*Er*,*<sup>m</sup>* <sup>=</sup> <sup>−</sup><sup>1</sup>

where Maslov index *l* = 4 for three dimensions is taken.

2

In AKP every PO can be coded by the sign of the heavy axis coordinate when the heavy axis is crossed by it. Note that number of the crossings must be even (2*nc*) for the orbit to close. In this note we shall denote the PO according to Gutzwiller's identification number along with the Bernoulli sequence of POS. (See Table 1 in ref. [3] which gives a complete list1 of POs up to *nc* = 5 for the anisotropy *γ* = 0.2. ) For instance, PO36(+ + − + + −) is the

The contribution of a single periodic orbit *r* to the DOS is estimated by a resummation of the

(*Sr* − 2*πh*¯(*m* + *l*/4))

*Sr*(*E*) = *Tr* √−2*<sup>E</sup>*

 *Tr* 2*πh*¯(*m* + 1)

We are aware that it is meant by (24) that the exact DOS with sharp delta function peaks on the energy axis corresponds to the sum of all PO contributions [20] (assuming convergence). It is the collective addition of all POs that gives the dos. But, still, it is amusing to observe that the localization of wave functions occurs around the classical periodic orbits as we will

<sup>1</sup> In [3, 5] an amazing approximation formula that gives a good estimate of the action of each periodic orbit from its symbolic code is presented. The trace formula has a difficulty coming from the proliferation of POs of long length. This approximation gives a nice way of estimating the sum. The table is created to fix the two parameters involved

in the approximation. We thank Professor Gutzwiller for informing us of this formula a few years ago.

2

*urh*¯ /2

<sup>2</sup> + (*urh*¯ /2)

*Sr* = 2*πh*¯(*m* + *l*/4) (27)

. (28)

, (29)

<sup>2</sup> . (26)

Let us start exploring the scars in AKP first by investigating the case of the fundamental periodic orbit FPO (+ −) which reduces the Kepler ellipse orbit in the limit *γ* = 1.

In Figure 1, we show the wave function squared in the *µν* plane and the Husimi distribution in the *µ p<sup>µ</sup>* plane. The FPO is shown by red line and compared with the probability distributions in the in the *µν* plane. At high anisotropy the orbit is largely distorted. Still at chosen energy levels (upper row) we find clear localization around the FPO for both anisotropies. In the lower row we have displayed other energy eigenstates. For these energy levels we see also characteristic probability distribution patterns but not around FPO. Now let us look at the Husimi distributions. In case of energies in the upper diagrams we see very clearly that around the Poincaré section of the FPO (the fixed points) the Husimi functions show clear scars, while in the lower we see anti-scars, the Husimi density is very low at the fixed points. It is clear that Husimi functions are superior observables. In this demonstration of scars we have scanned thousands of energy eigenstates and picked examples. Next task is to use the ability of POT predictions (24) to locate the scaring levels.

**Figure 1.** Scar and anti-scar phenomena with respect to the fundamental periodic orbit. The left set is for the anisotropy *γ* = 0.2 and right for *γ* = 0.6. In each set, the upper and lower row display prominent scar and anti-scar respectively, while the left and right columns exhibit the probability distribution on the *µν* plane and Husimi distribution on the *µ p<sup>µ</sup>* plane (*ν* = 0). The fundamental orbit is drawn by a red line on the *µν* plane and its Poincaré section on the *µ p<sup>µ</sup>* plane by red points. The respective eigenvalues are *E*786, *E*787, *E*438, *E*<sup>459</sup> in the m = 0, ℓ =even sector. Classical kinematical boundaries are shown by yellow circles.

#### *2.4.2. PO prediction and AKP Scars*

As for the FPO the POT works quite well. Thus for this test we have selected more complicated PO PO22 (+ + +−) and PO36 (+ + − + +−). These orbits wind around the heavy axis forth and back and presumably correspond to the bounce orbit in the billiard.<sup>2</sup> The top row in Fig.2 shows the prediction from POT – the contribution of the single orbit to the DOS (26). We observe clearly the peak regions of POT prediction contains at least one energy eigenstate which shows the scar of the orbit. On the other hand we have checked that the relevant orbit pattern does not appear in the non-peak region of the POT prediction.

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*<sup>µ</sup>* + *dp*<sup>2</sup> *ν*..

**2.5. Scars as analyzed by the scar strength function**

function (SSF) is presented [11]. It is defined as

*I PO <sup>n</sup>* = *PO*

In an extensive analysis of scars in the diamagnetic hydrogen, a tool called as scar strength

This quantity is exploiting to what extent a given PO is inducing localization along it in the Husimi function of a given energy eigenstate. Then spectral scar strength function is

> *n I PO*

Let explore the region of high anisotropy (*γ* = 0.2) where the classical phase space is occupied by the unstable periodic orbits and chaoticity is rather high. We explore this region

We start from FPO (+ −) in Fig. 3. The upper is the POT prediction curve, the middle is the SSF along with real eigenvalues and the bottom is as usual a direct comparison of FPO with wavefunctions squared as well as Husimi functions. The SSF is the quantum measure of scaring of a particular PO in consideration, while the FPO prediction is composed from purely classical information for the PO. When the curve of the contribution from a PO peaks, the SSF either peaks or reaches its minimum (10−10). The agreement in the energy values of the peaks (or dips) is quite remarkable. But we do not know why anti-scar occurs here. This anti-scar is interesting in that it produces a bright hallow just of the same size and position

Let us now examine the case of several POs simultaneously in Fig. 4. The profiles, the SSF, and the Fourier transforms to the action space are listed in three columns. As Wintgen et al. write as 'the scars are the rules rather than exception' [11] we find that particular energy eigenstates give salient high scar function value while the other states give very low value of order even 10<sup>−</sup>10. Further more the Fourier transform *<sup>I</sup>PO*(*S*) of *<sup>I</sup>PO*(*E*)shows sequential

and the action of the PO). They agree excellently; the POs live in quantum theory.

*<sup>r</sup>* and *SCl*

*µ*, *ν*, *pµ*, *p<sup>ν</sup>*

*<sup>n</sup> <sup>δ</sup>* (*<sup>E</sup>* <sup>−</sup> *En*)

 *PO d*4*s* −<sup>1</sup>

Classical and Quantum Correspondence in Anisotropic Kepler Problem

*dµ*<sup>2</sup> + *dν*<sup>2</sup> + *dp*<sup>2</sup>

*<sup>r</sup>* (the measured spacing of the orbit

*d*4*s WHus ψ<sup>n</sup>* 

where the integral is to be performed along the PO with *d*4*s* =

*I*

*PO* (*E*) = ∑

This shows how the given PO affects each energy eigenstate in one function.

*2.5.1. Scar strength functions*

introduced as

*2.5.2. The use of SSF IPO*(*E*)

by the ability of *IPO*(*E*).

of the scar but the central core is missing.

*2.5.3. IPO*(*E*) *for various POs and their Fourier transform*

peaks at equal ∆*S*. We compare in Fig. 5 ∆*SQM*

In this analysis the Husimi function again yields unmistakable information on the scaring.

**Figure 2.** *γ* = 0.6, *l* = even, *m* = 0. The upper diagrams: The red and green curves are contributions to the density of states *ρ*(*E*) from bouncing-type periodic orbits PO22 (+ + +−) and PO36 (+ + − + +−) respectively and the peak positions are compared with the *l* even *m* = 0 energy levels from matrix diagonalization (WMB with tensored harmonic oscillator basis). The lower: The quantum scars of these periodic orbits are exhibited on the probability distributions and the Husimi functions. (cf. Fig. 1).

<sup>2</sup> We thank Professor Toshiya Takami for explaining his articles [21,22] and pointing us this point.

#### **2.5. Scars as analyzed by the scar strength function**

#### *2.5.1. Scar strength functions*

10 Advances in Quantum Mechanics

Fig. 1).

*2.4.2. PO prediction and AKP Scars*

As for the FPO the POT works quite well. Thus for this test we have selected more complicated PO PO22 (+ + +−) and PO36 (+ + − + +−). These orbits wind around the heavy axis forth and back and presumably correspond to the bounce orbit in the billiard.<sup>2</sup> The top row in Fig.2 shows the prediction from POT – the contribution of the single orbit to the DOS (26). We observe clearly the peak regions of POT prediction contains at least one energy eigenstate which shows the scar of the orbit. On the other hand we have checked that the relevant orbit pattern does not appear in the non-peak region of the POT prediction. In this analysis the Husimi function again yields unmistakable information on the scaring.

**Figure 2.** *γ* = 0.6, *l* = even, *m* = 0. The upper diagrams: The red and green curves are contributions to the density of states *ρ*(*E*) from bouncing-type periodic orbits PO22 (+ + +−) and PO36 (+ + − + +−) respectively and the peak positions are compared with the *l* even *m* = 0 energy levels from matrix diagonalization (WMB with tensored harmonic oscillator basis). The lower: The quantum scars of these periodic orbits are exhibited on the probability distributions and the Husimi functions. (cf.

<sup>2</sup> We thank Professor Toshiya Takami for explaining his articles [21,22] and pointing us this point.

In an extensive analysis of scars in the diamagnetic hydrogen, a tool called as scar strength function (SSF) is presented [11]. It is defined as

$$I\_n^{PO} = \oint\_{PO} d^4s \,\mathcal{W}\_{\psi\_n}^{Hus} \left(\mu\_\prime \,\nu\_\prime \, p\_{\mu\prime} \, p\_\nu\right) \left(\oint\_{PO} d^4s\right)^{-1}$$

where the integral is to be performed along the PO with *d*4*s* = *dµ*<sup>2</sup> + *dν*<sup>2</sup> + *dp*<sup>2</sup> *<sup>µ</sup>* + *dp*<sup>2</sup> *ν*.. This quantity is exploiting to what extent a given PO is inducing localization along it in the Husimi function of a given energy eigenstate. Then spectral scar strength function is introduced as

$$I^{PO}\left(E\right) = \sum\_{n} I\_n^{PO} \delta\left(E - E\_n\right),$$

This shows how the given PO affects each energy eigenstate in one function.

#### *2.5.2. The use of SSF IPO*(*E*)

Let explore the region of high anisotropy (*γ* = 0.2) where the classical phase space is occupied by the unstable periodic orbits and chaoticity is rather high. We explore this region by the ability of *IPO*(*E*).

We start from FPO (+ −) in Fig. 3. The upper is the POT prediction curve, the middle is the SSF along with real eigenvalues and the bottom is as usual a direct comparison of FPO with wavefunctions squared as well as Husimi functions. The SSF is the quantum measure of scaring of a particular PO in consideration, while the FPO prediction is composed from purely classical information for the PO. When the curve of the contribution from a PO peaks, the SSF either peaks or reaches its minimum (10−10). The agreement in the energy values of the peaks (or dips) is quite remarkable. But we do not know why anti-scar occurs here. This anti-scar is interesting in that it produces a bright hallow just of the same size and position of the scar but the central core is missing.

#### *2.5.3. IPO*(*E*) *for various POs and their Fourier transform*

Let us now examine the case of several POs simultaneously in Fig. 4. The profiles, the SSF, and the Fourier transforms to the action space are listed in three columns. As Wintgen et al. write as 'the scars are the rules rather than exception' [11] we find that particular energy eigenstates give salient high scar function value while the other states give very low value of order even 10<sup>−</sup>10. Further more the Fourier transform *<sup>I</sup>PO*(*S*) of *<sup>I</sup>PO*(*E*)shows sequential peaks at equal ∆*S*. We compare in Fig. 5 ∆*SQM <sup>r</sup>* and *SCl <sup>r</sup>* (the measured spacing of the orbit and the action of the PO). They agree excellently; the POs live in quantum theory.

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Classical and Quantum Correspondence in Anisotropic Kepler Problem

**Figure 4.** *γ* = 0.2 Profiles of PO, the scar strength function, and Fourier transformation of scar strength function to the action

space.

E = - 0.002567 223 E = - 0.002398 239 E = - 0.002226 257 E = - 0.002073 275 E = - 0.001958 291

**Figure 3.** Contribution of FPO (+ −) to AKP. Upper two diagrams: The (+ −) contribution as a function of energy predicted by POT compared with the scar strength extraction from each of the energy eigenstates. Lower two diagrams: the scaring status of (+ −) at levels indicated by arrows are exhibited with respect to wave functions squared and Husimi functions. (cf. Fig. 1 and Fig. 2). Scar and anti-scar appear alternatively.

#### **2.6. Direct phase space observation of Scaring orbit**

The scar strength function is a useful tool which gives a list of numerical values which shows succinctly to which eigenstates the periodic orbit exerts its effect strongly. But we certainly want also visualized picture how the PO turns up in the 4 dimensional phase space. (Because *H* = const., the actual independent variables are three, and we choose *µ*, *ν*, *pµ*.) The sample pictures are shown in Fig.6.

### **3. Robustness of Scaring under the Variation of Anisotropy Parameter**

#### **3.1. Swap of the patterns under avoiding crossings**

It is well known that the patterns of wave functions (and of Husimi distributions) are swapped between the energy eigenstates via the avoiding level crossings, which is easy to demonstrate in terms a simple coupling model of two levels. Fig. 7 is a typical example of this phenomenon.

It is shown in [21,22] that with the aid of the diabatic transformation one can trace the localization on the transformed basis until very near to the minimum gap with an explicit evidence in the billiard scars. We have verified this issue in AKP. Furthermore it is conjectured that the long periodic orbits may interpolates two shorter orbits and they may be the cause of the avoiding crossings in this way. We are testing this conjecture in AKP.

12 Advances in Quantum Mechanics

Fig. 1 and Fig. 2). Scar and anti-scar appear alternatively.

pictures are shown in Fig.6.

this phenomenon.

**2.6. Direct phase space observation of Scaring orbit**

**3.1. Swap of the patterns under avoiding crossings**

γ=0.2 FPO(+-) 210-302th

E = - 0.002567 223 E = - 0.002398 239 E = - 0.002226 257 E = - 0.002073 275 E = - 0.001958 291

**Figure 3.** Contribution of FPO (+ −) to AKP. Upper two diagrams: The (+ −) contribution as a function of energy predicted by POT compared with the scar strength extraction from each of the energy eigenstates. Lower two diagrams: the scaring status of (+ −) at levels indicated by arrows are exhibited with respect to wave functions squared and Husimi functions. (cf.

The scar strength function is a useful tool which gives a list of numerical values which shows succinctly to which eigenstates the periodic orbit exerts its effect strongly. But we certainly want also visualized picture how the PO turns up in the 4 dimensional phase space. (Because *H* = const., the actual independent variables are three, and we choose *µ*, *ν*, *pµ*.) The sample

**3. Robustness of Scaring under the Variation of Anisotropy Parameter**

It is well known that the patterns of wave functions (and of Husimi distributions) are swapped between the energy eigenstates via the avoiding level crossings, which is easy to demonstrate in terms a simple coupling model of two levels. Fig. 7 is a typical example of

It is shown in [21,22] that with the aid of the diabatic transformation one can trace the localization on the transformed basis until very near to the minimum gap with an explicit evidence in the billiard scars. We have verified this issue in AKP. Furthermore it is conjectured that the long periodic orbits may interpolates two shorter orbits and they may be the cause of the avoiding crossings in this way. We are testing this conjecture in AKP.

**Figure 4.** *γ* = 0.2 Profiles of PO, the scar strength function, and Fourier transformation of scar strength function to the action space.

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**Figure 7.** Avoiding level crossing between 260th and 261th energy eigen states under the variation of the anisotropy *γ* ∈

2. On the other hand the peak locations (29) of the DOS as predicted by a single PO *change smoothly* with the anisotropy and the scar tends to be observed in the energy eigenstate

Aren't the two issues in contradiction? We have found that they can live together (within approximation of the fluctuation size). Most important point is that the swap of the localization patterns at avoiding crossing is in harmony with the transportation of them by the responsible PO orbits. Besides the POT prediction (29) does not imply the exact location

Let us explain this by Fig. 8. Here the anisotropy *γ* is varied from 0.6 to 0.7 with inclement 0.001. As for (1) we indeed observe both random fluctuation of energy levels as well as many

[0.6, 0.606]. Both the wave function squared and Husimi functions are swapped around the avoiding crossing.

It has some allowance as recognized by the width of the *modulation* of SSF[11].

around the peak position in the DOS as in Fig.2.

of theappearance of the scar.

**Figure 5.** Plot of (∆*SQM <sup>r</sup>* , *SCl <sup>r</sup>* ) for periodic orbits *<sup>r</sup>* = FPO, PO22, PO23, PO36, where <sup>∆</sup>*SQM <sup>r</sup>* is measured from the third column of Fig.4 and *SCl*

*<sup>r</sup>* is the action value of the classical orbit.

**Figure 6.** Two samples of density plot of Husimi functions in the 3 dimensional *<sup>µ</sup>* <sup>−</sup> *<sup>ν</sup>* <sup>−</sup> *<sup>p</sup><sup>µ</sup>* space. Left: *<sup>γ</sup>* =0.2, *<sup>W</sup>*Hus for *<sup>E</sup>*<sup>786</sup> <sup>=</sup> <sup>−</sup>0.0007201. Red orbit is the FPO (+ <sup>−</sup>). Right: *<sup>γ</sup>*=0.6, *<sup>W</sup>*Hus for *<sup>E</sup>*<sup>579</sup> <sup>=</sup> <sup>−</sup>0.0005681. Blue and green orbits are respectively PO23 (+ + −−) and PO37 (+ + − − + −).

#### **3.2. Robust association of localization with periodic orbits**

We have posed the following question in the introduction of this chapter.

1. Energy levels exhibit randomness at high anisotropy and change *their values randomly* repeating successive avoiding crossings when the anisotropy parameter is varied gradually.

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14 Advances in Quantum Mechanics

**Figure 5.** Plot of (∆*SQM*

column of Fig.4 and *SCl*

gradually.

*<sup>r</sup>* , *SCl*

respectively PO23 (+ + −−) and PO37 (+ + − − + −).

**3.2. Robust association of localization with periodic orbits**

We have posed the following question in the introduction of this chapter.

*<sup>r</sup>* is the action value of the classical orbit.

*<sup>r</sup>* ) for periodic orbits *<sup>r</sup>* = FPO, PO22, PO23, PO36, where <sup>∆</sup>*SQM*

**Figure 6.** Two samples of density plot of Husimi functions in the 3 dimensional *<sup>µ</sup>* <sup>−</sup> *<sup>ν</sup>* <sup>−</sup> *<sup>p</sup><sup>µ</sup>* space. Left: *<sup>γ</sup>* =0.2, *<sup>W</sup>*Hus for *<sup>E</sup>*<sup>786</sup> <sup>=</sup> <sup>−</sup>0.0007201. Red orbit is the FPO (+ <sup>−</sup>). Right: *<sup>γ</sup>*=0.6, *<sup>W</sup>*Hus for *<sup>E</sup>*<sup>579</sup> <sup>=</sup> <sup>−</sup>0.0005681. Blue and green orbits are

1. Energy levels exhibit randomness at high anisotropy and change *their values randomly* repeating successive avoiding crossings when the anisotropy parameter is varied

*<sup>r</sup>* is measured from the third

**Figure 7.** Avoiding level crossing between 260th and 261th energy eigen states under the variation of the anisotropy *γ* ∈ [0.6, 0.606]. Both the wave function squared and Husimi functions are swapped around the avoiding crossing.

2. On the other hand the peak locations (29) of the DOS as predicted by a single PO *change smoothly* with the anisotropy and the scar tends to be observed in the energy eigenstate around the peak position in the DOS as in Fig.2.

Aren't the two issues in contradiction? We have found that they can live together (within approximation of the fluctuation size). Most important point is that the swap of the localization patterns at avoiding crossing is in harmony with the transportation of them by the responsible PO orbits. Besides the POT prediction (29) does not imply the exact location of theappearance of the scar.

It has some allowance as recognized by the width of the *modulation* of SSF[11].

Let us explain this by Fig. 8. Here the anisotropy *γ* is varied from 0.6 to 0.7 with inclement 0.001. As for (1) we indeed observe both random fluctuation of energy levels as well as many avoided crossings. As for (2), we have picked the bouncing-type periodic orbit PO22 as an example. The predicted peak position (29) of its contribution to DOS varies with the change of *γ* as shown by a red (almost straight) curve. This PO22 produces a salient *cross-shaped scar* at *E*<sup>260</sup> (and *E*275) at *γ*=0.6. We have investigated how the cross-shaped scar travels in the spectrum space suffering many avoiding crossings. It reaches at *E*<sup>276</sup> (and *E*291) at *γ* =0.7 10.5772/55208

39

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and the track in between is enclosed by a belt shown by two dashed lines. We clearly observe that the belt is closely associated by the POT prediction curve. In this sense the association

Classical and Quantum Correspondence in Anisotropic Kepler Problem

We have presented ample examples of scaring phenomena for the first time in AKP. Especially we have found how the fluctuation of energy levels and smooth POT prediction for the scaring levels are compromised by using the advantage of AKP endowed by a chaoticity

Although the theme is old, the scaring phenomenon is fascinating and we hope this

Both KK and TS thank Professor Toshiya Takami for sharing his wisdom with us.

Department of Physics, School of Science and Technology, Meiji University, Japan

problem, Journal of Mathematical Physics 18: 806-823.

[1] Gutzwiller, M. C. (1977). Bernoulli sequences and trajectories in the anisotropic Kepler

[2] Devaney, R. L. (1979). Collision Orbits in the Anisotropic Kepler Problem, Inventions

[3] Gutzwiller, M. C. (1981). Periodic orbits in the anisotropic Kepler problem, in Devaney, R. L. & Nitecki, Z. H. (ed.) Classical Mechanics and Dynamical systems, Marcel Dekker,

[4] Gutzwiller, M. C. (1971). Periodic orbits and classical quantization conditions, Journal

[5] Gutzwiller, M. C. (1980). Classical quantization of a hamiltonian with ergodic behavior,

[6] Gutzwiller, M. C. (1982). The quantization of a classically ergodic system, Physica D 5:

[7] Gutzwiller, M. C. (1990). Chaos in Classical and Quantum Mechanics, Springer.

is robust.

**4. Conclusion**

changing parameter.

**Acknowledgement**

**Author details**

**References**

contribution fulfills a gap in the literature.

Keita Sumiya, Hisakazu Uchiyama, Kazuhiro Kubo and Tokuzo Shimada

math. 45: 221-251.

New York, pp. 69-90.

183-207.

of Mathematical Physics 12: 343-358.

Physical Review Letters 45: 150-153.

**Figure 8.** The spectrum lines in the wide interval *γ* ∈ [0.6, 0.7] investigated with increment 0.001. The cross-shaped scar by PO22 travels within a belt bounded by two dashed lines. The POT prediction (24) is exhibited by a red curve.

and the track in between is enclosed by a belt shown by two dashed lines. We clearly observe that the belt is closely associated by the POT prediction curve. In this sense the association is robust.

### **4. Conclusion**

16 Advances in Quantum Mechanics

276th

275th

274th

273th

262th

261th

260th

259th

258th

277th

γ=0.6

avoided crossings. As for (2), we have picked the bouncing-type periodic orbit PO22 as an example. The predicted peak position (29) of its contribution to DOS varies with the change of *γ* as shown by a red (almost straight) curve. This PO22 produces a salient *cross-shaped scar* at *E*<sup>260</sup> (and *E*275) at *γ*=0.6. We have investigated how the cross-shaped scar travels in the spectrum space suffering many avoiding crossings. It reaches at *E*<sup>276</sup> (and *E*291) at *γ* =0.7

γ=0.65

283th

284th

E PO 22(+++-)

282th

γ 0.65

**Figure 8.** The spectrum lines in the wide interval *γ* ∈ [0.6, 0.7] investigated with increment 0.001. The cross-shaped scar by

PO22 travels within a belt bounded by two dashed lines. The POT prediction (24) is exhibited by a red curve.

281th

278th

γ=0.7

277th

276th

275th

274th

266th

267th

268th

269th

270th

γ=0.65

We have presented ample examples of scaring phenomena for the first time in AKP. Especially we have found how the fluctuation of energy levels and smooth POT prediction for the scaring levels are compromised by using the advantage of AKP endowed by a chaoticity changing parameter.

Although the theme is old, the scaring phenomenon is fascinating and we hope this contribution fulfills a gap in the literature.

### **Acknowledgement**

Both KK and TS thank Professor Toshiya Takami for sharing his wisdom with us.

### **Author details**

Keita Sumiya, Hisakazu Uchiyama, Kazuhiro Kubo and Tokuzo Shimada

Department of Physics, School of Science and Technology, Meiji University, Japan

### **References**


[8] Wintgen, D., Marxer, H. & Briggs, J. S. (1987). Efficient quantisation scheme for the anisotropic Kepler problem, Journal of Physics A 20: L965-L968.

**Chapter 3**

**Provisional chapter**

**Charathéodory's "Royal Road" to the Calculus of**

**Quantum Physics**

**Quantum Physics**

http://dx.doi.org/10.5772/53843

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

Francisco De Zela

Francisco De Zela

10.5772/53843

**1. Introduction**

**Variations: A Possible Bridge Between Classical and**

**Variations: A Possible Bridge Between Classical and**

Constantin Carathéodory, a Greek-born, well-known German mathematician, is rarely mentioned in connection to physics. One of his most remarkable contributions to mathematics is his approach to the calculus of variations, the so-called Carathéodory's "royal road" [1]. Among physicists, Carathéodory's name is most frequently related to his contributions to the foundations of thermodynamics [2] and to topics of classical optics, though, as a pupil of Hermann Minkowski, he also worked on the development of especial relativity. In our opinion, however, Caratheodory's formulation of the variational problem deserves to be better known among physicists. For mathematicians, features like rigor and non-redundancy of basic postulates are of utmost importance. Among physicists, a more pragmatic attitude is usually behind efforts towards a theoretical construction, whose principal merit should be to offer an adequate description of Nature. Such a construction must provide us with predictive power. Rigor of the theoretical construction is necessary but not sufficient. Elegance – which is how non-redundancy and simplicity usually manifest themselves – can be sometimes just a welcome feature. Some other times, however, elegance has become a guiding principle when guessing at how Nature works. Nevertheless, once the basic principles of a theoretical construction have been identified, elegance may recede in favor of clarity, and redundancy might become acceptable. Such differences between the perspectives adopted by mathematicians and physicists have been presumably behind the different weight they have assigned to Caratheodory's achievements in the calculus of variations. To be sure, variational calculus does play a central role in physics, nowadays even more than ever before. It is by seeking for the appropriate Lagrangian that we hope to find out the most basic principles ruling physical behavior. Concepts like Feynman's path integral

> ©2012 De Zela, 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 De Zela; 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 De Zela, 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.

**Charathéodory's "Royal Road" to the Calculus of**


**Provisional chapter**

#### **Charathéodory's "Royal Road" to the Calculus of Variations: A Possible Bridge Between Classical and Quantum Physics Charathéodory's "Royal Road" to the Calculus of Variations: A Possible Bridge Between Classical and Quantum Physics**

Francisco De Zela Francisco De Zela

18 Advances in Quantum Mechanics

40 Advances in Quantum Mechanics


024101.

amg73/oslo2007.ppt.

Review Letters 60: 971-974.

[8] Wintgen, D., Marxer, H. & Briggs, J. S. (1987). Efficient quantisation scheme for the

[9] Wintgen, D. & Marxer, H. (1988). Level statistics of a quantized cantori system, Physical

[10] Heller, E. J. (1984). Bound-state eigenfunctions of classically chaotic hamiltonian

[11] Müller, K. & Wintgen, D. (1994). Scars in wavefunctions of the diamagnetic Kepler problem, Journal of Physics B: Atomic, Molecular and Optical Physics 27: 2693-2718.

[12] García-García, A. M. & Verbaarschot, J. J. M. (2003). Critical statistics in quantum chaos and Calogero-Sutherland model at finite temperature, Physical Review E 67: 046104-1

[13] García-García, A. M. (2007). Universality in quantum chaos and the one parameter scaling theory (a power point of a talk). http://www.tcm.phy.cam.ac.uk/

[14] García-García, A. M. & Wang, J. (2008). Universality in quantum chaos and the one-parameter scaling theory, Physical Review Letters 100: 070603-1 - 070603-4.

[15] Chen, Z. et al. (2009). Realization of Anisotropic Diamagnetic Kepler Problem in a Solid

[16] Zhou, W. (2010). Magnetic Field Control of the Quantum Chaotic Dynamics of Hydrogen Analogues in an Anisotropic Crystal Field, Physical Review Letters 105:

[17] Kubo, K. & Shimada, T. (2011). Theoretical Concepts of Quantum Mechanics, InTech.

[18] Kustaanheimo, P. & Stiefel, E. (1965). Perturbation theory of Kepler motion based on spinor regularization, Journal für die reine und angewandte Mathematik 218: 204-219.

[19] Husimi, K. (1940). Some formal properties of the density matrix, Proceedings of the

[20] Wintgen, D. (1988). Semiclassical Path-Integral Quantization of Nonintegrable

[21] Takami, T. (1992). Semiclassical Interpretation of Avoided Crossings for Classically

[22] Takami, T. (1995). Semiclassical study of avoided crossings, Physical Review E 52: 2434.

State Environment , Physical Review Letters 102: 244103.

Physico-Mathematical Society of Japan 22: 264-314.

Hamiltonian Systems, Physical Review Letters 61: 1803-1806.

Nonintegrable Systems, Physical Review Letters 68: 3371-3374.

systems: scars of periodic orbits, Physical Review Letters 53: 1515-1518.

anisotropic Kepler problem, Journal of Physics A 20: L965-L968.

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

http://dx.doi.org/10.5772/53843 10.5772/53843

### **1. Introduction**

Constantin Carathéodory, a Greek-born, well-known German mathematician, is rarely mentioned in connection to physics. One of his most remarkable contributions to mathematics is his approach to the calculus of variations, the so-called Carathéodory's "royal road" [1]. Among physicists, Carathéodory's name is most frequently related to his contributions to the foundations of thermodynamics [2] and to topics of classical optics, though, as a pupil of Hermann Minkowski, he also worked on the development of especial relativity. In our opinion, however, Caratheodory's formulation of the variational problem deserves to be better known among physicists. For mathematicians, features like rigor and non-redundancy of basic postulates are of utmost importance. Among physicists, a more pragmatic attitude is usually behind efforts towards a theoretical construction, whose principal merit should be to offer an adequate description of Nature. Such a construction must provide us with predictive power. Rigor of the theoretical construction is necessary but not sufficient. Elegance – which is how non-redundancy and simplicity usually manifest themselves – can be sometimes just a welcome feature. Some other times, however, elegance has become a guiding principle when guessing at how Nature works. Nevertheless, once the basic principles of a theoretical construction have been identified, elegance may recede in favor of clarity, and redundancy might become acceptable. Such differences between the perspectives adopted by mathematicians and physicists have been presumably behind the different weight they have assigned to Caratheodory's achievements in the calculus of variations. To be sure, variational calculus does play a central role in physics, nowadays even more than ever before. It is by seeking for the appropriate Lagrangian that we hope to find out the most basic principles ruling physical behavior. Concepts like Feynman's path integral

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 De Zela; 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 De Zela, 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 De Zela, licensee InTech. This is an open access chapter distributed under the terms of the Creative

have become basic tools for the calculation of probability amplitudes of different processes, as well as for suggesting new developments in quantum field theory. Symmetry, such a basic concept underlying those aspects of Nature which appear to us in the form of interactions among fundamental particles, is best accounted for within the framework of a variational principle.

10.5772/53843

43

http://dx.doi.org/10.5772/53843

mechanical: superconductivity and the response of a sample of charged particles to an external magnetic field. The London equations of superconductivity were originally understood as an *ad-hoc* assumption, with quantum mechanics lying at its roots. On the other hand, according to classical mechanics there can be no diamagnetism and no paramagnetism at all. We will deal with these two issues, showing how it is possible to classically derive the London equations and the existence of magnetic moments. This is not to say that there is a classical explanation of these phenomena. What is meant is that, specifically, the London equations of superconductivity can be derived from a classical Lagrangian. It is worth noting that a previous attempt in this direction, due to W. F. Edwards [5], proved false [6–8]. The failure was due to an improper application of the principle of least action. The approach presented here is free from any shortcomings. It leads to the London equations both in the relativistic and in the nonrelativistic domains. It should be stressed that this does not explain the appearance of the superconducting phase. It only shows how the London equations follow from a purely classical approach. Also the expulsion of a magnetic field from the interior of a superconductor, i.e., the Meissner effect, follows. That is, perfect diamagnetism can be explained classically, as has been shown recently [9] but under restricted conditions. This is in contradiction with the Bohr-van Leeuwen theorem, according to which there can be no classical magnetism [10]. This point has been recently discussed (see, e.g., [11]) and it has been shown that the Bohr-van Leeuwen theorem does not hold when one uses the Darwin Hamiltonian, which was proposed back in 1920. The Darwin Hamiltonian contains additional terms with respect to the standard one that is used to describe a charged particle interacting with an electromagnetic field. Applying Carathéodory's approach it can be shown that it is unnecessary to go beyond the standard Hamiltonian or Lagrangian to conclude that a magnetic response may be explained classically. The main point is that the Bohr-van Leeuwen theorem did not consider a constant of the motion which in Carathéodory's approach naturally arises. By considering this constant of the motion, the possibility of

Charathéodory's "Royal Road" to the Calculus of Variations: A Possible Bridge Between Classical and Quantum Physics

magnetic response in a sample of charged particles automatically appears.

Lagrangian density L. The Euler-Lagrange equations are, respectively,

*∂ ∂x<sup>µ</sup>*

*d dt <sup>∂</sup><sup>L</sup> ∂x*˙*i* <sup>−</sup> *<sup>∂</sup><sup>L</sup>*

), with *i* = 1, . . . , *n*, and

*∂*L *<sup>∂</sup>*(*∂µψi*)  <sup>−</sup> *<sup>∂</sup>*<sup>L</sup>

**2. Carathéodory's royal road**

**2.1. Preliminaries**

for a Lagrangian *L*(*t*, *x<sup>i</sup>*

, *x*˙ *i*

After dealing with the above two cases, the rest of the chapter will be devoted to show how gauge invariance can be considered within Carathéodory's framework. This may have some inspiring effect for future work aiming at exploring the quantum-classical correspondence.

Let us begin by recapitulating the approach usually employed in physics. For the sake of describing a particle's motion we use a variational principle based on a Lagrangian *L*. When describing the dynamics of a field we use instead a variational principle based on a

*<sup>∂</sup>xi* <sup>=</sup> 0, (1)

*∂ψ<sup>i</sup>* <sup>=</sup> 0, (2)

Within the domain of classical physics, only two fundamental interactions have been addressed: the gravitational and the electromagnetic interactions. The theoretical construction may correspondingly rest on two variational principles, one for gravitation and the other for electromagnetism. These principles lead to so-called "equations of motion": the Maxwell and the Lorentz equations for electromagnetism, and the Einstein and the geodesic equations for gravitation. All these differential equations can be derived as Euler-Lagrange equations from the appropriate Lagrangian or Lagrangian density.

The usual approach to variational calculus in physics starts by considering small variations of a curve which renders extremal the action integral *Ldt*, with *L* being the Lagrangian. This leads to the Euler-Lagrange equations of motion. By submitting *L* to a Legendre transformation one obtains the corresponding Hamiltonian, in terms of which the Hamilton equations of motion can be established. By considering canonical transformations of these equations, one arrives at the Hamilton-Jacobi equation for a scalar function *S*(*t*, *x*). It is last one that has been used to connect the classical approach with the quantum one, e.g., in Madelung's hydrodynamic model [3] or in Bohm's "hidden variables" approach [4]. This appears natural, because both the Hamilton-Jacobi and the Schrödinger equation rule the dynamics of quantities like *S*(*t*, *x*) and *ψ*(*t*, *x*), respectively, which are scalar fields. Their scalar nature is in fact irrelevant; they could be tensors and spinors. The relevant issue is that while the Euler-Lagrange and the Hamilton equations refer to a single path, quantum equations address a field. The quantum-classical connection thus requires making a field out of single paths, something which occurs by going to the Hamilton-Jacobi equation, or else by establishing a path-integral formulation, as Feynman did. The latter considers a family of trajectories and assigns a probability to each of them. Now, Carathéodory's approach has the advantage of addressing right from the start a field of extremals. In fact, as the calculus of variations shows, a solution of the extremal problem exists only when the sought-after extremal curve can be embedded in a field of similar extremals. Carathéodory exploited this fact by introducing the concepts of "equivalent variational problems" and the "complete figure". It is then possible to elegantly derive from a single statement the Euler-Lagrange and the Hamilton equations, as well as the Hamilton-Jacobi equation, all of them as field equations. The familiar Euler-Lagrange and Hamilton equations can be obtained afterwards by singling out a particular extremal of the field. But – as already stressed – it is not the inherent elegance of the formulation what drives our interest towards Carathéodory's approach. It is rather its potentiality as a bridge between classical and quantum formulations what should be brought to the fore. Indeed, Carathéodory's approach can provide new insights into the connection between classical and quantum formulations. These insights could go beyond those already known, which were obtained by extending the Hamilton-Jacobi equation with the inclusion of additional terms. By dealing with the other field equations that appear within Carathéodory's approach, one may hope to gain additional insight.

The present chapter, after discussing Carathéodory's approach, shows how one can classically explain two phenomena that have been understood as being exclusively quantum mechanical: superconductivity and the response of a sample of charged particles to an external magnetic field. The London equations of superconductivity were originally understood as an *ad-hoc* assumption, with quantum mechanics lying at its roots. On the other hand, according to classical mechanics there can be no diamagnetism and no paramagnetism at all. We will deal with these two issues, showing how it is possible to classically derive the London equations and the existence of magnetic moments. This is not to say that there is a classical explanation of these phenomena. What is meant is that, specifically, the London equations of superconductivity can be derived from a classical Lagrangian. It is worth noting that a previous attempt in this direction, due to W. F. Edwards [5], proved false [6–8]. The failure was due to an improper application of the principle of least action. The approach presented here is free from any shortcomings. It leads to the London equations both in the relativistic and in the nonrelativistic domains. It should be stressed that this does not explain the appearance of the superconducting phase. It only shows how the London equations follow from a purely classical approach. Also the expulsion of a magnetic field from the interior of a superconductor, i.e., the Meissner effect, follows. That is, perfect diamagnetism can be explained classically, as has been shown recently [9] but under restricted conditions. This is in contradiction with the Bohr-van Leeuwen theorem, according to which there can be no classical magnetism [10]. This point has been recently discussed (see, e.g., [11]) and it has been shown that the Bohr-van Leeuwen theorem does not hold when one uses the Darwin Hamiltonian, which was proposed back in 1920. The Darwin Hamiltonian contains additional terms with respect to the standard one that is used to describe a charged particle interacting with an electromagnetic field. Applying Carathéodory's approach it can be shown that it is unnecessary to go beyond the standard Hamiltonian or Lagrangian to conclude that a magnetic response may be explained classically. The main point is that the Bohr-van Leeuwen theorem did not consider a constant of the motion which in Carathéodory's approach naturally arises. By considering this constant of the motion, the possibility of magnetic response in a sample of charged particles automatically appears.

After dealing with the above two cases, the rest of the chapter will be devoted to show how gauge invariance can be considered within Carathéodory's framework. This may have some inspiring effect for future work aiming at exploring the quantum-classical correspondence.

### **2. Carathéodory's royal road**

#### **2.1. Preliminaries**

2 Quantum Mechanics

principle.

additional insight.

have become basic tools for the calculation of probability amplitudes of different processes, as well as for suggesting new developments in quantum field theory. Symmetry, such a basic concept underlying those aspects of Nature which appear to us in the form of interactions among fundamental particles, is best accounted for within the framework of a variational

Within the domain of classical physics, only two fundamental interactions have been addressed: the gravitational and the electromagnetic interactions. The theoretical construction may correspondingly rest on two variational principles, one for gravitation and the other for electromagnetism. These principles lead to so-called "equations of motion": the Maxwell and the Lorentz equations for electromagnetism, and the Einstein and the geodesic equations for gravitation. All these differential equations can be derived as Euler-Lagrange

The usual approach to variational calculus in physics starts by considering small variations of a curve which renders extremal the action integral *Ldt*, with *L* being the Lagrangian. This leads to the Euler-Lagrange equations of motion. By submitting *L* to a Legendre transformation one obtains the corresponding Hamiltonian, in terms of which the Hamilton equations of motion can be established. By considering canonical transformations of these equations, one arrives at the Hamilton-Jacobi equation for a scalar function *S*(*t*, *x*). It is last one that has been used to connect the classical approach with the quantum one, e.g., in Madelung's hydrodynamic model [3] or in Bohm's "hidden variables" approach [4]. This appears natural, because both the Hamilton-Jacobi and the Schrödinger equation rule the dynamics of quantities like *S*(*t*, *x*) and *ψ*(*t*, *x*), respectively, which are scalar fields. Their scalar nature is in fact irrelevant; they could be tensors and spinors. The relevant issue is that while the Euler-Lagrange and the Hamilton equations refer to a single path, quantum equations address a field. The quantum-classical connection thus requires making a field out of single paths, something which occurs by going to the Hamilton-Jacobi equation, or else by establishing a path-integral formulation, as Feynman did. The latter considers a family of trajectories and assigns a probability to each of them. Now, Carathéodory's approach has the advantage of addressing right from the start a field of extremals. In fact, as the calculus of variations shows, a solution of the extremal problem exists only when the sought-after extremal curve can be embedded in a field of similar extremals. Carathéodory exploited this fact by introducing the concepts of "equivalent variational problems" and the "complete figure". It is then possible to elegantly derive from a single statement the Euler-Lagrange and the Hamilton equations, as well as the Hamilton-Jacobi equation, all of them as field equations. The familiar Euler-Lagrange and Hamilton equations can be obtained afterwards by singling out a particular extremal of the field. But – as already stressed – it is not the inherent elegance of the formulation what drives our interest towards Carathéodory's approach. It is rather its potentiality as a bridge between classical and quantum formulations what should be brought to the fore. Indeed, Carathéodory's approach can provide new insights into the connection between classical and quantum formulations. These insights could go beyond those already known, which were obtained by extending the Hamilton-Jacobi equation with the inclusion of additional terms. By dealing with the other field equations that appear within Carathéodory's approach, one may hope to gain

The present chapter, after discussing Carathéodory's approach, shows how one can classically explain two phenomena that have been understood as being exclusively quantum

equations from the appropriate Lagrangian or Lagrangian density.

Let us begin by recapitulating the approach usually employed in physics. For the sake of describing a particle's motion we use a variational principle based on a Lagrangian *L*. When describing the dynamics of a field we use instead a variational principle based on a Lagrangian density L. The Euler-Lagrange equations are, respectively,

$$\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{x}^i}\right) - \frac{\partial L}{\partial x^i} = 0,\tag{1}$$

for a Lagrangian *L*(*t*, *x<sup>i</sup>* , *x*˙ *i* ), with *i* = 1, . . . , *n*, and

$$\frac{\partial}{\partial \mathbf{x}^{\mu}} \left( \frac{\partial \mathcal{L}}{\partial (\partial\_{\mu} \boldsymbol{\psi}^{i})} \right) - \frac{\partial \mathcal{L}}{\partial \boldsymbol{\psi}^{i}} = \mathbf{0},\tag{2}$$

for a Lagrangian density L(*ψ<sup>i</sup>* , *∂µψ<sup>i</sup>* ) that depends on *n* fields *ψ<sup>i</sup>* and their derivatives *∂µψ<sup>i</sup>* with respect to space-time coordinates *xµ*. The convention of summing over repeated indices has been used in Eq.(2), as we will do henceforth. The above equations are necessary conditions that are derivable from the action principle

$$
\delta I = 0,\tag{3}
$$

and on the right-hand side,

and replacing *∂L*/*∂x*˙

those being of the form {*x<sup>i</sup>*

independent variables: {*x<sup>i</sup>*

independent variables is {*x<sup>i</sup>*

original Hamiltonian,

splitting of *S*(*t*, *x<sup>i</sup>*

∑*i*=1,3 *k*2(*∂ψ*/*∂x<sup>i</sup>*

the orbits *pi* = *pi*(*x<sup>j</sup>*

They lead to equations for {*X<sup>i</sup>*

*K*(*t*, *X*, *P*). From the set of canonical variables {*x<sup>i</sup>*

together with

*dH* = *pidx*˙

*<sup>i</sup>* + *x*˙ *i*

, *pi*}→{*X<sup>i</sup>*

, *X<sup>i</sup>* }, {*x<sup>i</sup>*

, *<sup>α</sup>i*) as *<sup>S</sup>* <sup>=</sup> *Et* <sup>−</sup> <sup>∑</sup>*<sup>j</sup> Wj*(*x<sup>i</sup>*

considered first the case *∂H*/*∂t* = 0, with *H* = ∑*i*=1,3 *p*<sup>2</sup>

the relationship between the *pi* and the *x<sup>i</sup>*

*dpi* <sup>−</sup> *<sup>∂</sup><sup>L</sup> ∂x<sup>i</sup>*

Charathéodory's "Royal Road" to the Calculus of Variations: A Possible Bridge Between Classical and Quantum Physics

*∂H <sup>∂</sup><sup>t</sup>* <sup>=</sup> <sup>−</sup>*∂<sup>L</sup> ∂t*

A third way to deal with the motion problem is given by the *Hamilton-Jacobi* equation. In order to introduce it, one usually starts by considering canonical transformations, i.e.,

One considers then four types of transformations, in accordance to the chosen set of

the old to the new canonical variables can be afforded by a so-called "generating function" *S*, which depends on the chosen set of independent variables and the time *t*. The old and new Hamiltonians are related by *K* = *H* + *∂S*/*∂t*. If we succeed in finding a transformation such that *K* = 0, the Hamilton equations for *K* can be trivially solved. One is thus led to seek for a transformation whose generating function is such that *K* = 0. When the set of

have constant values *Pi* = *α<sup>i</sup>* in virtue of *K* = 0. This last equation reads, in terms of the

This is the Hamilton-Jacobi equation. It has played an important role beyond the context in which it originally arose, becoming a sort of bridge that links classical and quantum mechanics. As a first attempt to obtain a quantum-mechanical formalism it was Sommerfeld who, following Bohr, considered action-angle variables for the case of a conservative Hamiltonian, *∂H*/*∂t* = 0. This Hamiltonian was furthermore assumed to allow the

action-angle variables can be introduced as new canonical variables [12]. By imposing that the action variables are integer-multiples of a fundamental action, i.e., Planck's *h*, it was possible to obtain a first formulation of quantum mechanics. This version is known as "old quantum mechanics". A second attempt went along Schrödinger's reinterpretation of the left-hand-side of Eq.(10) as a Lagrangian density of a new variational principle. Schrödinger

*S* = *k* ln *ψ*, with *k* a constant. From the left-hand side of Eq.(10), after multiplying it by *ψ*2, Schrödinger obtained an expression that he took as a Lagrangian density: L =

one readily obtains the (time-independent) Schrödinger equation. The constant *k* could be

, *∂S*/*∂x<sup>i</sup>*

*∂S*/*∂t* + *H*(*t*, *x<sup>i</sup>*

, *Pi*}, {*pi*, *<sup>X</sup><sup>i</sup>*

, *pi*, *X<sup>i</sup>*

*dx<sup>i</sup>* <sup>−</sup> *<sup>∂</sup><sup>L</sup> ∂x*˙*<sup>i</sup> dx*˙ *<sup>i</sup>* − *∂L ∂t*

*<sup>i</sup>* by *pi*, after equating both sides we see that Eqs.(6) must hold true,

10.5772/53843

45

*dt*, (8)

http://dx.doi.org/10.5772/53843

. (9)

, *Pi*} only 2*<sup>n</sup>* of them are independent.

) = 0. (10)

*<sup>i</sup>* /2*m* + *V*, and introduced *ψ* through

, is such that

; *αi*). Restricting the treatment to cases where

, given through *pi* = *∂S*/*∂x<sup>i</sup>*

, *αj*) are either closed (*libration*-like) or else periodic (*rotation*-like),

)2/2*m* + (*V* − *E*)*ψ*2. Inserting this L into the Euler-Lagrange equations (2)

}, and {*pi*, *Pi*}. The transformation from

(*x*, *<sup>p</sup>*, *<sup>t</sup>*), *Pi*(*x*, *<sup>p</sup>*, *<sup>t</sup>*)} and leaving the action *<sup>I</sup>* invariant.

, *Pi*} that are similar to Eqs.(6) but with a new Hamiltonian,

, *Pi*} the *pi* are given by *pi* = *<sup>∂</sup>S*/*∂x<sup>i</sup>* , while the new momenta

with the action given by *I* = *Ldt* for the particle motion and *I* = L*d*4*x* for the field dynamics. The variation *δ* means that we consider different paths joining some fixed initial and end points – hypersurfaces in the case of L*d*4*x* – and seek for the path that affords *I* an extremal value. Curves which are solutions of the Euler-Lagrange equations are called *extremals*.

Let us concentrate on the case *I* = *Ldt* in what follows and sketch how the standard derivation of Eq.(1) is usually obtained: one takes the variation *δ Ldt* = *dt* (*∂L*/*∂x<sup>i</sup>* )*δx<sup>i</sup>* + (*∂L*/*∂x*˙ *i* )*δx*˙ *i* , and observing that *δx*˙ *<sup>i</sup>* = *d*(*δx<sup>i</sup>* )/*dt*, integration by parts gives *<sup>δ</sup><sup>I</sup>* = *dt ∂L*/*∂x<sup>i</sup>* − *d*(*∂L*/*∂x*˙ *i* )/*dt δx<sup>i</sup>* = 0, where we have considered that *δx<sup>i</sup>* = 0 at the common endpoints of all the paths involved in the variation. The arbitrariness of *δx<sup>i</sup>* leads to Eq.(1) as a necessary condition for *δI* to be zero.

The important case of a time-independent Lagrangian (*∂L*/*∂t* = 0) leads to the conservation of the quantity

$$\frac{\partial L}{\partial \dot{\mathbf{x}}^i} \dot{\mathbf{x}}^i - L \tag{4}$$

along an extremal, as can be seen by taking its time-derivative and using Eq.(1). By introducing the canonical momenta *pi* = *∂L*(*t*, *x*, *x*˙)/*∂x*˙ *<sup>i</sup>* and assuming that we can solve these equations for the *x*˙ *<sup>i</sup>* as functions of the new set of independent variables, *x*˙ *<sup>i</sup>* = *x*˙ *i* (*t*, *x<sup>i</sup>* , *pi*), we can define a Hamiltonian *H*(*t*, *x*, *p*) through the expression given by Eq.(4), written in terms of the new variables (*t*, *x<sup>i</sup>* , *pi*):

$$H(t, \mathbf{x}, p) = p\_i \dot{\mathbf{x}}^i(t, \mathbf{x}, p) - L(t, \mathbf{x}, \dot{\mathbf{x}}(t, \mathbf{x}, p)). \tag{5}$$

The Euler-Lagrange equations are then replaced by the *Hamilton equations*:

$$\frac{dx^i}{dt} = \frac{\partial H}{\partial p\_i}, \frac{dp\_i}{dt} = -\frac{\partial H}{\partial x^i}.\tag{6}$$

Eq.(5) can be seen as a Legendre transformation leading from the set (*x<sup>i</sup>* , *x*˙ *i* , *t*) to the set (*xi* , *pi*, *t*) by means of the function *H*(*t*, *x*, *p*). Taking the differential of *H*(*t*, *x*, *p*) on the left-hand side of Eq.(5),

$$dH = \frac{\partial H}{\partial t}dt + \frac{\partial H}{\partial x^i}dx^i + \frac{\partial H}{\partial p\_i}dp\_{i\prime} \tag{7}$$

<sup>44</sup> Advances in Quantum Mechanics Charathéodory's "Royal Road" to the Calculus of Variations: A Possible Bridge Between Classical and Quantum Physics 5 10.5772/53843 Charathéodory's "Royal Road" to the Calculus of Variations: A Possible Bridge Between Classical and Quantum Physics http://dx.doi.org/10.5772/53843 45

and on the right-hand side,

4 Quantum Mechanics

*extremals*.

gives *δI* = *dt*

of the quantity

equations for the *x*˙

of the new variables (*t*, *x<sup>i</sup>*

left-hand side of Eq.(5),

 *dt* (*∂L*/*∂x<sup>i</sup>*

(*xi*

for a Lagrangian density L(*ψ<sup>i</sup>*

, *∂µψ<sup>i</sup>*

conditions that are derivable from the action principle

)*δx<sup>i</sup>* + (*∂L*/*∂x*˙

*i* )*δx*˙ *i* 

leads to Eq.(1) as a necessary condition for *δI* to be zero.

introducing the canonical momenta *pi* = *∂L*(*t*, *x*, *x*˙)/*∂x*˙

, *pi*):

*H*(*t*, *x*, *p*) = *pix*˙

*∂L*/*∂x<sup>i</sup>* − *d*(*∂L*/*∂x*˙

with respect to space-time coordinates *xµ*. The convention of summing over repeated indices has been used in Eq.(2), as we will do henceforth. The above equations are necessary

with the action given by *I* = *Ldt* for the particle motion and *I* = L*d*4*x* for the field dynamics. The variation *δ* means that we consider different paths joining some fixed initial and end points – hypersurfaces in the case of L*d*4*x* – and seek for the path that affords *I* an extremal value. Curves which are solutions of the Euler-Lagrange equations are called

Let us concentrate on the case *I* = *Ldt* in what follows and sketch how the

, and observing that *δx*˙

at the common endpoints of all the paths involved in the variation. The arbitrariness of *δx<sup>i</sup>*

The important case of a time-independent Lagrangian (*∂L*/*∂t* = 0) leads to the conservation

along an extremal, as can be seen by taking its time-derivative and using Eq.(1). By

*<sup>i</sup>* as functions of the new set of independent variables, *x*˙

can define a Hamiltonian *H*(*t*, *x*, *p*) through the expression given by Eq.(4), written in terms

, *dpi*

, *pi*, *t*) by means of the function *H*(*t*, *x*, *p*). Taking the differential of *H*(*t*, *x*, *p*) on the

*dt* <sup>=</sup> <sup>−</sup>*∂<sup>H</sup> ∂x<sup>i</sup>*

> *∂H ∂pi*

*i*

The Euler-Lagrange equations are then replaced by the *Hamilton equations*:

Eq.(5) can be seen as a Legendre transformation leading from the set (*x<sup>i</sup>*

*dx<sup>i</sup> dt* <sup>=</sup> *<sup>∂</sup><sup>H</sup> ∂pi*

*dH* <sup>=</sup> *<sup>∂</sup><sup>H</sup> ∂t dt* + *∂H <sup>∂</sup>x<sup>i</sup> dx<sup>i</sup>* <sup>+</sup>

*∂L <sup>∂</sup>x*˙*<sup>i</sup> <sup>x</sup>*˙

standard derivation of Eq.(1) is usually obtained: one takes the variation *δ*

*i* )/*dt* 

) that depends on *n* fields *ψ<sup>i</sup>* and their derivatives *∂µψ<sup>i</sup>*

*<sup>i</sup>* = *d*(*δx<sup>i</sup>*

*δI* = 0, (3)

*δx<sup>i</sup>* = 0, where we have considered that *δx<sup>i</sup>* = 0

*<sup>i</sup>* − *L* (4)

(*t*, *x*, *p*) − *L*(*t*, *x*, *x*˙(*t*, *x*, *p*)). (5)

*<sup>i</sup>* and assuming that we can solve these

*<sup>i</sup>* = *x*˙ *i* (*t*, *x<sup>i</sup>*

. (6)

*dpi*, (7)

, *x*˙ *i*

, *t*) to the set

*Ldt* =

, *pi*), we

)/*dt*, integration by parts

$$dH = p\_i d\dot{\mathbf{x}}^i + \dot{\mathbf{x}}^i dp\_i - \frac{\partial L}{\partial \mathbf{x}^i} d\mathbf{x}^i - \frac{\partial L}{\partial \dot{\mathbf{x}}^i} d\dot{\mathbf{x}}^i - \frac{\partial L}{\partial t} dt,\tag{8}$$

and replacing *∂L*/*∂x*˙ *<sup>i</sup>* by *pi*, after equating both sides we see that Eqs.(6) must hold true, together with

$$\frac{\partial H}{\partial t} = -\frac{\partial L}{\partial t}.\tag{9}$$

A third way to deal with the motion problem is given by the *Hamilton-Jacobi* equation. In order to introduce it, one usually starts by considering canonical transformations, i.e., those being of the form {*x<sup>i</sup>* , *pi*}→{*X<sup>i</sup>* (*x*, *<sup>p</sup>*, *<sup>t</sup>*), *Pi*(*x*, *<sup>p</sup>*, *<sup>t</sup>*)} and leaving the action *<sup>I</sup>* invariant. They lead to equations for {*X<sup>i</sup>* , *Pi*} that are similar to Eqs.(6) but with a new Hamiltonian, *K*(*t*, *X*, *P*). From the set of canonical variables {*x<sup>i</sup>* , *pi*, *X<sup>i</sup>* , *Pi*} only 2*<sup>n</sup>* of them are independent. One considers then four types of transformations, in accordance to the chosen set of independent variables: {*x<sup>i</sup>* , *X<sup>i</sup>* }, {*x<sup>i</sup>* , *Pi*}, {*pi*, *<sup>X</sup><sup>i</sup>* }, and {*pi*, *Pi*}. The transformation from the old to the new canonical variables can be afforded by a so-called "generating function" *S*, which depends on the chosen set of independent variables and the time *t*. The old and new Hamiltonians are related by *K* = *H* + *∂S*/*∂t*. If we succeed in finding a transformation such that *K* = 0, the Hamilton equations for *K* can be trivially solved. One is thus led to seek for a transformation whose generating function is such that *K* = 0. When the set of independent variables is {*x<sup>i</sup>* , *Pi*} the *pi* are given by *pi* = *<sup>∂</sup>S*/*∂x<sup>i</sup>* , while the new momenta have constant values *Pi* = *α<sup>i</sup>* in virtue of *K* = 0. This last equation reads, in terms of the original Hamiltonian,

$$
\partial \mathbf{S} / \partial t + H(t, \mathbf{x}^l, \partial \mathbf{S} / \partial \mathbf{x}^l) = \mathbf{0}. \tag{10}
$$

This is the Hamilton-Jacobi equation. It has played an important role beyond the context in which it originally arose, becoming a sort of bridge that links classical and quantum mechanics. As a first attempt to obtain a quantum-mechanical formalism it was Sommerfeld who, following Bohr, considered action-angle variables for the case of a conservative Hamiltonian, *∂H*/*∂t* = 0. This Hamiltonian was furthermore assumed to allow the splitting of *S*(*t*, *x<sup>i</sup>* , *<sup>α</sup>i*) as *<sup>S</sup>* <sup>=</sup> *Et* <sup>−</sup> <sup>∑</sup>*<sup>j</sup> Wj*(*x<sup>i</sup>* ; *αi*). Restricting the treatment to cases where the relationship between the *pi* and the *x<sup>i</sup>* , given through *pi* = *∂S*/*∂x<sup>i</sup>* , is such that the orbits *pi* = *pi*(*x<sup>j</sup>* , *αj*) are either closed (*libration*-like) or else periodic (*rotation*-like), action-angle variables can be introduced as new canonical variables [12]. By imposing that the action variables are integer-multiples of a fundamental action, i.e., Planck's *h*, it was possible to obtain a first formulation of quantum mechanics. This version is known as "old quantum mechanics". A second attempt went along Schrödinger's reinterpretation of the left-hand-side of Eq.(10) as a Lagrangian density of a new variational principle. Schrödinger considered first the case *∂H*/*∂t* = 0, with *H* = ∑*i*=1,3 *p*<sup>2</sup> *<sup>i</sup>* /2*m* + *V*, and introduced *ψ* through *S* = *k* ln *ψ*, with *k* a constant. From the left-hand side of Eq.(10), after multiplying it by *ψ*2, Schrödinger obtained an expression that he took as a Lagrangian density: L = ∑*i*=1,3 *k*2(*∂ψ*/*∂x<sup>i</sup>* )2/2*m* + (*V* − *E*)*ψ*2. Inserting this L into the Euler-Lagrange equations (2) one readily obtains the (time-independent) Schrödinger equation. The constant *k* could be identified with ℏ by comparison with Bohr's energy levels in the case of the hydrogen atom (*V* ∼ 1/*r*). We recall that parallel to this approach, another one, due to Heisenberg, Born, Jordan and Dirac, was constructed out of a reformulation of the action-angle formalism applied to multiple periodic motions. This reformulation led to a formalism in which the Poisson brackets were replaced by commutators, and the canonical variables by operators.

10.5772/53843

47

): at each point

. In other

(*t*, *x*),

(*t*, *x*)

(*t*, *x<sup>j</sup>*

http://dx.doi.org/10.5772/53843

(*t*, *x*). Once we have *v<sup>i</sup>*

*<sup>L</sup>*∗*dt* = 0. Of course, not

(*t*, *x<sup>j</sup>* ):

(*t*, *x*(*t*)). (11)

*∂iS*)*dt* = 0. (12)

*∂iS* = 0 (13)

*∂iS*)*dt* = *L*(*t*, *x*, *x*˙)*dt*, this last integral

, *w<sup>i</sup>*

*<sup>∂</sup>v<sup>i</sup>* . (14)

, Eq.(14) gives *pi* = *∂S*(*t*, *x*)/*∂x<sup>i</sup>*

) − *∂tS* − *w<sup>i</sup>*

*∂iS* >

. If

b) An extremal curve exists only in case that it can be embedded in a whole set of extremals,

Charathéodory's "Royal Road" to the Calculus of Variations: A Possible Bridge Between Classical and Quantum Physics

(*t*) is thus equivalent to fixing *v<sup>i</sup>*

can be approached locally. To this end, observe that the extremals we are seeking, for

*Ldt* = 0, are also extremals of the modified, "*equivalent variational problem*",

*i*

Now, assume that we are dealing with a particular Lagrangian *<sup>L</sup>*∗(*t*, *<sup>x</sup>*, *<sup>v</sup>*), for which the following requirements are met: First, it is possible to find a vector field *v* such that *<sup>L</sup>*∗(*t*, *<sup>x</sup>*, *<sup>v</sup>*) = 0. Second, *<sup>L</sup>*∗(*t*, *<sup>x</sup>*, *<sup>w</sup>*) > 0 for any other field *<sup>w</sup>* � *<sup>v</sup>*. It is then easy to show that

every Lagrangian will satisfy the requirements we have put on *<sup>L</sup>*∗; but by making use of the freedom we have to change our original problem into an "*equivalent variational problem*", we

identically, the value zero being an extremal one with respect to variations of *v*. This happens for a *suitably* chosen *S*(*t*, *x*) that remains fixed in this context. The function *S*(*t*, *x*) must

> (*∂tS* + *x*˙ *i*

being calculated along an extremal curve. In other words, among all equivalent variational problems we seek the one for which the conditions put upon *<sup>L</sup>*<sup>∗</sup> are fulfilled. Thus, for the

0 for any other field *w* � *v*. In this way our variational problem becomes a local one: *v* has to be determined so as to afford an extremal value to the expression at the left-hand side of Eq.(13). Thus, taking the partial derivative of this expression with respect to *v* and equating

*<sup>∂</sup>x<sup>i</sup>* <sup>=</sup> *<sup>∂</sup>L*(*t*, *<sup>x</sup>*, *<sup>v</sup>*(*t*, *<sup>x</sup>*))

Eqs.(13) and (14) are referred to as the *fundamental equations* in Carathéodory's approach. From these two equations we can derive all known results of the calculus of variations.

*L*(*t*, *x*, *v*) − *∂tS* − *v<sup>i</sup>*

the extremals can be obtained by integration of Eq.(11). The task of finding *v<sup>i</sup>*

(*L* − *∂tS* − *x*˙

) to be tangent to the unique curve which goes through *x<sup>j</sup>*

Now, having a field of curves is equivalent to defining a vector field *v<sup>i</sup>*

*dx<sup>i</sup>* (*t*) *dt* <sup>=</sup> *<sup>v</sup><sup>i</sup>*

words, the curves that constitute the field are integral curves of *v<sup>i</sup>*

*δ* 

the integral curves of *v* are extremals of the variational problem *δ*

extremal value being, e.g., a minimum, it must hold Eq.(13), while *L*(*t*, *x<sup>i</sup>*

*∂S*

We see, for instance, that defining *pi* = *∂L*(*t*, *x*, *v*)/*∂v<sup>i</sup>*

let *<sup>L</sup>*<sup>∗</sup> = *<sup>L</sup>* − *dS*/*dt* and seek for a vector field *<sup>v</sup>* such that

be just the one for which the value of

it to zero we obtain

a so-called "Mayer field".

Finding all the extremals *x<sup>i</sup>*

(*t*, *x<sup>j</sup>*

(*L* − *dS*/*dt*)*dt* = 0. This can be written as

*x<sup>j</sup>* we just define *v<sup>i</sup>*

which *δ*

*δ* 

Coming back to the general action principle, we have so far followed the road usually employed by physicists. This road was build out of manifold contributions, made at different times and with different purposes. As a consequence, it lacks the unity and compactness that a mathematical theory usually has. At the beginning of the 20th century mathematicians were concerned with the construction and extension of a sound theory for the calculus of variations. It is in this context that Carathéodory made his contributions to the subject. They were thus naturally conceived from a mathematical viewpoint. Apparently, they added nothing new that could be of use for physicists, and so passed almost unnoticed to them. Our purpose here is to show how Carathéodory's formulation can provide physical insight and inspire new approaches. In the following, we give a short account of Carathéodory's approach. We will try to show the conceptual unity and potential usefulness that Carathéodory's formulation entails. Such a unity roots on the so-called *complete figure* that Carathéodory introduces as a central concept of his approach. It serves as the basis of a formulation in which the Euler-Lagrange, the Hamilton and the Hamilton-Jacobi equations appear as three alternative expressions of one and the same underlying concept.

#### **2.2. The non-homogeneous case**

Let us first consider the so-called *non-homogeneous* case, i.e., one in which the action principle – and with it the Euler-Lagrange equation – is not invariant under a change of the curve parameter. In physics, this parameter usually corresponds to time. By solving the equations of motion one obtains not only the geometrical path traced by the particle – or group of particles – being described, but also how, i.e., the rate at which this path is traveled. The *non-homogeneous* case applies to non-relativistic formulations.

The equation of motion follows from the variational principle *δ L*(*t*, *x<sup>i</sup>* , *x*˙ *i* )*dt* = 0. As physicists, we usually visualize the variational principle as expressing how Nature works: among all possible paths joining two given points, Nature chooses the one which affords *Ldt* an extremal value. In some sense, this presupposes a non-local behavior, as two distant points determine the extremal curve that should join them. This is reminiscent of the action-at-a-distance invoked by earlier formulations, in whose context the variational principle originally arose. The approach proposed by Carathéodory is more in accordance with our modern view of local interactions. He replaced the problem of finding an extremum for the action integral by one of finding a local extremal value for a function. Thus, the field concept is at the forefront, playing a major role.

Let us recall some important assumptions [1, 13–15] concerning the central problem of variational calculus:

a) To find an extremal curve *x<sup>i</sup>* = *x<sup>i</sup>* (*t*) that satisfies *δ Ldt* = 0 requires that we restrict ourselves to a simply-connected domain. Though apparently technical, this point might entail a profound physical significance.

6 Quantum Mechanics

identified with ℏ by comparison with Bohr's energy levels in the case of the hydrogen atom (*V* ∼ 1/*r*). We recall that parallel to this approach, another one, due to Heisenberg, Born, Jordan and Dirac, was constructed out of a reformulation of the action-angle formalism applied to multiple periodic motions. This reformulation led to a formalism in which the Poisson brackets were replaced by commutators, and the canonical variables by operators. Coming back to the general action principle, we have so far followed the road usually employed by physicists. This road was build out of manifold contributions, made at different times and with different purposes. As a consequence, it lacks the unity and compactness that a mathematical theory usually has. At the beginning of the 20th century mathematicians were concerned with the construction and extension of a sound theory for the calculus of variations. It is in this context that Carathéodory made his contributions to the subject. They were thus naturally conceived from a mathematical viewpoint. Apparently, they added nothing new that could be of use for physicists, and so passed almost unnoticed to them. Our purpose here is to show how Carathéodory's formulation can provide physical insight and inspire new approaches. In the following, we give a short account of Carathéodory's approach. We will try to show the conceptual unity and potential usefulness that Carathéodory's formulation entails. Such a unity roots on the so-called *complete figure* that Carathéodory introduces as a central concept of his approach. It serves as the basis of a formulation in which the Euler-Lagrange, the Hamilton and the Hamilton-Jacobi equations

appear as three alternative expressions of one and the same underlying concept.

*non-homogeneous* case applies to non-relativistic formulations.

concept is at the forefront, playing a major role.

a) To find an extremal curve *x<sup>i</sup>* = *x<sup>i</sup>*

entail a profound physical significance.

variational calculus:

The equation of motion follows from the variational principle *δ*

Let us first consider the so-called *non-homogeneous* case, i.e., one in which the action principle – and with it the Euler-Lagrange equation – is not invariant under a change of the curve parameter. In physics, this parameter usually corresponds to time. By solving the equations of motion one obtains not only the geometrical path traced by the particle – or group of particles – being described, but also how, i.e., the rate at which this path is traveled. The

physicists, we usually visualize the variational principle as expressing how Nature works: among all possible paths joining two given points, Nature chooses the one which affords *Ldt* an extremal value. In some sense, this presupposes a non-local behavior, as two distant points determine the extremal curve that should join them. This is reminiscent of the action-at-a-distance invoked by earlier formulations, in whose context the variational principle originally arose. The approach proposed by Carathéodory is more in accordance with our modern view of local interactions. He replaced the problem of finding an extremum for the action integral by one of finding a local extremal value for a function. Thus, the field

Let us recall some important assumptions [1, 13–15] concerning the central problem of

(*t*) that satisfies *δ*

ourselves to a simply-connected domain. Though apparently technical, this point might

*L*(*t*, *x<sup>i</sup>*

, *x*˙ *i*

*Ldt* = 0 requires that we restrict

)*dt* = 0. As

**2.2. The non-homogeneous case**

b) An extremal curve exists only in case that it can be embedded in a whole set of extremals, a so-called "Mayer field".

Now, having a field of curves is equivalent to defining a vector field *v<sup>i</sup>* (*t*, *x<sup>j</sup>* ): at each point *x<sup>j</sup>* we just define *v<sup>i</sup>* (*t*, *x<sup>j</sup>* ) to be tangent to the unique curve which goes through *x<sup>j</sup>* . In other words, the curves that constitute the field are integral curves of *v<sup>i</sup>* (*t*, *x<sup>j</sup>* ):

$$\frac{d\mathbf{x}^i(t)}{dt} = \mathbf{v}^i(t, \mathbf{x}(t)). \tag{11}$$

Finding all the extremals *x<sup>i</sup>* (*t*) is thus equivalent to fixing *v<sup>i</sup>* (*t*, *x*). Once we have *v<sup>i</sup>* (*t*, *x*), the extremals can be obtained by integration of Eq.(11). The task of finding *v<sup>i</sup>* (*t*, *x*) can be approached locally. To this end, observe that the extremals we are seeking, for which *δ Ldt* = 0, are also extremals of the modified, "*equivalent variational problem*", *δ* (*L* − *dS*/*dt*)*dt* = 0. This can be written as

$$
\delta \int (L - \partial\_t \mathcal{S} - \dot{\mathfrak{x}}^i \partial\_i \mathcal{S}) dt = 0. \tag{12}
$$

Now, assume that we are dealing with a particular Lagrangian *<sup>L</sup>*∗(*t*, *<sup>x</sup>*, *<sup>v</sup>*), for which the following requirements are met: First, it is possible to find a vector field *v* such that *<sup>L</sup>*∗(*t*, *<sup>x</sup>*, *<sup>v</sup>*) = 0. Second, *<sup>L</sup>*∗(*t*, *<sup>x</sup>*, *<sup>w</sup>*) > 0 for any other field *<sup>w</sup>* � *<sup>v</sup>*. It is then easy to show that the integral curves of *v* are extremals of the variational problem *δ <sup>L</sup>*∗*dt* = 0. Of course, not every Lagrangian will satisfy the requirements we have put on *<sup>L</sup>*∗; but by making use of the freedom we have to change our original problem into an "*equivalent variational problem*", we let *<sup>L</sup>*<sup>∗</sup> = *<sup>L</sup>* − *dS*/*dt* and seek for a vector field *<sup>v</sup>* such that

$$L(t, x, v) - \partial\_t S - v^i \partial\_i S = 0 \tag{13}$$

identically, the value zero being an extremal one with respect to variations of *v*. This happens for a *suitably* chosen *S*(*t*, *x*) that remains fixed in this context. The function *S*(*t*, *x*) must be just the one for which the value of (*∂tS* + *x*˙ *i ∂iS*)*dt* = *L*(*t*, *x*, *x*˙)*dt*, this last integral being calculated along an extremal curve. In other words, among all equivalent variational problems we seek the one for which the conditions put upon *<sup>L</sup>*<sup>∗</sup> are fulfilled. Thus, for the extremal value being, e.g., a minimum, it must hold Eq.(13), while *L*(*t*, *x<sup>i</sup>* , *w<sup>i</sup>* ) − *∂tS* − *w<sup>i</sup> ∂iS* > 0 for any other field *w* � *v*. In this way our variational problem becomes a local one: *v* has to be determined so as to afford an extremal value to the expression at the left-hand side of Eq.(13). Thus, taking the partial derivative of this expression with respect to *v* and equating it to zero we obtain

$$\frac{\partial \mathcal{S}}{\partial \mathbf{x}^i} = \frac{\partial L(t, \mathbf{x}, v(t, \mathbf{x}))}{\partial v^i}. \tag{14}$$

Eqs.(13) and (14) are referred to as the *fundamental equations* in Carathéodory's approach. From these two equations we can derive all known results of the calculus of variations. We see, for instance, that defining *pi* = *∂L*(*t*, *x*, *v*)/*∂v<sup>i</sup>* , Eq.(14) gives *pi* = *∂S*(*t*, *x*)/*∂x<sup>i</sup>* . If we now introduce, by means of a Legendre transformation, the Hamiltonian *H*(*t*, *x*, *p*) = *vi* (*t*, *<sup>x</sup>*, *<sup>p</sup>*)*pi* − *<sup>L</sup>*(*t*, *<sup>x</sup>*, *<sup>v</sup>*(*t*, *<sup>x</sup>*, *<sup>p</sup>*)), Eq.(13) reads

$$\left(\partial\_l S + H\left(t, x^i, \partial\_i S\right)\right) = 0,\tag{15}$$

10.5772/53843

49

From the *fundamental equations*, (17) and (18), we can derive all known results also in this

Charathéodory's "Royal Road" to the Calculus of Variations: A Possible Bridge Between Classical and Quantum Physics

*<sup>∂</sup>xµ∂x<sup>ν</sup>* <sup>=</sup> *<sup>∂</sup>*2*<sup>S</sup>*

which are, as we will shortly see, at the very basis of the Euler-Lagrange equations. Indeed,

*∂S*

*<sup>∂</sup>xσ∂v<sup>µ</sup>* <sup>+</sup>

*∂*2*L ∂vτ∂v<sup>µ</sup>*

*∂*2*L ∂vτ∂v<sup>µ</sup>*

*∂v<sup>τ</sup>*

*∂v<sup>τ</sup>*

*<sup>∂</sup>x<sup>σ</sup>* <sup>+</sup> *<sup>v</sup><sup>σ</sup> <sup>∂</sup>*2*<sup>S</sup>*

*<sup>∂</sup>xν∂x<sup>µ</sup>* , (19)

*<sup>∂</sup>xµ∂x<sup>σ</sup>* . (21)

*<sup>∂</sup>xµ∂x<sup>σ</sup>* . (20)

http://dx.doi.org/10.5772/53843

*<sup>∂</sup>x<sup>σ</sup>* , (22)

*<sup>∂</sup>x<sup>σ</sup> <sup>v</sup>σ*. (23)

. (24)

*<sup>∂</sup>x<sup>i</sup>* <sup>=</sup> 0, (25)

case. In particular, we see that *S*(*x*) must satisfy the integrability conditions

from Eq.(17) we obtain, by taking the derivative with respect to *xµ*,

*∂L ∂v<sup>σ</sup>* *∂v<sup>σ</sup> <sup>∂</sup>x<sup>µ</sup>* <sup>=</sup> *<sup>∂</sup>v<sup>σ</sup> ∂x<sup>µ</sup>*

*∂L*

*<sup>∂</sup>x<sup>µ</sup>* <sup>=</sup> *<sup>v</sup><sup>σ</sup> <sup>∂</sup>*2*<sup>S</sup>*

*<sup>∂</sup>xσ∂x<sup>µ</sup>* <sup>=</sup> *<sup>∂</sup>*2*<sup>L</sup>*

*<sup>∂</sup>xσ∂v<sup>µ</sup>* <sup>+</sup>

If we now evaluate this last relation along a single extremal, *dxµ*/*dτ* = *vµ*(*x*(*τ*)), we obtain, after recognizing the right hand side of Eq.(23) as *d*(*∂L*/*∂vµ*)/*dτ*, the Euler-Lagrange

Eq.(23) is therefore more general than the Euler-Lagrange equation. The latter follows from

but with the important difference that now the curve-parameter *t* is fixed: the solution of Eq.(25) provides us not only with the geometrical shape of the extremal curve, but also with

 *∂L ∂x*˙*<sup>µ</sup>* 

*∂L <sup>∂</sup>x<sup>µ</sup>* <sup>+</sup>

*∂*2*S*

*<sup>∂</sup>xµ∂x<sup>σ</sup>* <sup>=</sup> *<sup>∂</sup>*2*<sup>S</sup>*

*∂L*

For the non-homogeneous case we obtain a similar result

*<sup>∂</sup>x<sup>µ</sup>* <sup>=</sup> *<sup>v</sup><sup>σ</sup> <sup>∂</sup>*2*<sup>L</sup>*

*∂L <sup>∂</sup>x<sup>µ</sup>* <sup>=</sup> *<sup>d</sup> dτ*

*d dt <sup>∂</sup><sup>L</sup> ∂x*˙*<sup>i</sup>* − *∂L*

On using Eq.(18), Eq.(20) reduces to

From Eqs.(18) and (19) we thus obtain

Eq.(23); but not the other way around.

the rate at which this curve is traced back.

so that

equation:

*∂*2*S*

which is the Hamilton-Jacobi equation. In this way we obtain an equation for *S*, the auxiliary function that was so far undetermined. It is also straightforward to deduce the Euler-Lagrange and the Hamilton equations within the present approach. For the sake of brevity, we will show how to derive the Euler-Lagrange equations in the homogeneous case only. The non-homogeneous case can be treated along similar lines.

#### **2.3. The homogeneous case**

Let us turn into the so-called homogeneous problem, the one appropriate for a relativistic formulation. In relativity, we consider a space-time continuum described by four variables *xµ*. Our variational principle is of the same form as before, i.e., *δ Ldτ* = 0; but we require it to be invariant under Lorentz transformations and under parameter changes. Indeed, all we need in order to fix the motion is the geometrical shape of the extremal curve *xµ*(*τ*) in space-time, so that the parameter *τ* has no physical meaning and the theory must be invariant under arbitrary changes of it. This is achieved when *L* does not depend explicitly on *τ* and, furthermore, it is homogeneous of first degree in the generalized velocities *x*˙ *µ* : *L*(*xµ*, *αx*˙ *<sup>µ</sup>*) = *αL*(*xµ*, *x*˙ *<sup>µ</sup>*), for *α* ≥ 0. From this requirement, it follows the identity

$$\dot{\mathfrak{x}}^{\mu} \frac{\partial L(\mathfrak{x}, \dot{\mathfrak{x}})}{\partial \dot{\mathfrak{x}}^{\mu}} = L,\tag{16}$$

which holds true for homogeneous Lagrangians. This property, however, precludes us from introducing a Hamiltonian in a similar manner as we did in the non-homogeneous case. We come back to this point later on.

As before, we seek also now for a velocity field *v*(*x*) and a function *S*(*xµ*), such that

$$L(\mathfrak{x}, \upsilon) - \upsilon^{\mu} \partial\_{\mu} S = 0,\tag{17}$$

the value zero being an extremal one with respect to *v*, for a suitably chosen *S*(*x*) that remains fixed in this context. For a maximum, for example, it must hold *<sup>L</sup>*(*xµ*, *<sup>w</sup>µ*) − *<sup>w</sup>µ∂µ<sup>S</sup>* < 0 for any other field1 *w* � *v*. Differentiating the left-hand side of Eq.(17) with respect to *v* and equating the result to zero we get

$$\frac{\partial S}{\partial \mathbf{x}^{\mu}} = \frac{\partial L(\mathbf{x}, v)}{\partial v^{\mu}}. \tag{18}$$

<sup>1</sup> The considered fields *w<sup>α</sup>* are essentially different from *vα*. A field *w<sup>α</sup>* = *φvα*, with *φ* a scalar function, is essentially the same as *vα*.

From the *fundamental equations*, (17) and (18), we can derive all known results also in this case. In particular, we see that *S*(*x*) must satisfy the integrability conditions

$$\frac{\partial^2 S}{\partial \mathbf{x}^\mu \partial \mathbf{x}^\nu} = \frac{\partial^2 S}{\partial \mathbf{x}^\nu \partial \mathbf{x}^\mu} \,, \tag{19}$$

which are, as we will shortly see, at the very basis of the Euler-Lagrange equations. Indeed, from Eq.(17) we obtain, by taking the derivative with respect to *xµ*,

$$
\frac{\partial L}{\partial \mathbf{x}^{\mu}} + \frac{\partial L}{\partial \mathbf{z}^{\sigma}} \frac{\partial \mathbf{z}^{\sigma}}{\partial \mathbf{x}^{\mu}} = \frac{\partial \mathbf{z}^{\sigma}}{\partial \mathbf{x}^{\mu}} \frac{\partial \mathbf{S}}{\partial \mathbf{x}^{\sigma}} + \mathbf{z}^{\sigma} \frac{\partial^{2} \mathbf{S}}{\partial \mathbf{x}^{\mu} \partial \mathbf{x}^{\sigma}}.\tag{20}
$$

On using Eq.(18), Eq.(20) reduces to

$$\frac{\partial L}{\partial \mathbf{x}^{\mu}} = v^{\sigma} \frac{\partial^2 S}{\partial \mathbf{x}^{\mu} \partial \mathbf{x}^{\sigma}}. \tag{21}$$

From Eqs.(18) and (19) we thus obtain

$$\frac{\partial^2 S}{\partial x^\mu \partial x^\sigma} = \frac{\partial^2 S}{\partial x^\sigma \partial x^\mu} = \frac{\partial^2 L}{\partial x^\sigma \partial v^\mu} + \frac{\partial^2 L}{\partial v^\tau \partial v^\mu} \frac{\partial v^\tau}{\partial x^\sigma} \tag{22}$$

so that

8 Quantum Mechanics

(*t*, *<sup>x</sup>*, *<sup>p</sup>*)*pi* − *<sup>L</sup>*(*t*, *<sup>x</sup>*, *<sup>v</sup>*(*t*, *<sup>x</sup>*, *<sup>p</sup>*)), Eq.(13) reads

**2.3. The homogeneous case**

*<sup>µ</sup>*) = *αL*(*xµ*, *x*˙

come back to this point later on.

equating the result to zero we get

the same as *vα*.

*vi*

*L*(*xµ*, *αx*˙

we now introduce, by means of a Legendre transformation, the Hamiltonian *H*(*t*, *x*, *p*) =

which is the Hamilton-Jacobi equation. In this way we obtain an equation for *S*, the auxiliary function that was so far undetermined. It is also straightforward to deduce the Euler-Lagrange and the Hamilton equations within the present approach. For the sake of brevity, we will show how to derive the Euler-Lagrange equations in the homogeneous case

Let us turn into the so-called homogeneous problem, the one appropriate for a relativistic formulation. In relativity, we consider a space-time continuum described by four variables

it to be invariant under Lorentz transformations and under parameter changes. Indeed, all we need in order to fix the motion is the geometrical shape of the extremal curve *xµ*(*τ*) in space-time, so that the parameter *τ* has no physical meaning and the theory must be invariant under arbitrary changes of it. This is achieved when *L* does not depend explicitly on *τ* and, furthermore, it is homogeneous of first degree in the generalized velocities *x*˙

*<sup>µ</sup>*), for *α* ≥ 0. From this requirement, it follows the identity

= 0, (15)

*Ldτ* = 0; but we require

*<sup>∂</sup>x*˙*<sup>µ</sup>* <sup>=</sup> *<sup>L</sup>*, (16)

*<sup>L</sup>*(*x*, *<sup>v</sup>*) − *<sup>v</sup>µ∂µ<sup>S</sup>* = 0, (17)

*<sup>∂</sup>v<sup>µ</sup>* . (18)

*µ* :

 *t*, *x<sup>i</sup>* , *∂iS* 

*∂tS* + *H*

only. The non-homogeneous case can be treated along similar lines.

*xµ*. Our variational principle is of the same form as before, i.e., *δ*

*x*˙

*<sup>µ</sup> ∂L*(*x*, *x*˙)

As before, we seek also now for a velocity field *v*(*x*) and a function *S*(*xµ*), such that

*∂S*

which holds true for homogeneous Lagrangians. This property, however, precludes us from introducing a Hamiltonian in a similar manner as we did in the non-homogeneous case. We

the value zero being an extremal one with respect to *v*, for a suitably chosen *S*(*x*) that remains fixed in this context. For a maximum, for example, it must hold *<sup>L</sup>*(*xµ*, *<sup>w</sup>µ*) − *<sup>w</sup>µ∂µ<sup>S</sup>* < 0 for any other field1 *w* � *v*. Differentiating the left-hand side of Eq.(17) with respect to *v* and

*<sup>∂</sup>x<sup>µ</sup>* <sup>=</sup> *<sup>∂</sup>L*(*x*, *<sup>v</sup>*)

<sup>1</sup> The considered fields *w<sup>α</sup>* are essentially different from *vα*. A field *w<sup>α</sup>* = *φvα*, with *φ* a scalar function, is essentially

$$\frac{\partial L}{\partial \mathbf{x}^{\mu}} = \boldsymbol{\upsilon}^{\sigma} \frac{\partial^2 L}{\partial \mathbf{x}^{\sigma} \partial \boldsymbol{\upsilon}^{\mu}} + \frac{\partial^2 L}{\partial \boldsymbol{\upsilon}^{\tau} \partial \boldsymbol{\upsilon}^{\mu}} \frac{\partial \boldsymbol{\upsilon}^{\tau}}{\partial \mathbf{x}^{\sigma}} \boldsymbol{\upsilon}^{\sigma}. \tag{23}$$

If we now evaluate this last relation along a single extremal, *dxµ*/*dτ* = *vµ*(*x*(*τ*)), we obtain, after recognizing the right hand side of Eq.(23) as *d*(*∂L*/*∂vµ*)/*dτ*, the Euler-Lagrange equation:

$$\frac{\partial L}{\partial \mathbf{x}^{\mu}} = \frac{d}{d\tau} \left( \frac{\partial L}{\partial \dot{\mathbf{x}}^{\mu}} \right) . \tag{24}$$

Eq.(23) is therefore more general than the Euler-Lagrange equation. The latter follows from Eq.(23); but not the other way around.

For the non-homogeneous case we obtain a similar result

$$\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{\mathbf{x}}^i}\right) - \frac{\partial L}{\partial \mathbf{x}^i} = \mathbf{0},\tag{25}$$

but with the important difference that now the curve-parameter *t* is fixed: the solution of Eq.(25) provides us not only with the geometrical shape of the extremal curve, but also with the rate at which this curve is traced back.

#### **2.4. The arbitrariness of the curve parameter**

Let us see how the arbitrariness of the curve parameter *τ* manifests itself when dealing with fields of extremals. It is usual to take advantage of such an arbitrariness in order to simplify the equations of motion. It is well known that in the cases of electromagnetism, for which *L*(*x*, *x*˙) = *mc*(*ηµνx*˙ *µx*˙ *<sup>ν</sup>*)1/2 + *<sup>e</sup> <sup>c</sup> Aµ*(*x*)*x*˙ *<sup>µ</sup>*, and gravitation, for which *L*(*x*, *x*˙)=(*gµν*(*x*)*x*˙ *µx*˙ *<sup>ν</sup>*)1/2, by choosing *τ* such that (*ηµνx*˙ *µx*˙ *<sup>ν</sup>*)1/2 = 1, and (*gµν*(*x*)*x*˙ *µx*˙ *<sup>ν</sup>*)1/2 = 1, respectively, the equations of motion acquire a simple form. We are so led to ask whether the field *v* satisfying the fundamental Eqs.(17) and (18) has a corresponding arbitrariness. That this is indeed the case can be seen as follows. We wish to prove that in case *v<sup>µ</sup>* satisfies Eqs.(17) and (18), so does *w<sup>µ</sup>* = *φvµ*, with *φ*(*x*) > 0 an arbitrary, scalar function. From the homogeneity of the Lagrangian we have *L*(*xµ*, *φvµ*) = *φL*(*xµ*, *vµ*), so that it is seen at once that *w<sup>µ</sup>* satisfies Eq.(17) if *v<sup>µ</sup>* does. Indeed, multiplying Eq.(17) by *φ*(*x*) > 0 leads to

$$\phi(\mathbf{x})\left(L(\mathbf{x},\mathbf{v}) - \mathbf{v}^{\mu}\partial\_{\mu}S\right) = L(\mathbf{x},\phi\mathbf{v}) - \left(\phi\mathbf{v}^{\mu}\right)\partial\_{\mu}S = L(\mathbf{x},\mathbf{w}) - \mathbf{w}^{\mu}\partial\_{\mu}S.\tag{26}$$

The Lagrangian of the "equivalent variational problem" is *<sup>L</sup>*<sup>∗</sup> <sup>=</sup> *<sup>L</sup>* <sup>−</sup> *<sup>v</sup>µ∂µS*. Clearly, *<sup>L</sup>*∗(*x*, *<sup>φ</sup>v*) = *<sup>φ</sup>L*∗(*x*, *<sup>v</sup>*), and hence it follows that

$$\frac{\partial L^\*(\mathbf{x}, w)}{\partial v^{\mu}} = \frac{\partial L^\*(\mathbf{x}, w)}{\partial w^{\nu}} \frac{\partial w^{\nu}}{\partial v^{\mu}} = \frac{\partial L^\*(\mathbf{x}, w)}{\partial w^{\mu}} \phi. \tag{27}$$

10.5772/53843

51

, (30)

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*<sup>µ</sup>*. (31)

= 0. (32)

= 0. (33)

Euler-Lagrange equation, Eq.(25), one gets

the other half of the Hamilton equations.

[1, 13] in which this material is discussed at length.

**3.1. The London equations of superconductivity**

four-potential *Aµ*, whose components are *φ*(*t*, *x<sup>i</sup>*

*∂ ∂x<sup>µ</sup>*

> *∂v<sup>ν</sup> <sup>∂</sup>x<sup>µ</sup>* <sup>−</sup> *<sup>∂</sup>v<sup>µ</sup>*

*x*˙ *µ∂L*/*∂x*˙

**theorem**

general,

we get

*dpi*

*dt* <sup>=</sup> <sup>−</sup>*∂<sup>H</sup> ∂x<sup>i</sup>*

Charathéodory's "Royal Road" to the Calculus of Variations: A Possible Bridge Between Classical and Quantum Physics

In the homogeneous case, as already mentioned, the corresponding expression for *H*, i.e.,

to introduce a Hamiltonian in a number of ways. Carathéodory's approach leads to an infinite set of Hamiltonians, from which we can choose the most suitable one for the problem at hand. We will not go into the details here, but refer the reader to the standard literature

**3. Electromagnetism: The London equations and the Bohr-van Leeuwen**

As mentioned before, there are only two interactions relevant to classical physics:

*µx*˙

Here, *ηµν* = *diag*(+1, −1, −1, −1) is the Minkowski metric tensor and summation over repeated indices from 0 to 3 is understood. The electromagnetic field is given by the

We are now in a position to show how the London equations follow as a *logical* consequence of the relations presented above, when we use Eq.(31). From Eqs.(18) and (19) we obtain, in

This equation can be used to obtain the relativistic version of the London equations: As stated before, because the Lagrangian is homogeneous of first order in *v*, this vector field can be chosen so as to satisfy (*vµvµ*)1/2 = 1 in the region of interest. From Eq.(32) and Eq.(31)

*∂A<sup>ν</sup>*

This condition leads to the London equations, if we go to the non-relativistic limit, *v*2/*c*<sup>2</sup> ≪ 1. Indeed, after multiplication by *ne*, with *n* meaning a uniform particle density, Eq.(33) can be

*<sup>∂</sup>x<sup>µ</sup>* <sup>−</sup> *<sup>∂</sup>A<sup>µ</sup> ∂x<sup>ν</sup>* 

 − *∂ ∂x<sup>ν</sup>*

*e mc*<sup>2</sup> *<sup>ν</sup>*)1/2 +

*e c*

) and **A**(*t*, *x<sup>i</sup>*

 *L*(*x*, *v*(*x*)) *∂v<sup>µ</sup>*

*Aµ*(*x*)*x*˙

).

electromagnetism and gravitation. In electromagnetism, the Lagrangian is given by

*L*(*x*, *x*˙) = *mc*(*ηµνx*˙

 *L*(*x*, *v*(*x*)) *∂v<sup>ν</sup>*

*<sup>∂</sup>x<sup>ν</sup>* <sup>+</sup>

*<sup>µ</sup>* − *L*, vanishes identically by virtue of Eq.(16) . It is nonetheless generally possible

On the other hand,

$$\frac{\partial L^\*(\mathbf{x}, w)}{\partial v^\mu} = \frac{\partial}{\partial v^\mu} \left( \phi L^\*(\mathbf{x}, v) \right) = \phi \frac{\partial L^\*(\mathbf{x}, v)}{\partial v^\mu} = \phi \left( \frac{\partial L}{\partial v^\mu} - \frac{\partial S}{\partial \mathbf{x}^\mu} \right) = 0,\tag{28}$$

on account of Eq.(17). In view of Eq.(27) we have then that *<sup>∂</sup>L*∗(*x*, *<sup>w</sup>*)/*∂w<sup>µ</sup>* <sup>=</sup> 0. In summary, Eqs.(17,18) hold with *v* being replaced by *w*, so that both velocity fields solve our variational problem for the same *S*(*x*). We have thus the freedom to choose *φ* according to our convenience. The integral curves of *vµ*(*x*) and *wµ*(*x*) coincide with each other, differing only in their parametrization.

#### **2.5. Hamiltonians**

The introduction of a Hamiltonian offers no problem in the non-homogeneous case, where it was defined as *H*(*x<sup>i</sup>* , *pi*) ≡ *<sup>x</sup>*˙ *i* (*t*, *<sup>x</sup>*, *<sup>p</sup>*)*pi* − *<sup>L</sup>*(*t*, *<sup>x</sup>*, *<sup>x</sup>*˙(*t*, *<sup>x</sup>*, *<sup>p</sup>*)), with *pi* ≡ *<sup>∂</sup>L*/*∂x*˙ *i* ; the condition for solving *x*˙ *<sup>i</sup>* in terms of (*x<sup>j</sup>* , *pj*) being assumed to be fulfilled: det(*∂*2*L*/*∂x*˙ *i ∂x*˙ *<sup>j</sup>* ) �= 0. It is then straightforward [1, 13] to obtain

$$\frac{\partial H}{\partial p\_i} = \dot{\mathbf{x}}^i = \frac{d\mathbf{x}^i}{dt} \,\tag{29}$$

which constitute half of the Hamilton equations. It is also easy to sow that *∂H*/*∂t* = −*∂L*/*∂t* and *∂H*/*∂x<sup>i</sup>* = −*∂L*/*∂x<sup>i</sup>* . Using this last result together with *pi* = *∂L*/*∂x*˙ *<sup>i</sup>* in the <sup>50</sup> Advances in Quantum Mechanics Charathéodory's "Royal Road" to the Calculus of Variations: A Possible Bridge Between Classical and Quantum Physics 11 10.5772/53843 Charathéodory's "Royal Road" to the Calculus of Variations: A Possible Bridge Between Classical and Quantum Physics http://dx.doi.org/10.5772/53843 51

Euler-Lagrange equation, Eq.(25), one gets

10 Quantum Mechanics

*L*(*x*, *x*˙) = *mc*(*ηµνx*˙

**2.4. The arbitrariness of the curve parameter**

*<sup>ν</sup>*)1/2 + *<sup>e</sup>*

*<sup>L</sup>*(*x*, *<sup>v</sup>*) − *<sup>v</sup>µ∂µ<sup>S</sup>*

*<sup>∂</sup>L*∗(*x*, *<sup>w</sup>*)

*<sup>L</sup>*∗(*x*, *<sup>φ</sup>v*) = *<sup>φ</sup>L*∗(*x*, *<sup>v</sup>*), and hence it follows that

*<sup>c</sup> Aµ*(*x*)*x*˙

*µx*˙

Eq.(17) if *v<sup>µ</sup>* does. Indeed, multiplying Eq.(17) by *φ*(*x*) > 0 leads to

*<sup>∂</sup>v<sup>µ</sup>* <sup>=</sup> *<sup>∂</sup>L*∗(*x*, *<sup>w</sup>*)

*<sup>∂</sup>v<sup>µ</sup>* (*φL*∗(*x*, *<sup>v</sup>*)) <sup>=</sup> *<sup>φ</sup>*

*∂w<sup>ν</sup>*

*µx*˙

by choosing *τ* such that (*ηµνx*˙

*φ*(*x*) 

*<sup>∂</sup>L*∗(*x*, *<sup>w</sup>*)

only in their parametrization.

**2.5. Hamiltonians**

it was defined as *H*(*x<sup>i</sup>*

for solving *x*˙

*<sup>∂</sup>v<sup>µ</sup>* <sup>=</sup> *<sup>∂</sup>*

, *pi*) ≡ *<sup>x</sup>*˙ *i*

*<sup>i</sup>* in terms of (*x<sup>j</sup>*

then straightforward [1, 13] to obtain

−*∂L*/*∂t* and *∂H*/*∂x<sup>i</sup>* = −*∂L*/*∂x<sup>i</sup>*

On the other hand,

Let us see how the arbitrariness of the curve parameter *τ* manifests itself when dealing with fields of extremals. It is usual to take advantage of such an arbitrariness in order to simplify the equations of motion. It is well known that in the cases of electromagnetism, for which

*<sup>ν</sup>*)1/2 = 1, and (*gµν*(*x*)*x*˙

equations of motion acquire a simple form. We are so led to ask whether the field *v* satisfying the fundamental Eqs.(17) and (18) has a corresponding arbitrariness. That this is indeed the case can be seen as follows. We wish to prove that in case *v<sup>µ</sup>* satisfies Eqs.(17) and (18), so does *w<sup>µ</sup>* = *φvµ*, with *φ*(*x*) > 0 an arbitrary, scalar function. From the homogeneity of the Lagrangian we have *L*(*xµ*, *φvµ*) = *φL*(*xµ*, *vµ*), so that it is seen at once that *w<sup>µ</sup>* satisfies

The Lagrangian of the "equivalent variational problem" is *<sup>L</sup>*<sup>∗</sup> <sup>=</sup> *<sup>L</sup>* <sup>−</sup> *<sup>v</sup>µ∂µS*. Clearly,

*∂w<sup>ν</sup>*

*<sup>∂</sup>L*∗(*x*, *<sup>v</sup>*) *<sup>∂</sup>v<sup>µ</sup>* <sup>=</sup> *<sup>φ</sup>*

on account of Eq.(17). In view of Eq.(27) we have then that *<sup>∂</sup>L*∗(*x*, *<sup>w</sup>*)/*∂w<sup>µ</sup>* <sup>=</sup> 0. In summary, Eqs.(17,18) hold with *v* being replaced by *w*, so that both velocity fields solve our variational problem for the same *S*(*x*). We have thus the freedom to choose *φ* according to our convenience. The integral curves of *vµ*(*x*) and *wµ*(*x*) coincide with each other, differing

The introduction of a Hamiltonian offers no problem in the non-homogeneous case, where

*∂H ∂pi*

= *x*˙

which constitute half of the Hamilton equations. It is also easy to sow that *∂H*/*∂t* =

*<sup>i</sup>* <sup>=</sup> *dx<sup>i</sup>*

(*t*, *<sup>x</sup>*, *<sup>p</sup>*)*pi* − *<sup>L</sup>*(*t*, *<sup>x</sup>*, *<sup>x</sup>*˙(*t*, *<sup>x</sup>*, *<sup>p</sup>*)), with *pi* ≡ *<sup>∂</sup>L*/*∂x*˙

. Using this last result together with *pi* = *∂L*/*∂x*˙

, *pj*) being assumed to be fulfilled: det(*∂*2*L*/*∂x*˙

*<sup>∂</sup>v<sup>µ</sup>* <sup>=</sup> *<sup>∂</sup>L*∗(*x*, *<sup>w</sup>*)

*<sup>µ</sup>*, and gravitation, for which *L*(*x*, *x*˙)=(*gµν*(*x*)*x*˙

*µx*˙

= *<sup>L</sup>*(*x*, *<sup>φ</sup>v*) − (*φvµ*) *∂µ<sup>S</sup>* = *<sup>L</sup>*(*x*, *<sup>w</sup>*) − *<sup>w</sup>µ∂µS*. (26)

 *∂L <sup>∂</sup>v<sup>µ</sup>* <sup>−</sup> *<sup>∂</sup><sup>S</sup> ∂x<sup>µ</sup>*  *µx*˙ *<sup>ν</sup>*)1/2,

*<sup>ν</sup>*)1/2 = 1, respectively, the

*<sup>∂</sup>w<sup>µ</sup> <sup>φ</sup>*. (27)

*i*

*i ∂x*˙ *<sup>j</sup>*

*dt* , (29)

= 0, (28)

; the condition

) �= 0. It is

*<sup>i</sup>* in the

$$\frac{dp\_i}{dt} = -\frac{\partial H}{\partial x^i} \tag{30}$$

the other half of the Hamilton equations.

In the homogeneous case, as already mentioned, the corresponding expression for *H*, i.e., *x*˙ *µ∂L*/*∂x*˙ *<sup>µ</sup>* − *L*, vanishes identically by virtue of Eq.(16) . It is nonetheless generally possible to introduce a Hamiltonian in a number of ways. Carathéodory's approach leads to an infinite set of Hamiltonians, from which we can choose the most suitable one for the problem at hand. We will not go into the details here, but refer the reader to the standard literature [1, 13] in which this material is discussed at length.

### **3. Electromagnetism: The London equations and the Bohr-van Leeuwen theorem**

#### **3.1. The London equations of superconductivity**

As mentioned before, there are only two interactions relevant to classical physics: electromagnetism and gravitation. In electromagnetism, the Lagrangian is given by

$$L(\mathbf{x}, \dot{\mathbf{x}}) = mc(\eta\_{\mu\nu}\dot{\mathbf{x}}^{\mu}\dot{\mathbf{x}}^{\nu})^{1/2} + \frac{e}{c}A\_{\mu}(\mathbf{x})\dot{\mathbf{x}}^{\mu}.\tag{31}$$

Here, *ηµν* = *diag*(+1, −1, −1, −1) is the Minkowski metric tensor and summation over repeated indices from 0 to 3 is understood. The electromagnetic field is given by the four-potential *Aµ*, whose components are *φ*(*t*, *x<sup>i</sup>* ) and **A**(*t*, *x<sup>i</sup>* ).

We are now in a position to show how the London equations follow as a *logical* consequence of the relations presented above, when we use Eq.(31). From Eqs.(18) and (19) we obtain, in general,

$$\frac{\partial}{\partial \mathbf{x}^{\mu}} \left( \frac{L(\mathbf{x}, v(\mathbf{x}))}{\partial v^{\nu}} \right) - \frac{\partial}{\partial \mathbf{x}^{\nu}} \left( \frac{L(\mathbf{x}, v(\mathbf{x}))}{\partial v^{\mu}} \right) = \mathbf{0}. \tag{32}$$

This equation can be used to obtain the relativistic version of the London equations: As stated before, because the Lagrangian is homogeneous of first order in *v*, this vector field can be chosen so as to satisfy (*vµvµ*)1/2 = 1 in the region of interest. From Eq.(32) and Eq.(31) we get

$$
\frac{\partial v\_{\nu}}{\partial \mathbf{x}^{\mu}} - \frac{\partial v\_{\mu}}{\partial \mathbf{x}^{\nu}} + \frac{e}{mc^{2}} \left( \frac{\partial A\_{\nu}}{\partial \mathbf{x}^{\mu}} - \frac{\partial A\_{\mu}}{\partial \mathbf{x}^{\nu}} \right) = \mathbf{0}.\tag{33}
$$

This condition leads to the London equations, if we go to the non-relativistic limit, *v*2/*c*<sup>2</sup> ≪ 1. Indeed, after multiplication by *ne*, with *n* meaning a uniform particle density, Eq.(33) can be brought into the form:

$$
\frac{
\partial \dot{j}\_\nu
}{
\partial \mathbf{x}^\mu
} - \frac{
\partial \dot{j}\_\mu
}{
\partial \mathbf{x}^\nu
} + \frac{m\varepsilon^2}{mc^2} \left( \frac{
\partial A\_\nu
}{
\partial \mathbf{x}^\mu
} - \frac{
\partial A\_\mu
}{
\partial \mathbf{x}^\nu
} \right) = \mathbf{0},
\tag{34}
$$

10.5772/53843

53

. (40)

http://dx.doi.org/10.5772/53843

*<sup>µ</sup>E<sup>µ</sup>* = 0, with

with *<sup>F</sup>νµ* = *∂νA<sup>µ</sup>* − *∂µAν*, which relates to **<sup>E</sup>** and **<sup>B</sup>** by *Ei* = *<sup>F</sup>*0*<sup>i</sup>* and *Bi* = −*ǫijkFjk*/2, with *<sup>ǫ</sup>ijk* the totally antisymmetric symbol and latin indices running from 1 to 3. The nonrelativistic

Charathéodory's "Royal Road" to the Calculus of Variations: A Possible Bridge Between Classical and Quantum Physics

*m* **<sup>E</sup>** <sup>+</sup> **<sup>v</sup>** *c* ×**B** 

*<sup>µ</sup>*)/*dτ* − *∂L*/*∂x<sup>µ</sup>* being the Euler vector[13]. Such a result follows from Eq.(16).

The left-hand side of this equation is the *convective derivative*, which reduces to *d***v**/*dt* by restriction to a single extremal. Analogously, Eq.(39) becomes the well-known Lorentz equation when evaluated along a single extremal: *dxµ*/*ds* = *vµ*(*x*(*s*)). In this case, *vµ*(*x*(*s*))*∂vν*(*x*(*s*))/*∂x<sup>µ</sup>* = *dvν*(*s*)/*ds*. Thus, we see that the Lorentz equation for a single particle follows from the more general Eq.(39). For *µ* = 0 Eq.(39) gives an equation which can be derived from Eq.(40) by scalar multiplication with **v**. This is the energy equation. It is worth mentioning that this last fact is a particular manifestation of a well-known result valid for *homogeneous* Lagrangians: only *n* − 1 out of the *n* Euler-Lagrange

Some remarks are in place here. Our derivation of the London equations brings into evidence that they have a validity that goes beyond their original scope. They cannot be seen by themselves as characterizing the phenomenon of superconductivity. Instead, they describe a "dust" of charged particles moving along the extremals of the Lagrangian given by Eq.(31). The field *A<sup>µ</sup>* under which these particles move could be produced by external sources, or else be the field resulting from the superposition of some external fields with those produced by the charges themselves. In this last case, the Maxwell and London equations constitute a self-consistent system. Only under special circumstances, the system of charges can be in a state of collective motion that may be described by the field of extremals obeying Eq.(39). This is the superconducting phase, for which quantum aspects are known to play a fundamental role[17]. However, *once* the phase transition from the normal to the superconducting state has occurred, it becomes possible to describe some aspects of the superconducting state by classical means. This is a case analogous to the one encountered in laser theory. Indeed, several features of a lasing system can be understood within a semi-classical laser theory, whereby the electromagnetic field is treated as a classical, non-quantized field. Perhaps some plasmas could reach the limit of perfect conductivity. However, in order to produce a Meissner-like effect some conditions should be met. It is necessary, for instance, that the available free-energy of the plasma is sufficient to overcome the magnetic field energy, so that the magnetic field can be driven out of the plasma [5–8]. The so-called helicity of the system should also play a role, attaining the value zero for the superconducting state to be

In any case, we see that Carathéodory's approach can be a fruitful one in physics. In the case of superconductivity, from the sole assumption that the Lagrangian be given by Eq.(31) one can derive all the equations that were more or less guessed, in the course of almost twenty five years, since Kamerlingh Onnes discovered superconductivity in 1911, until the London model was proposed, in 1935. But beyond this, there are other aspects that can be illuminated

by following Carathéodory's approach, as we shall see next.

*∂***v**

*<sup>∂</sup><sup>t</sup>* <sup>+</sup> (**<sup>v</sup>** · ∇) **<sup>v</sup>** <sup>=</sup> *<sup>e</sup>*

equations are independent from each other in this case, due to the identity *x*˙

limit of Eq.(39) reads

*<sup>E</sup><sup>µ</sup>* ≡ *<sup>d</sup>*(*∂L*/*∂x*˙

reached [9].

where *<sup>j</sup><sup>µ</sup>* ≡ *nevµ*. In the non-relativistic limit Eq.(34) reduces, for *<sup>µ</sup>*, *<sup>ν</sup>* = *<sup>i</sup>*, *<sup>k</sup>* = 1, 2, 3, to

$$\frac{\partial j\_k}{\partial \mathbf{x}^i} - \frac{\partial j\_i}{\partial \mathbf{x}^k} = -\frac{ne^2}{mc} \left( \frac{\partial A\_k}{\partial \mathbf{x}^i} - \frac{\partial A\_i}{\partial \mathbf{x}^k} \right) \tag{35}$$

where we have used *x*˙ *<sup>µ</sup>*(*τ*) = *<sup>γ</sup>*(1, **<sup>v</sup>**(*t*)/*c*) with *<sup>γ</sup>* <sup>≡</sup> (<sup>1</sup> <sup>−</sup> **<sup>v</sup>**2/*c*2)−1/2 <sup>≈</sup> 1. In three-vector notation this equation reads

$$
\nabla \times \mathbf{j} = -\frac{ne^2}{mc} \nabla \times \mathbf{A} = -\frac{ne^2}{mc} \mathbf{B},
\tag{36}
$$

which is the London equation [16]. Eq. (36), together with the steady-state Maxwell equation, ∇ × **B** = (4*π*/*c*)**j**, lead to ∇2**B** = 4*πne*2/*mc*<sup>2</sup> **B**, from which the Meissner effect follows. By considering now the case *µ* = 0, *ν* = *k* = 1, 2, 3 in Eq.(34), we obtain

$$\frac{\partial j\_k}{\partial \mathbf{x}^0} - \frac{\partial j\_0}{\partial \mathbf{x}^k} = \frac{ne^2}{mc^2} \left( \frac{\partial A\_0}{\partial \mathbf{x}^k} - \frac{\partial A\_k}{\partial \mathbf{x}^0} \right). \tag{37}$$

Multiplying this equation by −*c*<sup>2</sup> and using three-vector notation it reads, with *<sup>j</sup>*<sup>0</sup> = *<sup>c</sup>ρ*,

$$
\frac{\partial \mathbf{j}}{\partial t} + c^2 \nabla \rho = \frac{ne^2}{m} \mathbf{E}.\tag{38}
$$

This equation was also postulated by the London brothers as part of the phenomenological description of superconductors. It was guessed as a relativistic generalization of the equation that should hold for a perfect conductor. Without the *ρ*-term (which in our case vanishes due to the assumed uniformity of *n*) it is nothing but the Newton, or "acceleration" equation for charges moving under the force *e***E**. The *ρ*-term was originally conceived as a relativistic "time-like supplement " to the current **j** [16]. We see that the London equations are in fact the non-relativistic limit of an integrability condition, Eq.(33), which follows from the variational principle *δ Lds* = 0 alone. The physical content of this procedure appears when we interpret the integral curves of *v*(*x*) as streamlines of an ideal fluid. By contracting Eq.(33) with *v<sup>µ</sup>* and using *vµ∂νv<sup>µ</sup>* = 0 (which follows from *vµv<sup>µ</sup>* = 1) we obtain

$$
\upsilon^{\mu}\frac{\partial\upsilon\_{\nu}}{\partial x^{\mu}} = \frac{e}{mc^{2}}F\_{\nu\mu}\upsilon^{\mu}\,,\tag{39}
$$

12 Quantum Mechanics

brought into the form:

where we have used *x*˙

variational principle *δ*

notation this equation reads

∇ × **B** = (4*π*/*c*)**j**, lead to ∇2**B** =

*∂j<sup>ν</sup> <sup>∂</sup>x<sup>µ</sup>* <sup>−</sup> *<sup>∂</sup>j<sup>µ</sup>*

> *∂jk <sup>∂</sup>x<sup>i</sup>* <sup>−</sup> *<sup>∂</sup>ji*

*<sup>∂</sup>x<sup>ν</sup>* <sup>+</sup>

*ne*<sup>2</sup> *mc*<sup>2</sup>

where *<sup>j</sup><sup>µ</sup>* ≡ *nevµ*. In the non-relativistic limit Eq.(34) reduces, for *<sup>µ</sup>*, *<sup>ν</sup>* = *<sup>i</sup>*, *<sup>k</sup>* = 1, 2, 3, to

*mc*

*<sup>∂</sup>x<sup>k</sup>* <sup>=</sup> <sup>−</sup>*ne*<sup>2</sup>

∇ × **<sup>j</sup>** <sup>=</sup> <sup>−</sup>*ne*<sup>2</sup>

By considering now the case *µ* = 0, *ν* = *k* = 1, 2, 3 in Eq.(34), we obtain

*∂jk <sup>∂</sup>x*<sup>0</sup> <sup>−</sup> *<sup>∂</sup>j*<sup>0</sup> *mc*

4*πne*2/*mc*<sup>2</sup>

*<sup>∂</sup>x<sup>k</sup>* <sup>=</sup> *ne*<sup>2</sup> *mc*<sup>2</sup>

*∂***j** *∂t*

with *v<sup>µ</sup>* and using *vµ∂νv<sup>µ</sup>* = 0 (which follows from *vµv<sup>µ</sup>* = 1) we obtain

*<sup>v</sup><sup>µ</sup> <sup>∂</sup>v<sup>ν</sup> <sup>∂</sup>x<sup>µ</sup>* <sup>=</sup> *<sup>e</sup>*

Multiplying this equation by −*c*<sup>2</sup> and using three-vector notation it reads, with *<sup>j</sup>*<sup>0</sup> = *<sup>c</sup>ρ*,

<sup>+</sup> *<sup>c</sup>*2∇*<sup>ρ</sup>* <sup>=</sup> *ne*<sup>2</sup>

This equation was also postulated by the London brothers as part of the phenomenological description of superconductors. It was guessed as a relativistic generalization of the equation that should hold for a perfect conductor. Without the *ρ*-term (which in our case vanishes due to the assumed uniformity of *n*) it is nothing but the Newton, or "acceleration" equation for charges moving under the force *e***E**. The *ρ*-term was originally conceived as a relativistic "time-like supplement " to the current **j** [16]. We see that the London equations are in fact the non-relativistic limit of an integrability condition, Eq.(33), which follows from the

we interpret the integral curves of *v*(*x*) as streamlines of an ideal fluid. By contracting Eq.(33)

which is the London equation [16]. Eq. (36), together with the steady-state Maxwell equation,

*∂A<sup>ν</sup>*

*<sup>∂</sup>x<sup>µ</sup>* <sup>−</sup> *<sup>∂</sup>A<sup>µ</sup> ∂x<sup>ν</sup>* 

*∂Ak*

∇×**<sup>A</sup>** <sup>=</sup> <sup>−</sup>*ne*<sup>2</sup>

*∂A*<sup>0</sup>

*m*

*Lds* = 0 alone. The physical content of this procedure appears when

*<sup>∂</sup>x<sup>k</sup>* <sup>−</sup> *<sup>∂</sup>Ak ∂x*<sup>0</sup> 

*<sup>∂</sup>x<sup>i</sup>* <sup>−</sup> *<sup>∂</sup>Ai ∂x<sup>k</sup>* 

*<sup>µ</sup>*(*τ*) = *<sup>γ</sup>*(1, **<sup>v</sup>**(*t*)/*c*) with *<sup>γ</sup>* <sup>≡</sup> (<sup>1</sup> <sup>−</sup> **<sup>v</sup>**2/*c*2)−1/2 <sup>≈</sup> 1. In three-vector

*mc*

= 0, (34)

, (35)

**B**, (36)

. (37)

**E**. (38)

*mc*<sup>2</sup> *<sup>F</sup>νµvµ*, (39)

**B**, from which the Meissner effect follows.

with *<sup>F</sup>νµ* = *∂νA<sup>µ</sup>* − *∂µAν*, which relates to **<sup>E</sup>** and **<sup>B</sup>** by *Ei* = *<sup>F</sup>*0*<sup>i</sup>* and *Bi* = −*ǫijkFjk*/2, with *<sup>ǫ</sup>ijk* the totally antisymmetric symbol and latin indices running from 1 to 3. The nonrelativistic limit of Eq.(39) reads

$$\frac{\partial \mathbf{v}}{\partial t} + \left(\mathbf{v} \cdot \nabla\right)\mathbf{v} = \frac{e}{m} \left(\mathbf{E} + \frac{\mathbf{v}}{c} \times \mathbf{B}\right). \tag{40}$$

The left-hand side of this equation is the *convective derivative*, which reduces to *d***v**/*dt* by restriction to a single extremal. Analogously, Eq.(39) becomes the well-known Lorentz equation when evaluated along a single extremal: *dxµ*/*ds* = *vµ*(*x*(*s*)). In this case, *vµ*(*x*(*s*))*∂vν*(*x*(*s*))/*∂x<sup>µ</sup>* = *dvν*(*s*)/*ds*. Thus, we see that the Lorentz equation for a single particle follows from the more general Eq.(39). For *µ* = 0 Eq.(39) gives an equation which can be derived from Eq.(40) by scalar multiplication with **v**. This is the energy equation. It is worth mentioning that this last fact is a particular manifestation of a well-known result valid for *homogeneous* Lagrangians: only *n* − 1 out of the *n* Euler-Lagrange equations are independent from each other in this case, due to the identity *x*˙ *<sup>µ</sup>E<sup>µ</sup>* = 0, with *<sup>E</sup><sup>µ</sup>* ≡ *<sup>d</sup>*(*∂L*/*∂x*˙ *<sup>µ</sup>*)/*dτ* − *∂L*/*∂x<sup>µ</sup>* being the Euler vector[13]. Such a result follows from Eq.(16).

Some remarks are in place here. Our derivation of the London equations brings into evidence that they have a validity that goes beyond their original scope. They cannot be seen by themselves as characterizing the phenomenon of superconductivity. Instead, they describe a "dust" of charged particles moving along the extremals of the Lagrangian given by Eq.(31). The field *A<sup>µ</sup>* under which these particles move could be produced by external sources, or else be the field resulting from the superposition of some external fields with those produced by the charges themselves. In this last case, the Maxwell and London equations constitute a self-consistent system. Only under special circumstances, the system of charges can be in a state of collective motion that may be described by the field of extremals obeying Eq.(39). This is the superconducting phase, for which quantum aspects are known to play a fundamental role[17]. However, *once* the phase transition from the normal to the superconducting state has occurred, it becomes possible to describe some aspects of the superconducting state by classical means. This is a case analogous to the one encountered in laser theory. Indeed, several features of a lasing system can be understood within a semi-classical laser theory, whereby the electromagnetic field is treated as a classical, non-quantized field. Perhaps some plasmas could reach the limit of perfect conductivity. However, in order to produce a Meissner-like effect some conditions should be met. It is necessary, for instance, that the available free-energy of the plasma is sufficient to overcome the magnetic field energy, so that the magnetic field can be driven out of the plasma [5–8]. The so-called helicity of the system should also play a role, attaining the value zero for the superconducting state to be reached [9].

In any case, we see that Carathéodory's approach can be a fruitful one in physics. In the case of superconductivity, from the sole assumption that the Lagrangian be given by Eq.(31) one can derive all the equations that were more or less guessed, in the course of almost twenty five years, since Kamerlingh Onnes discovered superconductivity in 1911, until the London model was proposed, in 1935. But beyond this, there are other aspects that can be illuminated by following Carathéodory's approach, as we shall see next.

#### **3.2. Beyond the London equations**

Let us address the case when the charge density *ρ* is not constant, as previously assumed. There is a close relationship between the norm of our velocity field, i.e., *φ*(*x*) = *<sup>v</sup>µv<sup>µ</sup>*1/2, and *ρ*(*x*). It can be shown that it is always possible to choose *v<sup>µ</sup>* so that the continuity equation *∂µj <sup>µ</sup>* = 0 holds. Here, *j <sup>µ</sup>* :<sup>=</sup> *<sup>ρ</sup>v<sup>µ</sup>* and *<sup>ρ</sup>* <sup>=</sup> *necφ*<sup>−</sup>1, *<sup>n</sup>* being a free parameter whose dimensions are 1/volume. Indeed, in view of the aforementioned possibility of changing the field *v<sup>µ</sup>* by *w<sup>µ</sup>* = *φ*˜*vµ*, we can always satisfy the continuity equation. For, if *∂µj <sup>µ</sup>* = −*f* , we may choose ˜*j <sup>µ</sup>* = *φ*˜*j <sup>µ</sup>* such that *∂µ* ˜*j <sup>µ</sup>* = *φ∂*˜ *<sup>µ</sup>j <sup>µ</sup>* + *j µ∂µφ*˜ = 0. Putting *ψ* = log *φ*˜, we need to solve *j µ∂µψ* = *f* , which is always possible.

Coming back to our Lagrangian of Eq.(31), by replacing it in Eq.(18), we obtain

$$w\_{\mu} = \frac{\Phi}{mc} \left( \partial\_{\mu} S - \frac{e}{c} A\_{\mu} \right) , \tag{41}$$

10.5772/53843

55

*<sup>µ</sup>* = *ρ∂µv<sup>µ</sup>* +

http://dx.doi.org/10.5772/53843

*<sup>µ</sup>* = 0, and we

*<sup>ν</sup>* to be the same as

As we saw before, *<sup>v</sup><sup>µ</sup>* can be chosen so that *<sup>j</sup><sup>µ</sup>* = *necw<sup>µ</sup>* = (*nec*/*φ*)*v<sup>µ</sup>* ≡ *<sup>ρ</sup>*(*x*)*v<sup>µ</sup>* satisfies

Charathéodory's "Royal Road" to the Calculus of Variations: A Possible Bridge Between Classical and Quantum Physics

has no dimensions and *n* is a free parameter such that *ne* has dimensions of charge per unit volume. While *n* is a constant, *ρ*(*x*) is a non-uniform charge density. Thus, the scalar field

*<sup>µ</sup>* = 0 implies a restriction on *∂µvµ*. To see this, observe that *∂µj*

with *∂µv<sup>µ</sup>* = *f* . On the other hand, from *vµv<sup>µ</sup>* = *φ*<sup>2</sup> it follows that *φ∂µφ* = *vν∂µvν*. Eq.(48)

We could argue that the second term on the right hand side is not physical, because we could choose *φ* = 1, as we did before, getting Eq.(39). However, such a choice is not available any

Coming back to Eq.(47), we see that it implies that *∂µA*′*<sup>µ</sup>* <sup>=</sup> 0, i.e., *<sup>A</sup>*′*<sup>µ</sup>* is in the Lorentz

before, we are assuming that *Fµν* is generated by the same currents upon which this field is acting. That is, we are considering a closed system of charges and fields. We have then,

*∂µFµν* <sup>=</sup> *<sup>A</sup>*′*<sup>ν</sup>* <sup>=</sup> <sup>4</sup>*<sup>π</sup>*

*<sup>A</sup>*′*<sup>ν</sup>* <sup>=</sup> <sup>−</sup> *mc*

*ne*<sup>2</sup> *<sup>j</sup>*

*∂µ∂µ<sup>S</sup>* <sup>≡</sup> *<sup>S</sup>* <sup>=</sup> *<sup>e</sup>*

*<sup>µ</sup>* <sup>=</sup> *<sup>A</sup><sup>µ</sup>* <sup>−</sup> (*c*/*e*) *∂µS*, the scalar function *<sup>S</sup>* must satisfy

*<sup>µ</sup>* = *ρvµ*, so that the above considerations apply.

*c*

*c j*

*<sup>v</sup><sup>ν</sup>* <sup>=</sup> *<sup>e</sup><sup>φ</sup>*

*vµ φ*

*vµv<sup>ν</sup>*

It is also worth noting that instead of Eq.(39) we have now

Let us consider now Maxwell equations, *∂µFµν* = (4*π*/*c*)*j*

*mc*<sup>2</sup> *<sup>F</sup>νµv<sup>µ</sup>* <sup>+</sup>

 *<sup>v</sup><sup>µ</sup> <sup>∂</sup> <sup>∂</sup>x<sup>µ</sup>* ln *<sup>φ</sup>*

longer when we invoke charge (or matter) conservation. In such a case, *∂µj*

*<sup>v</sup>µv<sup>µ</sup>*1/2, the norm of the velocity field, is related to the density *<sup>ρ</sup>*(*x*) by *<sup>ρ</sup>* <sup>=</sup> *necφ*<sup>−</sup>1.

*<sup>µ</sup>* = 0. The factor *nec* is included for dimensional purposes: *c*/*φ*

*∂µφ* = *f* , (48)

*mc*<sup>2</sup> *<sup>F</sup>νµv<sup>µ</sup>* <sup>+</sup> *f vν*. (50)

*∂µAµ*. (51)

*<sup>ν</sup>*, (52)

*<sup>ν</sup>*, (53)

*<sup>ν</sup>*. If we take *j*

<sup>2</sup>*φ*<sup>2</sup> (*∂µv<sup>ν</sup>* <sup>+</sup> *∂νvµ*) = *∂σvσ*. (49)

the continuity equation *∂µj*

*vµ∂µρ* = 0. This can be rewritten as

*<sup>v</sup><sup>µ</sup> <sup>∂</sup>v<sup>ν</sup>*

*<sup>µ</sup>* with *v<sup>µ</sup>* by *j*

using *<sup>F</sup>µν* <sup>=</sup> *<sup>∂</sup>µA*′*<sup>ν</sup>* <sup>−</sup> *<sup>∂</sup>νA*′*<sup>µ</sup>* and *∂µA*′*<sup>µ</sup>* <sup>=</sup> 0,

*<sup>∂</sup>x<sup>µ</sup>* <sup>=</sup> *<sup>e</sup><sup>φ</sup>*

*φ* =

Note that *∂µj*

then implies that

must relate *j*

gauge. Because of *<sup>A</sup>*′

while from Eq. (47) we get

with *φ* := *<sup>v</sup>µv<sup>µ</sup>*1/2. Using the gauge freedom of *<sup>A</sup><sup>µ</sup>* we may replace this field by

$$A'\_{\mu} = A\_{\mu} - \frac{c}{e} \partial\_{\mu} S\_{\prime} \tag{42}$$

in which case Eq.(41) reads

$$
v\_{\mu} = -\phi \left(\frac{\varepsilon}{mc^2}\right) A'\_{\mu}.\tag{43}$$

From this equation and *vµv<sup>µ</sup>* = *φ*<sup>2</sup> we get

$$A\_{\mu}^{\prime}A^{\prime \mu} = \left(\frac{mc^2}{e}\right)^2. \tag{44}$$

Eq.(32) applied to the present case gives

$$\frac{\partial}{\partial \mathbf{x}^{\mu}} \left( \frac{v\_{\nu}}{\Phi} \right) - \frac{\partial}{\partial \mathbf{x}^{\nu}} \left( \frac{v\_{\mu}}{\Phi} \right) + \frac{e}{mc^{2}} \left( \frac{\partial A\_{\nu}}{\partial \mathbf{x}^{\mu}} - \frac{\partial A\_{\mu}}{\partial \mathbf{x}^{\nu}} \right) = \mathbf{0}. \tag{45}$$

It is clear that this equation holds for *<sup>A</sup>*′ *<sup>µ</sup>* as well. Eq.(43) is a particular solution of this equation. By Fourier-transforming Eq.(45) we obtain, with *w<sup>µ</sup>* := *vµ*/*φ*,

$$k^{\mu}w^{\nu} - k^{\nu}w^{\mu} = -\frac{e}{mc^2} \left( k^{\mu}A^{\nu} - k^{\nu}A^{\mu} \right). \tag{46}$$

As for the Fourier-transformed version of Eq.(43), it is given by

$$w\_{\mu}(k) = -\frac{e}{mc^2}A\_{\mu}'(k). \tag{47}$$

As we saw before, *<sup>v</sup><sup>µ</sup>* can be chosen so that *<sup>j</sup><sup>µ</sup>* = *necw<sup>µ</sup>* = (*nec*/*φ*)*v<sup>µ</sup>* ≡ *<sup>ρ</sup>*(*x*)*v<sup>µ</sup>* satisfies the continuity equation *∂µj <sup>µ</sup>* = 0. The factor *nec* is included for dimensional purposes: *c*/*φ* has no dimensions and *n* is a free parameter such that *ne* has dimensions of charge per unit volume. While *n* is a constant, *ρ*(*x*) is a non-uniform charge density. Thus, the scalar field *φ* = *<sup>v</sup>µv<sup>µ</sup>*1/2, the norm of the velocity field, is related to the density *<sup>ρ</sup>*(*x*) by *<sup>ρ</sup>* <sup>=</sup> *necφ*<sup>−</sup>1. Note that *∂µj <sup>µ</sup>* = 0 implies a restriction on *∂µvµ*. To see this, observe that *∂µj <sup>µ</sup>* = *ρ∂µv<sup>µ</sup>* + *vµ∂µρ* = 0. This can be rewritten as

> *vµ φ ∂µφ* = *f* , (48)

with *∂µv<sup>µ</sup>* = *f* . On the other hand, from *vµv<sup>µ</sup>* = *φ*<sup>2</sup> it follows that *φ∂µφ* = *vν∂µvν*. Eq.(48) then implies that

$$\frac{\upsilon^{\mu}\upsilon^{\nu}}{2\mathfrak{g}^{2}}(\partial\_{\mu}\upsilon\_{\nu}+\partial\_{\nu}\upsilon\_{\mu})=\partial\_{\sigma}\upsilon^{\sigma}.\tag{49}$$

It is also worth noting that instead of Eq.(39) we have now

$$
\upsilon^{\mu}\frac{\partial\upsilon\_{\nu}}{\partial x^{\mu}} = \frac{e\phi}{mc^{2}}F\_{\nu\mu}\upsilon^{\mu} + \left(\upsilon^{\mu}\frac{\partial}{\partial x^{\mu}}\ln\phi\right)\upsilon\_{\nu} = \frac{e\phi}{mc^{2}}F\_{\nu\mu}\upsilon^{\mu} + f\upsilon\_{\nu}.\tag{50}
$$

We could argue that the second term on the right hand side is not physical, because we could choose *φ* = 1, as we did before, getting Eq.(39). However, such a choice is not available any longer when we invoke charge (or matter) conservation. In such a case, *∂µj <sup>µ</sup>* = 0, and we must relate *j <sup>µ</sup>* with *v<sup>µ</sup>* by *j <sup>µ</sup>* = *ρvµ*, so that the above considerations apply.

Coming back to Eq.(47), we see that it implies that *∂µA*′*<sup>µ</sup>* <sup>=</sup> 0, i.e., *<sup>A</sup>*′*<sup>µ</sup>* is in the Lorentz gauge. Because of *<sup>A</sup>*′ *<sup>µ</sup>* <sup>=</sup> *<sup>A</sup><sup>µ</sup>* <sup>−</sup> (*c*/*e*) *∂µS*, the scalar function *<sup>S</sup>* must satisfy

$$
\partial\_{\mu}\partial^{\mu}\mathcal{S} \equiv \Box \mathcal{S} = \frac{e}{c}\partial\_{\mu}A^{\mu}.\tag{51}
$$

Let us consider now Maxwell equations, *∂µFµν* = (4*π*/*c*)*j <sup>ν</sup>*. If we take *j <sup>ν</sup>* to be the same as before, we are assuming that *Fµν* is generated by the same currents upon which this field is acting. That is, we are considering a closed system of charges and fields. We have then, using *<sup>F</sup>µν* <sup>=</sup> *<sup>∂</sup>µA*′*<sup>ν</sup>* <sup>−</sup> *<sup>∂</sup>νA*′*<sup>µ</sup>* and *∂µA*′*<sup>µ</sup>* <sup>=</sup> 0,

$$
\partial\_{\mu} F^{\mu \nu} = \Box A^{\prime \nu} = \frac{4\pi}{c} j^{\nu} \,, \tag{52}
$$

while from Eq. (47) we get

14 Quantum Mechanics

equation *∂µj*

may choose ˜*j*

with *φ* :=

in which case Eq.(41) reads

From this equation and *vµv<sup>µ</sup>* = *φ*<sup>2</sup> we get

Eq.(32) applied to the present case gives

*∂ ∂x<sup>µ</sup>*

It is clear that this equation holds for *<sup>A</sup>*′

 *v<sup>ν</sup> φ* − *∂ ∂x<sup>ν</sup>*

solve *j*

**3.2. Beyond the London equations**

*<sup>µ</sup>* = *φ*˜*j*

*<sup>µ</sup>* = 0 holds. Here, *j*

*µ∂µψ* = *f* , which is always possible.

*<sup>µ</sup>* such that *∂µ* ˜*j*

Let us address the case when the charge density *ρ* is not constant, as previously assumed.

and *ρ*(*x*). It can be shown that it is always possible to choose *v<sup>µ</sup>* so that the continuity

dimensions are 1/volume. Indeed, in view of the aforementioned possibility of changing the

*<sup>µ</sup>* + *j*

*<sup>v</sup>µv<sup>µ</sup>*1/2. Using the gauge freedom of *<sup>A</sup><sup>µ</sup>* we may replace this field by

 *e mc*<sup>2</sup> *A*′

 *mc*<sup>2</sup> *e*

2

*∂A<sup>ν</sup>*

*<sup>∂</sup>x<sup>µ</sup>* <sup>−</sup> *<sup>∂</sup>A<sup>µ</sup> ∂x<sup>ν</sup>* 

*<sup>µ</sup>* as well. Eq.(43) is a particular solution of this

*mc*<sup>2</sup> (*kµA<sup>ν</sup>* <sup>−</sup> *<sup>k</sup>νAµ*). (46)

*<sup>µ</sup>*(*k*). (47)

*e*

*<sup>µ</sup>* <sup>=</sup> *<sup>A</sup><sup>µ</sup>* <sup>−</sup> *<sup>c</sup>*

*<sup>µ</sup>* :<sup>=</sup> *<sup>ρ</sup>v<sup>µ</sup>* and *<sup>ρ</sup>* <sup>=</sup> *necφ*<sup>−</sup>1, *<sup>n</sup>* being a free parameter whose

*µ∂µφ*˜ = 0. Putting *ψ* = log *φ*˜, we need to

, (41)

*∂µS*, (42)

*<sup>µ</sup>*. (43)

. (44)

= 0. (45)

*<sup>v</sup>µv<sup>µ</sup>*1/2,

*<sup>µ</sup>* = −*f* , we

There is a close relationship between the norm of our velocity field, i.e., *φ*(*x*) =

field *v<sup>µ</sup>* by *w<sup>µ</sup>* = *φ*˜*vµ*, we can always satisfy the continuity equation. For, if *∂µj*

*<sup>µ</sup>* = *φ∂*˜ *<sup>µ</sup>j*

Coming back to our Lagrangian of Eq.(31), by replacing it in Eq.(18), we obtain

*<sup>v</sup><sup>µ</sup>* <sup>=</sup> *<sup>φ</sup> mc ∂µ<sup>S</sup>* <sup>−</sup> *<sup>e</sup> c A<sup>µ</sup>* 

*A*′

*<sup>v</sup><sup>µ</sup>* = −*<sup>φ</sup>*

*A*′ *<sup>µ</sup>A*′*<sup>µ</sup>* <sup>=</sup>

equation. By Fourier-transforming Eq.(45) we obtain, with *w<sup>µ</sup>* := *vµ*/*φ*,

As for the Fourier-transformed version of Eq.(43), it is given by

*<sup>k</sup>µw<sup>ν</sup>* <sup>−</sup> *<sup>k</sup>νw<sup>µ</sup>* <sup>=</sup> <sup>−</sup> *<sup>e</sup>*

*<sup>w</sup>µ*(*k*) = <sup>−</sup> *<sup>e</sup>*

*mc*<sup>2</sup> *<sup>A</sup>*′

 *v<sup>µ</sup> φ* + *e mc*<sup>2</sup>

$$A^{\prime \nu} = -\frac{mc}{ne^2} j^{\nu},\tag{53}$$

so that we can write Eq. (52) as

$$
\Box \mathfrak{j}^{\nu} = -\frac{4\pi m e^2}{m c^2} \mathfrak{j}^{\nu} \equiv -\frac{1}{\lambda\_L^2} \mathfrak{j}^{\nu} \, , \tag{54}
$$

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57

The Bohr-van Leeuwen theorem addresses a sample of charged particles subjected to a uniform magnetic field **B**. The nonrelativistic Lagrangian of the system is *L* =

Charathéodory's "Royal Road" to the Calculus of Variations: A Possible Bridge Between Classical and Quantum Physics

with *dNτ* a properly normalized volume element in configuration space. We see that the terms in *L* that depend on magnetic potentials are linear in the velocities, so that the integrand in *Z* turns out to be independent of magnetic potentials. The Bohr-van Leeuwen theorem then follows: because *Z* is independent of magnetic potentials, there is no effect on the system in response to **B**. This prediction changes when we take into account the constant of motion, Eq.(58), or equivalently, *G* := (*e***B**/2*mc*) · (**r** × **v**) =*c*2. For a sample of identical particles we define *Gs* :<sup>=</sup> <sup>∑</sup>*i*(*e***B**/2*mc*) · (**r***i*×**v***i*) <sup>≡</sup> <sup>∑</sup>*<sup>i</sup>* <sup>ω</sup>*<sup>L</sup>* · (**r***i*×**v***i*), with ω*<sup>L</sup>* the Larmor frequency. The phase density *D* for the corresponding Hamiltonian

the partition function that normalizes *<sup>D</sup>* and *<sup>β</sup>* = (*kBT*)<sup>−</sup>1. Both *<sup>λ</sup>* and *<sup>β</sup>* are Lagrange multipliers, introduced to take account of the restrictions imposed by Eq. (58) and the fixed

> −*β*(**p***<sup>i</sup>* − (*e*/*c*)**A***i*)<sup>2</sup> 2*m*

The single-particle velocity distribution that can be obtained from *D* is proportional to

*βm*ω*L*×**<sup>r</sup>**

This gives the mean velocity at **<sup>r</sup>**. We have thus �**v**� = (*λ*/*βm*)ω*L*×**r**, which determines the value of the Lagrange multiplier as *λ* = *βm*. The phase density can finally be written as

with **<sup>M</sup>** <sup>≡</sup> <sup>∑</sup>*i*(*e*/2*c*)**r***i*×**v***<sup>i</sup>* naturally arising as the magnetic moment of the system. A magnetic response shows up therefore also classically, contrary to what the original version of the Bohr-van Leeuwen theorem stated. It has been shown before that this theorem does not hold whenever the magnetic field produced by the moving charges is taken into account. Such a

*m* 2 **v***i*

<sup>2</sup> + **B** · **M**

2 + *λ*2 <sup>2</sup>*β<sup>m</sup>* (ω*L*×**r**)

*<sup>i</sup>* + (*ei*/*c*)**v***i*·**A**. We can take **<sup>A</sup>** <sup>=</sup> **<sup>B</sup>** <sup>×</sup> **<sup>r</sup>**/2. The partition function is given by

 

*∂*2*L ∂x*˙*i∂x*˙ *<sup>j</sup>*

<sup>2</sup> is given by *<sup>D</sup>* <sup>=</sup> *<sup>Z</sup>*−<sup>1</sup> exp(−*β<sup>H</sup>* <sup>−</sup> *<sup>λ</sup>Gs*), with *<sup>Z</sup>* being

+ *<sup>λ</sup>*(ω*<sup>L</sup>* · **<sup>r</sup>***i*×**v***i*)

2  . (60)

. (61)

, (62)

2

. In our case,

 

*dNτ*, (59)

http://dx.doi.org/10.5772/53843

∑*N*

*i*=1(*mi*/2)**v**<sup>2</sup>

*Z* =

*<sup>H</sup>* <sup>=</sup> <sup>∑</sup>*i*(2*m*)−1(**p***<sup>i</sup>* <sup>−</sup> (*ei*/*c*)**A***i*)

mean energy, respectively. Thus,

*<sup>D</sup>* <sup>=</sup> <sup>1</sup>

*<sup>Z</sup>* exp

exp

∑ *i*

− *βm* 2 **<sup>v</sup>** <sup>−</sup> *<sup>λ</sup>*

*<sup>D</sup>* <sup>=</sup> <sup>1</sup>

we do not need to modify the standard Lagrangian.

*<sup>Z</sup>* exp

−*β* ∑ *i*

field is included in the Darwin Lagrangian [11], which is correct to order (*v*/*c*)

+ ∞

−∞ ... + ∞

−∞

exp −*β* ∑ *i x*˙ *<sup>i</sup> ∂L <sup>∂</sup>x*˙*<sup>i</sup>* <sup>−</sup> *<sup>L</sup>*

in which we have identified the London penetration length *λL*. This equation can be rewritten in the form of the Klein-Gordon equation:

$$\left(\Box + \lambda\_L^{-2}\right)j^\nu = 0,\tag{55}$$

with *λ<sup>L</sup>* replacing *λ<sup>C</sup>* = *h*¯ /*mc*, the Compton wavelength that appears in the Klein-Gordon equation. For the steady-state (*∂*<sup>0</sup> *j <sup>ν</sup>* = 0), Eq.(54) reads

$$
\nabla^2 j^\nu = +\frac{1}{\lambda\_L^2} j^\nu. \tag{56}
$$

Taking the usual configuration of a superconductor filling half the space (*z* > 0), the solution of this equation (satisfying appropriate boundary conditions: lim*z*→∞ *j <sup>ν</sup>*(*z*) = 0) is

$$j^{\nu}(z) = \exp\left(-\frac{z}{\lambda\_L}\right) j^{\nu}(0). \tag{57}$$

In general, however, Eq.(55) admits several other solutions that depend on the assumed boundary conditions. Note that Eq.(55) corresponds to a field-free case of the Klein-Gordon equation. This is because *j <sup>ν</sup>* ∼ *<sup>A</sup>*′*ν*, so that electromagnetic fields and current share the same dynamics. This is a consequence of having assumed that the Euler-Lagrange equations (written as field equations) and Maxwell equations conform a closed system. Notably, *<sup>A</sup>*′*<sup>ν</sup>* behaves like a source-free Proca field [18] whose mass (in units of inverse length) is fixed by *λL*.

#### **3.3. The Bohr-van Leeuwen theorem**

Dropping the prime, Eq.(53) gives *j <sup>ν</sup>* = −(*ne*2/*mc*)*Aν*, which can be rewritten as *v<sup>ν</sup>* = − *e*/*mc*2 *<sup>A</sup>ν*, with *<sup>v</sup>νv<sup>ν</sup>* = 1. We get thus *<sup>v</sup>νA<sup>ν</sup>* = −*mc*2/*e*, which in the nonrelativistic limit reads

$$\mathbf{v} \cdot \mathbf{A} = \frac{mc^3}{e} . \tag{58}$$

This condition is important for the following reason. Our considerations have confirmed the possibility of classical diamagnetism, in contradiction with the Bohr-van Leeuwen theorem. Therefore, this theorem should be modified. Eq.(58) represents a constant of the motion that must be taken into account when constructing the phase density for a system of charged particles. The original version of the Bohr-van Leeuwen theorem did not consider condition (58). We will show next how this condition modifies the theorem.

16 Quantum Mechanics

so that we can write Eq. (52) as

in the form of the Klein-Gordon equation:

equation. For the steady-state (*∂*<sup>0</sup> *j*

equation. This is because *j*

**3.3. The Bohr-van Leeuwen theorem**

Dropping the prime, Eq.(53) gives *j*

*λL*.

−

reads

*e*/*mc*2

*j*

*<sup>ν</sup>* <sup>=</sup> <sup>−</sup>4*πne*<sup>2</sup> *mc*<sup>2</sup> *<sup>j</sup>*

 <sup>+</sup> *<sup>λ</sup>*−<sup>2</sup> *L j*

in which we have identified the London penetration length *λL*. This equation can be rewritten

with *λ<sup>L</sup>* replacing *λ<sup>C</sup>* = *h*¯ /*mc*, the Compton wavelength that appears in the Klein-Gordon

*<sup>ν</sup>* = 0), Eq.(54) reads

*<sup>ν</sup>* = +

Taking the usual configuration of a superconductor filling half the space (*z* > 0), the solution

 − *z λL j*

In general, however, Eq.(55) admits several other solutions that depend on the assumed boundary conditions. Note that Eq.(55) corresponds to a field-free case of the Klein-Gordon

same dynamics. This is a consequence of having assumed that the Euler-Lagrange equations (written as field equations) and Maxwell equations conform a closed system. Notably, *<sup>A</sup>*′*<sup>ν</sup>* behaves like a source-free Proca field [18] whose mass (in units of inverse length) is fixed by

**<sup>v</sup>** · **<sup>A</sup>** <sup>=</sup> *mc*<sup>3</sup>

This condition is important for the following reason. Our considerations have confirmed the possibility of classical diamagnetism, in contradiction with the Bohr-van Leeuwen theorem. Therefore, this theorem should be modified. Eq.(58) represents a constant of the motion that must be taken into account when constructing the phase density for a system of charged particles. The original version of the Bohr-van Leeuwen theorem did not consider condition

*<sup>A</sup>ν*, with *<sup>v</sup>νv<sup>ν</sup>* = 1. We get thus *<sup>v</sup>νA<sup>ν</sup>* = −*mc*2/*e*, which in the nonrelativistic limit

*<sup>ν</sup>* ∼ *<sup>A</sup>*′*ν*, so that electromagnetic fields and current share the

*<sup>ν</sup>* = −(*ne*2/*mc*)*Aν*, which can be rewritten as *v<sup>ν</sup>* =

*<sup>e</sup>* . (58)

1 *λ*2 *L j*

∇<sup>2</sup> *j*

*<sup>ν</sup>*(*z*) = exp

of this equation (satisfying appropriate boundary conditions: lim*z*→∞ *j*

*j*

(58). We will show next how this condition modifies the theorem.

*<sup>ν</sup>* ≡ − 1 *λ*2 *L j*

*<sup>ν</sup>*, (54)

*<sup>ν</sup>* = 0, (55)

*<sup>ν</sup>*. (56)

*<sup>ν</sup>*(*z*) = 0) is

*<sup>ν</sup>*(0). (57)

The Bohr-van Leeuwen theorem addresses a sample of charged particles subjected to a uniform magnetic field **B**. The nonrelativistic Lagrangian of the system is *L* = ∑*N i*=1(*mi*/2)**v**<sup>2</sup> *<sup>i</sup>* + (*ei*/*c*)**v***i*·**A**. We can take **<sup>A</sup>** <sup>=</sup> **<sup>B</sup>** <sup>×</sup> **<sup>r</sup>**/2. The partition function is given by

$$Z = \int\_{-\infty}^{+\infty} \dots \int\_{-\infty}^{+\infty} \exp\left[-\beta \left(\sum\_{i} \dot{\mathbf{x}}^{i} \frac{\partial L}{\partial \dot{\mathbf{x}}^{i}} - L\right)\right] \left|\frac{\partial^{2} L}{\partial \dot{\mathbf{x}}^{i} \partial \dot{\mathbf{x}}^{j}}\right| d^{N} \tau,\tag{59}$$

with *dNτ* a properly normalized volume element in configuration space. We see that the terms in *L* that depend on magnetic potentials are linear in the velocities, so that the integrand in *Z* turns out to be independent of magnetic potentials. The Bohr-van Leeuwen theorem then follows: because *Z* is independent of magnetic potentials, there is no effect on the system in response to **B**. This prediction changes when we take into account the constant of motion, Eq.(58), or equivalently, *G* := (*e***B**/2*mc*) · (**r** × **v**) =*c*2. For a sample of identical particles we define *Gs* :<sup>=</sup> <sup>∑</sup>*i*(*e***B**/2*mc*) · (**r***i*×**v***i*) <sup>≡</sup> <sup>∑</sup>*<sup>i</sup>* <sup>ω</sup>*<sup>L</sup>* · (**r***i*×**v***i*), with ω*<sup>L</sup>* the Larmor frequency. The phase density *D* for the corresponding Hamiltonian *<sup>H</sup>* <sup>=</sup> <sup>∑</sup>*i*(2*m*)−1(**p***<sup>i</sup>* <sup>−</sup> (*ei*/*c*)**A***i*) <sup>2</sup> is given by *<sup>D</sup>* <sup>=</sup> *<sup>Z</sup>*−<sup>1</sup> exp(−*β<sup>H</sup>* <sup>−</sup> *<sup>λ</sup>Gs*), with *<sup>Z</sup>* being the partition function that normalizes *<sup>D</sup>* and *<sup>β</sup>* = (*kBT*)<sup>−</sup>1. Both *<sup>λ</sup>* and *<sup>β</sup>* are Lagrange multipliers, introduced to take account of the restrictions imposed by Eq. (58) and the fixed mean energy, respectively. Thus,

$$D = \frac{1}{Z} \exp\left[\sum\_{i} \left(\frac{-\beta(\mathbf{p}\_i - (e/c)\mathbf{A}\_i)^2}{2m} + \lambda(\boldsymbol{\omega}\_L \cdot \mathbf{r}\_i \times \mathbf{v}\_i)\right)\right].\tag{60}$$

The single-particle velocity distribution that can be obtained from *D* is proportional to

$$\exp\left[-\frac{\beta m}{2}\left(\mathbf{v} - \frac{\lambda}{\beta m}\omega\_L \times \mathbf{r}\right)^2 + \frac{\lambda^2}{2\beta m}\left(\omega\_L \times \mathbf{r}\right)^2\right].\tag{61}$$

This gives the mean velocity at **<sup>r</sup>**. We have thus �**v**� = (*λ*/*βm*)ω*L*×**r**, which determines the value of the Lagrange multiplier as *λ* = *βm*. The phase density can finally be written as

$$D = \frac{1}{Z} \exp\left[-\beta \left(\sum\_{i} \frac{m}{2} \mathbf{v}\_i^2 + \mathbf{B} \cdot \mathbf{M}\right)\right],\tag{62}$$

with **<sup>M</sup>** <sup>≡</sup> <sup>∑</sup>*i*(*e*/2*c*)**r***i*×**v***<sup>i</sup>* naturally arising as the magnetic moment of the system. A magnetic response shows up therefore also classically, contrary to what the original version of the Bohr-van Leeuwen theorem stated. It has been shown before that this theorem does not hold whenever the magnetic field produced by the moving charges is taken into account. Such a field is included in the Darwin Lagrangian [11], which is correct to order (*v*/*c*) 2 . In our case, we do not need to modify the standard Lagrangian.

### **4. Hamilton-Jacobi equations without Hamiltonian**

We have already mentioned that for homogeneous Lagrangians the definition of a Hamiltonian is precluded by the vanishing of *x*˙ *µ∂L*/*∂x*˙ *<sup>µ</sup>* − *L*. It is nonetheless possible to introduce a Hamiltonian in a number of ways. Carathéodory's approach leads to an infinite set of Hamiltonians, from which we can choose the most suitable one for dealing with the problem at hand. Here, we focus on the two Lagrangians of interest to us, given by *L* = *mc*(*ηµνx*˙ *µx*˙ *<sup>ν</sup>*)1/2 + *eAµ*(*x*)*x*˙ *<sup>µ</sup>*/*c* for electromagnetism and

$$L(\mathbf{x}, \dot{\mathbf{x}}) = \left( g\_{\mu\nu}(\mathbf{x}) \dot{\mathbf{x}}^{\mu} \dot{\mathbf{x}}^{\nu} \right)^{1/2} \tag{63}$$

10.5772/53843

59

http://dx.doi.org/10.5772/53843

is therefore no physical reason to put a term like *m*2*c*<sup>2</sup> on the right-hand side of Eq.(65). To be sure, for all practical purposes it is irrelevant that we set *any* constant on the right-hand side of Eq.(65), as this constant will drop afterwards in the equations describing the motion. But, as a matter of principle, the mass of a test particle should not appear in an equation

Charathéodory's "Royal Road" to the Calculus of Variations: A Possible Bridge Between Classical and Quantum Physics

Gauge invariance is presently understood as a key principle that lies at the root of fundamental interactions. An equation like Schrödinger's (or Dirac's) for a free electron is invariant under the transformation *ψ* → exp(*iα*)*ψ*, for constant *α*. This is in accordance with the physical meaning of the wave-function and the way it enters in all expressions related to measurable quantities. However, one expects that Nature should respect such an invariance not only *globally*, i.e., with constant *α*, but also *locally*, with *α* a function of time and position. It is, so to say, by recourse to the appropriate interaction that Nature manages to reach this goal. For achieving invariance under the *U*(1) transformation *ψ* → exp(*iα*)*ψ*, it is necessary to introduce a *gauge field*, in this case a field represented by *Aµ*(*x*), which couples to the particle. The equation for a free particle is correspondingly changed into one in which *A<sup>µ</sup>* appears. In this context, gauge invariance means invariance under the simultaneous change *<sup>ψ</sup>* → exp(*iα*)*<sup>ψ</sup>* and an appropriate one for *<sup>A</sup>µ*. This last one must be so designed that the equation now containing *A<sup>µ</sup>* remains invariant. The change of *A<sup>µ</sup>* turns out to be *<sup>A</sup><sup>µ</sup>* → *<sup>A</sup><sup>µ</sup>* − (ℏ*c*/*e*)*∂µα*, which is the one corresponding to a gauge transformation of the *electromagnetic* field. Hence, one is led to interpret electromagnetic interactions as a consequence of local *U*(1)-invariance. Other fundamental interactions stem from similar gauge invariances: *SU*(2) × *U*(1) gives rise to electroweak interactions, *SU*(3) to the strong

In this Section we want to show how gauge invariance leads, within the classical context, to considerations paralleling those of quantum mechanics. Carathéodory's formulation will be particularly useful to this end. Let us start with the electromagnetic case. Replacing the

> *e c*

Now, the observable predictions we can make concern the integral curves of the velocity field

*<sup>µ</sup>* <sup>=</sup> *<sup>A</sup><sup>µ</sup>* <sup>−</sup> *<sup>c</sup>*

whenever a simultaneous change in *<sup>S</sup>* is undertaken. This change is given by *<sup>S</sup>* → *<sup>S</sup>*<sup>∗</sup> = *S* + *W*. It leaves Eq.(67) unchanged, for a fixed *vµ*(*x*). Eq.(66), to which the velocity field

result could have suggested such a conclusion, in view of the relationship *ψ* ∼ exp(*iS*/¯*h*). Indeed, a change *<sup>ψ</sup>* <sup>→</sup> *<sup>ψ</sup>*<sup>∗</sup> <sup>=</sup> exp(*iα*)*<sup>ψ</sup>* means that *<sup>ψ</sup>*<sup>∗</sup> <sup>∼</sup> exp(*iS*∗/¯*h*), with *<sup>S</sup>*<sup>∗</sup> <sup>=</sup> *<sup>S</sup>* <sup>+</sup> *<sup>W</sup>*, where

*e*

*<sup>A</sup>µv<sup>µ</sup>* − *<sup>v</sup>µ∂µ<sup>S</sup>* = 0. (67)

*∂µW*, (68)

*<sup>µ</sup>*. The quantum-mechanical counterpart of this

which describes how it moves under the sole action of gravity.

**5. Gauge invariance in electromagnetism and gravitation**

interaction [20], and local Lorentz invariance to gravitation [21, 22].

*mc*(*vµvµ*)1/2 +

*A*∗

Lagrangian of Eq.(31) in the fundamental Eq.(17), we get

*vµ*. This field remains invariant under the replacement

*<sup>v</sup>µ*(*x*) belongs, is also fulfilled with *<sup>S</sup>*<sup>∗</sup> and *<sup>A</sup>*<sup>∗</sup>

*W* = ℏ*α*.

for gravitation. We will prove that in these two particular cases it is possible to derive the equation which the function *S*(*x*) has to satisfy, *without* having to introduce a Hamiltonian.

Let us start with gravitation. From Eq.(63) with *v<sup>µ</sup>* replacing *x*˙ *<sup>µ</sup>*, it follows that

$$\frac{\partial L}{\partial v^{\mu}} = \frac{1}{L} g\_{\mu\nu} v^{\nu}. \tag{64}$$

Using *gµν gνσ* = *δ<sup>σ</sup> <sup>µ</sup>* this equation leads to *<sup>v</sup><sup>ν</sup>* <sup>=</sup> *Lgµν <sup>p</sup>µ*, with *<sup>p</sup><sup>µ</sup>* <sup>≡</sup> *<sup>∂</sup>L*/*∂vµ*. Considering that *L*<sup>2</sup> = *gµνvµv<sup>ν</sup>* = *L*2*gµν p<sup>µ</sup> pν*, it follows *gµν p<sup>µ</sup> p<sup>ν</sup>* = 1. And because *∂S*/*∂x<sup>µ</sup>* = *∂L*/*∂v<sup>µ</sup>* = *pµ*, we obtain the Hamilton-Jacobi equation for *S*:

$$g^{\mu\nu}(\mathbf{x})\frac{\partial \mathcal{S}}{\partial \mathbf{x}^{\mu}}\frac{\partial \mathcal{S}}{\partial \mathbf{x}^{\nu}} = 1.\tag{65}$$

In the electromagnetic case the corresponding Lagrangian leads, by the same token, to *v<sup>µ</sup>* = (*∂µ<sup>S</sup>* − *<sup>e</sup> <sup>c</sup> <sup>A</sup>µ*)*φ*/*mc* with *<sup>φ</sup>* <sup>≡</sup> (*ηµνvµvν*)1/2, again as a consequence of *<sup>∂</sup>S*/*∂x<sup>µ</sup>* <sup>=</sup> *<sup>∂</sup>L*/*∂vµ*. By replacing the above expression of *v<sup>µ</sup>* in *φ*<sup>2</sup> = *ηµνvµvν*, it follows the Hamilton-Jacobi equation

$$
\eta^{\mu\nu} \left( \frac{\partial S}{\partial \mathbf{x}^{\mu}} - \frac{e}{c} A\_{\mu} \right) \left( \frac{\partial S}{\partial \mathbf{x}^{\nu}} - \frac{e}{c} A\_{\nu} \right) = m^2 c^2. \tag{66}
$$

We remark that there was no need to choose *v* so as to satisfy either *φ* = *const*. in the electromagnetic case, or *L* = *const*. in the gravitational case, as it is usually done for obtaining the respective Euler-Lagrange equations in their simplest forms. As a consequence, the constants appearing on the right-hand sides of Eqs.(65) and (66) are independent of the way by which we decide to fix the parameter *τ* of the extremal curves. Let us remark that it is not unusual to find in textbooks Eq.(65) written with *m*2*c*<sup>2</sup> instead of the 1 on the right-hand side (see, e.g. [19]). This occurs because Eq.(65) is usually introduced as a generalization of Eq.(66), with *A<sup>µ</sup>* = 0 (field-free case). Invoking the equivalence principle, one replaces *ηµν* by *gµν* and so arrives at the equation which is supposed to describe a "free" particle moving in a curved space-time region. Now, the metric tensor *gµν* embodies all the information that determines how a test particle moves under gravity, irrespective of its inertial mass *m*. There is therefore no physical reason to put a term like *m*2*c*<sup>2</sup> on the right-hand side of Eq.(65). To be sure, for all practical purposes it is irrelevant that we set *any* constant on the right-hand side of Eq.(65), as this constant will drop afterwards in the equations describing the motion. But, as a matter of principle, the mass of a test particle should not appear in an equation which describes how it moves under the sole action of gravity.

### **5. Gauge invariance in electromagnetism and gravitation**

18 Quantum Mechanics

*L* = *mc*(*ηµνx*˙

Using *gµν gνσ* = *δ<sup>σ</sup>*

(*∂µ<sup>S</sup>* − *<sup>e</sup>*

*µx*˙

**4. Hamilton-Jacobi equations without Hamiltonian**

Let us start with gravitation. From Eq.(63) with *v<sup>µ</sup>* replacing *x*˙

Hamiltonian is precluded by the vanishing of *x*˙

*<sup>ν</sup>*)1/2 + *eAµ*(*x*)*x*˙

we obtain the Hamilton-Jacobi equation for *S*:

*ηµν*

 *∂S <sup>∂</sup>x<sup>µ</sup>* <sup>−</sup> *<sup>e</sup> c A<sup>µ</sup>*

We have already mentioned that for homogeneous Lagrangians the definition of a

to introduce a Hamiltonian in a number of ways. Carathéodory's approach leads to an infinite set of Hamiltonians, from which we can choose the most suitable one for dealing with the problem at hand. Here, we focus on the two Lagrangians of interest to us, given by

*<sup>µ</sup>*/*c* for electromagnetism and

for gravitation. We will prove that in these two particular cases it is possible to derive the equation which the function *S*(*x*) has to satisfy, *without* having to introduce a Hamiltonian.

*L*<sup>2</sup> = *gµνvµv<sup>ν</sup>* = *L*2*gµν p<sup>µ</sup> pν*, it follows *gµν p<sup>µ</sup> p<sup>ν</sup>* = 1. And because *∂S*/*∂x<sup>µ</sup>* = *∂L*/*∂v<sup>µ</sup>* = *pµ*,

In the electromagnetic case the corresponding Lagrangian leads, by the same token, to *v<sup>µ</sup>* =

replacing the above expression of *v<sup>µ</sup>* in *φ*<sup>2</sup> = *ηµνvµvν*, it follows the Hamilton-Jacobi equation

 *∂S <sup>∂</sup>x<sup>ν</sup>* <sup>−</sup> *<sup>e</sup> c A<sup>ν</sup>* 

We remark that there was no need to choose *v* so as to satisfy either *φ* = *const*. in the electromagnetic case, or *L* = *const*. in the gravitational case, as it is usually done for obtaining the respective Euler-Lagrange equations in their simplest forms. As a consequence, the constants appearing on the right-hand sides of Eqs.(65) and (66) are independent of the way by which we decide to fix the parameter *τ* of the extremal curves. Let us remark that it is not unusual to find in textbooks Eq.(65) written with *m*2*c*<sup>2</sup> instead of the 1 on the right-hand side (see, e.g. [19]). This occurs because Eq.(65) is usually introduced as a generalization of Eq.(66), with *A<sup>µ</sup>* = 0 (field-free case). Invoking the equivalence principle, one replaces *ηµν* by *gµν* and so arrives at the equation which is supposed to describe a "free" particle moving in a curved space-time region. Now, the metric tensor *gµν* embodies all the information that determines how a test particle moves under gravity, irrespective of its inertial mass *m*. There

*∂S*

*<sup>c</sup> <sup>A</sup>µ*)*φ*/*mc* with *<sup>φ</sup>* <sup>≡</sup> (*ηµνvµvν*)1/2, again as a consequence of *<sup>∂</sup>S*/*∂x<sup>µ</sup>* <sup>=</sup> *<sup>∂</sup>L*/*∂vµ*. By

*gµν*(*x*)*x*˙

*L*(*x*, *x*˙) =

*∂L <sup>∂</sup>v<sup>µ</sup>* <sup>=</sup> <sup>1</sup>

*<sup>g</sup>µν*(*x*) *<sup>∂</sup><sup>S</sup> ∂x<sup>µ</sup>* *µ∂L*/*∂x*˙

*µx*˙

*<sup>µ</sup>* this equation leads to *<sup>v</sup><sup>ν</sup>* <sup>=</sup> *Lgµν <sup>p</sup>µ*, with *<sup>p</sup><sup>µ</sup>* <sup>≡</sup> *<sup>∂</sup>L*/*∂vµ*. Considering that

*<sup>µ</sup>* − *L*. It is nonetheless possible

*<sup>ν</sup>*1/2 (63)

*<sup>µ</sup>*, it follows that

*<sup>L</sup> <sup>g</sup>µνvν*. (64)

*<sup>∂</sup>x<sup>ν</sup>* <sup>=</sup> 1. (65)

= *m*2*c*2. (66)

Gauge invariance is presently understood as a key principle that lies at the root of fundamental interactions. An equation like Schrödinger's (or Dirac's) for a free electron is invariant under the transformation *ψ* → exp(*iα*)*ψ*, for constant *α*. This is in accordance with the physical meaning of the wave-function and the way it enters in all expressions related to measurable quantities. However, one expects that Nature should respect such an invariance not only *globally*, i.e., with constant *α*, but also *locally*, with *α* a function of time and position. It is, so to say, by recourse to the appropriate interaction that Nature manages to reach this goal. For achieving invariance under the *U*(1) transformation *ψ* → exp(*iα*)*ψ*, it is necessary to introduce a *gauge field*, in this case a field represented by *Aµ*(*x*), which couples to the particle. The equation for a free particle is correspondingly changed into one in which *A<sup>µ</sup>* appears. In this context, gauge invariance means invariance under the simultaneous change *<sup>ψ</sup>* → exp(*iα*)*<sup>ψ</sup>* and an appropriate one for *<sup>A</sup>µ*. This last one must be so designed that the equation now containing *A<sup>µ</sup>* remains invariant. The change of *A<sup>µ</sup>* turns out to be *<sup>A</sup><sup>µ</sup>* → *<sup>A</sup><sup>µ</sup>* − (ℏ*c*/*e*)*∂µα*, which is the one corresponding to a gauge transformation of the *electromagnetic* field. Hence, one is led to interpret electromagnetic interactions as a consequence of local *U*(1)-invariance. Other fundamental interactions stem from similar gauge invariances: *SU*(2) × *U*(1) gives rise to electroweak interactions, *SU*(3) to the strong interaction [20], and local Lorentz invariance to gravitation [21, 22].

In this Section we want to show how gauge invariance leads, within the classical context, to considerations paralleling those of quantum mechanics. Carathéodory's formulation will be particularly useful to this end. Let us start with the electromagnetic case. Replacing the Lagrangian of Eq.(31) in the fundamental Eq.(17), we get

$$mc(\upsilon\_{\mu}\upsilon^{\mu})^{1/2} + \frac{e}{c}A\_{\mu}\upsilon^{\mu} - \upsilon^{\mu}\partial\_{\mu}\mathcal{S} = 0.\tag{67}$$

Now, the observable predictions we can make concern the integral curves of the velocity field *vµ*. This field remains invariant under the replacement

$$A^\*\_{\mu} = A\_{\mu} - \frac{c}{e} \partial\_{\mu} W\_{\prime} \tag{68}$$

whenever a simultaneous change in *<sup>S</sup>* is undertaken. This change is given by *<sup>S</sup>* → *<sup>S</sup>*<sup>∗</sup> = *S* + *W*. It leaves Eq.(67) unchanged, for a fixed *vµ*(*x*). Eq.(66), to which the velocity field *<sup>v</sup>µ*(*x*) belongs, is also fulfilled with *<sup>S</sup>*<sup>∗</sup> and *<sup>A</sup>*<sup>∗</sup> *<sup>µ</sup>*. The quantum-mechanical counterpart of this result could have suggested such a conclusion, in view of the relationship *ψ* ∼ exp(*iS*/¯*h*). Indeed, a change *<sup>ψ</sup>* <sup>→</sup> *<sup>ψ</sup>*<sup>∗</sup> <sup>=</sup> exp(*iα*)*<sup>ψ</sup>* means that *<sup>ψ</sup>*<sup>∗</sup> <sup>∼</sup> exp(*iS*∗/¯*h*), with *<sup>S</sup>*<sup>∗</sup> <sup>=</sup> *<sup>S</sup>* <sup>+</sup> *<sup>W</sup>*, where *W* = ℏ*α*.

Now, we are naturally led to ask about a similar invariance in the gravitational case. Here, Eq.(17) reads

$$(\mathcal{g}\_{\mu\nu}\upsilon^{\mu}\upsilon^{\nu})^{1/2} - \upsilon^{\mu}\partial\_{\mu}\mathcal{S} = 0,\tag{69}$$

with Ω*<sup>µ</sup>*

Using Γ*<sup>ν</sup>*

field equations:

*λσ* :<sup>=</sup> <sup>Γ</sup>*<sup>µ</sup>*

*νσ* = <sup>1</sup>

where, we recall, *Rµν* = *R<sup>σ</sup>*

of the sought-after equations. It is given by

Unfortunately, any contraction of *W<sup>λ</sup>*

principle in the general form *δ*

*gµν*(*x*, *v*) := <sup>1</sup>

Finsler spaces.

*λσ* <sup>+</sup> <sup>1</sup> <sup>2</sup> (*δ µ <sup>σ</sup> ∂λ*<sup>Λ</sup> + *<sup>δ</sup>*

*µ*

integrability conditions for the above equation are identically satisfied.

In fact, Weyl arrived at a relation like Eq.(74) but having the expression *wτδ<sup>ν</sup>*

a gradient. Indeed, after writing Eq.(74) in Weyl's form, with *w<sup>µ</sup>* replacing <sup>1</sup>

both sides of this equation with respect to *ν* and *τ*, thereby obtaining *w<sup>σ</sup>* = <sup>1</sup>

*<sup>R</sup>µν* <sup>−</sup> <sup>1</sup> 2

contraction of *λ* and *σ*, and *Tµν* means the energy-momentum tensor.

tensor that is invariant under Eq.(74), i.e., a candidate for replacing *R<sup>λ</sup>*

*µνσ* = *<sup>R</sup><sup>λ</sup>*

*µνσ* <sup>−</sup> <sup>1</sup> 4 *δλ*

*W<sup>λ</sup>*

setting of equations analogous to those of Einstein.

<sup>2</sup> *∂σ* ln *<sup>g</sup>*, with *<sup>g</sup>* <sup>=</sup><sup>|</sup> det(*gµν*) <sup>|</sup>, we get *<sup>w</sup><sup>σ</sup>* <sup>=</sup> *∂σ*(ln(*g*∗/*g*))/10.

right hand side, with *w<sup>σ</sup>* taken to be a covariant vector. Now, it is easy to see that *w<sup>σ</sup>* must be

Charathéodory's "Royal Road" to the Calculus of Variations: A Possible Bridge Between Classical and Quantum Physics

If we take geodesics as the only observable objects, then it is natural to seek transformations that leave them invariant. Such transformations are given by Eq.(74). However, a transformation of the metric tensor that fulfils Eq.(74) does not leave invariant Einstein's

If our transformations do not leave Eq.(76) invariant but we insist in viewing geodesic invariance as a fundamental requirement, then we are led to ask for alternative equations for the gravitational field. These equations should be invariant under Eq.(74). Weyl found a

One could argue that it remains still open the possibility of changing our very starting point, so that we should look for a Lagrangian which does not depend on a metric tensor. A natural candidate for this would be an affine connection (the Christoffel symbols being a special case). However, we can show that, even if we start from very general assumptions, we will end up with a Lagrangian like that of Eq. (63). That is, if we take our variational

Lorentz transformations, then *L* must be of the form (*gµν*(*x*)*vµvν*)1/2. The requirement of invariance under local Lorentz transformations follows from the principle of equivalence: at any given point we can choose our coordinate system so that a body subjected only to gravity appears to move freely in a small neighborhood of the given point. This requirement leads to the particular form of *L* just given, as can be seen as follows [24]: From the homogeneity of *L* with respect to *v* it follows that we can write *L* in the form *L* = (*gµν*(*x*, *v*)*vµvν*)1/2, with

of Finsler spaces [25]. But local Lorentz invariance implies that *gµν* is independent of *v*, as we shall see, so that we end up within the framework of Riemann spaces, a special case of

<sup>2</sup> *<sup>∂</sup>*2*L*2(*x*, *<sup>v</sup>*)/*∂vµ∂vν*. This puts our variational problem within the framework

*<sup>λ</sup>∂σ*Λ), and it can be straightforwardly proved that the

*gµν* = κ*Tµν*, (76)

*µνσ* is the Ricci tensor stemming from the Riemann tensor *R<sup>λ</sup>*

*<sup>σ</sup>Rµν* <sup>−</sup> *<sup>δ</sup><sup>λ</sup>*

*<sup>ν</sup> <sup>R</sup>µσ*

*µνσ* vanishes identically, thereby precluding an alternative

*L*(*x*, *v*)*dτ* = 0, and require that *L* is invariant under local

10.5772/53843

61

*<sup>τ</sup>* on the

*µνσ* by

*<sup>σ</sup>* + *wσδ<sup>ν</sup>*

http://dx.doi.org/10.5772/53843

<sup>2</sup> *∂µ*Λ, we contract

<sup>5</sup> (Γ∗*<sup>ν</sup> νσ* <sup>−</sup> <sup>Γ</sup>*<sup>ν</sup> νσ*).

*µνσ* as the starting point

. (77)

and we ask how a simultaneous change of *gµν* and *S* might be, in order that *v<sup>µ</sup>* remains fixed and with it the field of extremals. In the present case, it is better to start with Eq.(23) instead of Eq.(69). The reasons will become clear in what follows. Working out Eq.(23) for the present Lagrangian we obtain, after some manipulations,

$$v^{\tau}\frac{\partial v^{\nu}}{\partial \mathbf{x}^{\tau}} + g^{\mu \upsilon}(\partial\_{\tau}g\_{\mu \sigma} - \frac{1}{2}\partial\_{\mu}g\_{\sigma \tau})v^{\sigma}v^{\tau} = \frac{\partial(\ln L(\mathbf{x}, v(\mathbf{x})))}{\partial \mathbf{x}^{\tau}}v^{\nu}v^{\tau} = \frac{\partial \Phi(\mathbf{x})}{\partial \mathbf{x}^{\tau}}v^{\nu}v^{\tau},\tag{70}$$

where Φ(*x*) = ln *L*(*x*, *v*(*x*)). The right-hand side of Eq.(70) can be written in the form 1 <sup>2</sup> (*δ<sup>ν</sup> σ∂τ*<sup>Φ</sup> <sup>+</sup> *<sup>δ</sup><sup>ν</sup> τ∂σ*Φ)*vσvτ*. This suggests us to symmetrize the coefficient of *<sup>v</sup>σv<sup>τ</sup>* on the left-hand side, thereby obtaining

$$g^{\mu\nu}(\partial\_{\tau}g\_{\mu\sigma}-\frac{1}{2}\partial\_{\mu}g\_{\sigma\tau})v^{\sigma}v^{\tau} = \frac{1}{2}g^{\mu\nu}(\partial\_{\tau}g\_{\mu\sigma}+\partial\_{\sigma}g\_{\mu\tau}-\partial\_{\mu}g\_{\sigma\tau})v^{\sigma}v^{\tau} \equiv \Gamma^{\nu}\_{\sigma\tau}v^{\sigma}v^{\tau},\tag{71}$$

with Γ*<sup>ν</sup> στ* the Christoffel symbols. Eq.(70) then reads

$$
v^{\tau}\frac{\partial v^{\nu}}{\partial \mathbf{x}^{\tau}} + \Gamma^{\nu}\_{\sigma \tau}v^{\sigma}v^{\tau} = \frac{1}{2}(\delta^{\nu}\_{\sigma}\partial\_{\tau}\Phi + \delta^{\nu}\_{\tau}\partial\_{\sigma}\Phi)v^{\sigma}v^{\tau}.\tag{72}$$

Note that if we choose *v* such that *L* = *const*., then Eq.(72) becomes the usual geodesic equation, when it is calculated along an extremal curve, *dxµ*/*dτ* = *vµ*(*x*(*τ*)):

$$\frac{d\dot{x}^{\mu}}{d\tau} + \Gamma^{\mu}\_{\sigma\rho} \frac{d\mathbf{x}^{\sigma}}{d\tau} \frac{d\mathbf{x}^{\rho}}{d\tau} = 0. \tag{73}$$

If *L* � *const*., we obtain a geodesic equation with a right-hand side of the form (*d f* /*dτ*)*x*˙ *ν*. In both cases we obtain the same curves – geodesics – but with a different parametrization.

Now, assume that a change, *<sup>g</sup>µσ* <sup>→</sup> *<sup>g</sup>*<sup>∗</sup> *µσ* can be found, so that the corresponding <sup>Γ</sup>∗*<sup>ν</sup> τσ* satisfy

$$
\Gamma^{\ast \nu}\_{\tau \sigma} - \Gamma^{\nu}\_{\tau \sigma} = \frac{1}{2} \left( \delta^{\nu}\_{\sigma} \partial\_{\tau} \Lambda(\mathfrak{x}) + \delta^{\nu}\_{\tau} \partial\_{\sigma} \Lambda(\mathfrak{x}) \right), \tag{74}
$$

with Λ(*x*) being arbitrary. Such a change leads to an equation equivalent to Eq.(70), with Φ being replaced by <sup>Φ</sup><sup>∗</sup> = <sup>Φ</sup> + <sup>Λ</sup>, and hence to the same extremals. In this way we recover an old result due to Weyl: if Christoffel symbols are related to each other by Eq.(74), then they have the same geodesics [23]. Given *<sup>g</sup>µν* and <sup>Λ</sup>, it is always possible to find a *<sup>g</sup>*<sup>∗</sup> *µν* satisfying Eq.(74). This is because this equation can be put in the form

$$
\partial\_{\lambda} \mathbf{g}^{\*\mu\nu} = -\mathbf{g}^{\*\nu\sigma} \Omega^{\mu}\_{\lambda\sigma} - \mathbf{g}^{\*\mu\sigma} \Omega^{\nu}\_{\lambda\sigma\prime} \tag{75}
$$

20 Quantum Mechanics

Eq.(17) reads

1 <sup>2</sup> (*δ<sup>ν</sup>*

with Γ*<sup>ν</sup>*

*<sup>v</sup><sup>τ</sup> <sup>∂</sup>v<sup>ν</sup>*

left-hand side, thereby obtaining

*<sup>g</sup>µν*(*∂τ <sup>g</sup>µσ* <sup>−</sup> <sup>1</sup>

*σ∂τ*<sup>Φ</sup> <sup>+</sup> *<sup>δ</sup><sup>ν</sup>*

Now, we are naturally led to ask about a similar invariance in the gravitational case. Here,

and we ask how a simultaneous change of *gµν* and *S* might be, in order that *v<sup>µ</sup>* remains fixed and with it the field of extremals. In the present case, it is better to start with Eq.(23) instead of Eq.(69). The reasons will become clear in what follows. Working out Eq.(23) for

*∂µgστ*)*vσv<sup>τ</sup>* <sup>=</sup> *<sup>∂</sup>*(ln *<sup>L</sup>*(*x*, *<sup>v</sup>*(*x*)))

*τ∂σ*Φ)*vσvτ*. This suggests us to symmetrize the coefficient of *<sup>v</sup>σv<sup>τ</sup>* on the

*σ∂τ*<sup>Φ</sup> <sup>+</sup> *<sup>δ</sup><sup>ν</sup>*

*dx<sup>ρ</sup>*

*σ∂τ*Λ(*x*) + *<sup>δ</sup><sup>ν</sup>*

*λσ* <sup>−</sup> *<sup>g</sup>*∗*µσ*Ω*<sup>ν</sup>*

*<sup>g</sup>µν*(*∂τ <sup>g</sup>µσ* + *∂σgµτ* − *∂µgστ*)*vσv<sup>τ</sup>* ≡ <sup>Γ</sup>*<sup>ν</sup>*

*µσ* can be found, so that the corresponding <sup>Γ</sup>∗*<sup>ν</sup>*

where Φ(*x*) = ln *L*(*x*, *v*(*x*)). The right-hand side of Eq.(70) can be written in the form

the present Lagrangian we obtain, after some manipulations,

2

*∂µgστ*)*vσv<sup>τ</sup>* <sup>=</sup> <sup>1</sup>

*στ* the Christoffel symbols. Eq.(70) then reads

*<sup>v</sup><sup>τ</sup> <sup>∂</sup>v<sup>ν</sup> <sup>∂</sup>x<sup>τ</sup>* <sup>+</sup> <sup>Γ</sup>*<sup>ν</sup>*

> Γ∗*<sup>ν</sup> τσ* <sup>−</sup> <sup>Γ</sup>*<sup>ν</sup>*

Eq.(74). This is because this equation can be put in the form

2

*στvσv<sup>τ</sup>* <sup>=</sup> <sup>1</sup>

equation, when it is calculated along an extremal curve, *dxµ*/*dτ* = *vµ*(*x*(*τ*)):

+ Γ*<sup>µ</sup> σρ dx<sup>σ</sup> dτ*

*dx*˙ *µ dτ*

*τσ* <sup>=</sup> <sup>1</sup> <sup>2</sup> (*δ<sup>ν</sup>*

have the same geodesics [23]. Given *<sup>g</sup>µν* and <sup>Λ</sup>, it is always possible to find a *<sup>g</sup>*<sup>∗</sup>

*∂λg*∗*µν* <sup>=</sup> <sup>−</sup>*g*∗*νσ*Ω*<sup>µ</sup>*

2 (*δ<sup>ν</sup>*

Note that if we choose *v* such that *L* = *const*., then Eq.(72) becomes the usual geodesic

If *L* � *const*., we obtain a geodesic equation with a right-hand side of the form (*d f* /*dτ*)*x*˙

In both cases we obtain the same curves – geodesics – but with a different parametrization.

with Λ(*x*) being arbitrary. Such a change leads to an equation equivalent to Eq.(70), with Φ being replaced by <sup>Φ</sup><sup>∗</sup> = <sup>Φ</sup> + <sup>Λ</sup>, and hence to the same extremals. In this way we recover an old result due to Weyl: if Christoffel symbols are related to each other by Eq.(74), then they

*<sup>∂</sup>x<sup>τ</sup>* <sup>+</sup> *<sup>g</sup>µν*(*∂τ <sup>g</sup>µσ* <sup>−</sup> <sup>1</sup>

2

Now, assume that a change, *<sup>g</sup>µσ* <sup>→</sup> *<sup>g</sup>*<sup>∗</sup>

(*gµνvµvν*)1/2 − *<sup>v</sup>µ∂µ<sup>S</sup>* = 0, (69)

*<sup>∂</sup>x<sup>τ</sup> <sup>v</sup>νv<sup>τ</sup>* <sup>=</sup> *<sup>∂</sup>*Φ(*x*)

*<sup>∂</sup>x<sup>τ</sup> <sup>v</sup>νvτ*, (70)

*στvσvτ*, (71)

*ν*.

*τσ* satisfy

*µν* satisfying

*τ∂σ*Φ)*vσvτ*. (72)

*<sup>d</sup><sup>τ</sup>* <sup>=</sup> 0. (73)

*τ∂σ*Λ(*x*)), (74)

*λσ*, (75)

with Ω*<sup>µ</sup> λσ* :<sup>=</sup> <sup>Γ</sup>*<sup>µ</sup> λσ* <sup>+</sup> <sup>1</sup> <sup>2</sup> (*δ µ <sup>σ</sup> ∂λ*<sup>Λ</sup> + *<sup>δ</sup> µ <sup>λ</sup>∂σ*Λ), and it can be straightforwardly proved that the integrability conditions for the above equation are identically satisfied.

In fact, Weyl arrived at a relation like Eq.(74) but having the expression *wτδ<sup>ν</sup> <sup>σ</sup>* + *wσδ<sup>ν</sup> <sup>τ</sup>* on the right hand side, with *w<sup>σ</sup>* taken to be a covariant vector. Now, it is easy to see that *w<sup>σ</sup>* must be a gradient. Indeed, after writing Eq.(74) in Weyl's form, with *w<sup>µ</sup>* replacing <sup>1</sup> <sup>2</sup> *∂µ*Λ, we contract both sides of this equation with respect to *ν* and *τ*, thereby obtaining *w<sup>σ</sup>* = <sup>1</sup> <sup>5</sup> (Γ∗*<sup>ν</sup> νσ* <sup>−</sup> <sup>Γ</sup>*<sup>ν</sup> νσ*). Using Γ*<sup>ν</sup> νσ* = <sup>1</sup> <sup>2</sup> *∂σ* ln *<sup>g</sup>*, with *<sup>g</sup>* <sup>=</sup><sup>|</sup> det(*gµν*) <sup>|</sup>, we get *<sup>w</sup><sup>σ</sup>* <sup>=</sup> *∂σ*(ln(*g*∗/*g*))/10.

If we take geodesics as the only observable objects, then it is natural to seek transformations that leave them invariant. Such transformations are given by Eq.(74). However, a transformation of the metric tensor that fulfils Eq.(74) does not leave invariant Einstein's field equations:

$$R\_{\mu\nu} - \frac{1}{2}\mathbf{g}\_{\mu\nu} = \varkappa T\_{\mu\nu} \tag{76}$$

where, we recall, *Rµν* = *R<sup>σ</sup> µνσ* is the Ricci tensor stemming from the Riemann tensor *R<sup>λ</sup> µνσ* by contraction of *λ* and *σ*, and *Tµν* means the energy-momentum tensor.

If our transformations do not leave Eq.(76) invariant but we insist in viewing geodesic invariance as a fundamental requirement, then we are led to ask for alternative equations for the gravitational field. These equations should be invariant under Eq.(74). Weyl found a tensor that is invariant under Eq.(74), i.e., a candidate for replacing *R<sup>λ</sup> µνσ* as the starting point of the sought-after equations. It is given by

$$\mathcal{W}^{\lambda}\_{\mu\nu\sigma} = \mathcal{R}^{\lambda}\_{\mu\nu\sigma} - \frac{1}{4} \left( \delta^{\lambda}\_{\sigma} \mathcal{R}\_{\mu\nu} - \delta^{\lambda}\_{\nu} \mathcal{R}\_{\mu\sigma} \right). \tag{77}$$

Unfortunately, any contraction of *W<sup>λ</sup> µνσ* vanishes identically, thereby precluding an alternative setting of equations analogous to those of Einstein.

One could argue that it remains still open the possibility of changing our very starting point, so that we should look for a Lagrangian which does not depend on a metric tensor. A natural candidate for this would be an affine connection (the Christoffel symbols being a special case). However, we can show that, even if we start from very general assumptions, we will end up with a Lagrangian like that of Eq. (63). That is, if we take our variational principle in the general form *δ L*(*x*, *v*)*dτ* = 0, and require that *L* is invariant under local Lorentz transformations, then *L* must be of the form (*gµν*(*x*)*vµvν*)1/2. The requirement of invariance under local Lorentz transformations follows from the principle of equivalence: at any given point we can choose our coordinate system so that a body subjected only to gravity appears to move freely in a small neighborhood of the given point. This requirement leads to the particular form of *L* just given, as can be seen as follows [24]: From the homogeneity of *L* with respect to *v* it follows that we can write *L* in the form *L* = (*gµν*(*x*, *v*)*vµvν*)1/2, with *gµν*(*x*, *v*) := <sup>1</sup> <sup>2</sup> *<sup>∂</sup>*2*L*2(*x*, *<sup>v</sup>*)/*∂vµ∂vν*. This puts our variational problem within the framework of Finsler spaces [25]. But local Lorentz invariance implies that *gµν* is independent of *v*, as we shall see, so that we end up within the framework of Riemann spaces, a special case of Finsler spaces.

A transformation in the tangent space, *<sup>v</sup>* <sup>→</sup> *<sup>w</sup>*, defined through *<sup>w</sup><sup>µ</sup>* <sup>=</sup> <sup>∼</sup> Λ *µ <sup>ν</sup> vν*, is a local Lorentz transformation if it satisfies *<sup>g</sup>µν*(*x*, *<sup>v</sup>*) = *<sup>g</sup>λσ*(*x*, *<sup>v</sup>*)Λ*<sup>λ</sup> <sup>ν</sup>* (*x*)Λ*<sup>σ</sup> <sup>ν</sup>* (*x*) at any *fixed* point *<sup>x</sup>*. Here, <sup>Λ</sup>*<sup>µ</sup> ν* means the inverse of <sup>∼</sup> Λ *µ <sup>ν</sup>* . Invariance of *L* under local Lorentz transformations means that *L*(*xµ*, *wµ*) = *L*(*xµ*, ∼ Λ *µ <sup>ν</sup> vν*) = *L*(*xµ*, *vµ*). From this equality, by taking partial derivatives with respect to *v*, we obtain the two following equations:

$$\frac{\partial L(\mathbf{x}, w)}{\partial v^{\mu}} \Lambda^{\mu}\_{\nu} = \frac{\partial L(\mathbf{x}, v)}{\partial v^{\nu}} \tag{78}$$

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illustration of the capabilities of Carathéodory's approach, we have dealt with the two fundamental interactions of classical physics: electromagnetism and gravitation. We have seen that the London equations of superconductivity can be formally derived from the standard Lagrangian of a particle interacting with a prescribed electromagnetic field. The London equations have therefore not a distinctive quantum-mechanical origin, as it is often assumed. This does not mean, however, that we can explain superconductivity by recourse to classical physics alone. The conditions under which a system of charged particles behaves as described by the standard, classical Lagrangian, might be explainable only through quantum

Charathéodory's "Royal Road" to the Calculus of Variations: A Possible Bridge Between Classical and Quantum Physics

In the gravitational case, we recovered Weyl's results about the invariance of geodesics under some special transformation of the Christoffel symbols. Carathéodory's fundamental equations led us to formulate Weyl's result without having to resort to the tools of differential geometry. Furthermore, we have seen that the Lagrangian *L* = (*gµν*(*x*)*vµvν*)1/2 is a direct consequence of the assumption of local Lorentz invariance. The underlying principle that led us to state the appropriate questions was the principle of gauge invariance, something

In summary, Carathéodory's approach to variational calculus represents an alternative way to introduce some of the most basic principles of classical physics. It unifies different aspects that otherwise appear to be independent from one another, and it can help us in our quest

Partial financial support from DGI-PUCP and from the Science Department (PUCP) is

Departamento de Ciencias, Sección Física, Pontificia Universidad Católica del Perú, Lima,

[1] C. Carathéodory, *Calculus of Variations and Partial Differential Equations of the First Order.*

[2] O. Redlich, "Fundamental Thermodynamics since Carathéodory", *Rev. Mod. Phys.* vol.

[3] P. R. Holland, *The quantum theory of motion: an account of the de Broglie-Bohm causal interpretation of quantum mechanics*, Cambridge University Press, Cambridge 1993.

[4] D. Bohm, "A suggested interpretation of the quantum theory in terms of 'hidden'

*Part II: Calculus of Variations*, Holden-Day, San Francisco, 1967.

variables", *Phys. Rev.* vol. 85, pp. 166-179, 1952.

mechanics.

usually tied to a quantal approach.

**Acknowledgements**

gratefully acknowledged.

40 pp. 556-563, 1968.

**Author details** Francisco De Zela

Peru

**References**

for delimiting the quantum-classical correspondence.

$$\frac{\partial^2 L(\mathbf{x}, w)}{\partial v^\mu \partial v^\nu} \Lambda^\mu\_{\sigma} \Lambda^\nu\_{\tau} = \frac{\partial^2 L(\mathbf{x}, v)}{\partial v^\sigma \partial v^\tau} \tag{79}$$

When these equations are substituted into the identity

$$g\_{\mu\nu}(\mathbf{x}, \upsilon) = \frac{1}{2} \frac{\partial^2 L^2}{\partial \upsilon^\mu \partial \upsilon^\nu} = \frac{\partial L}{\partial \upsilon^\mu} \frac{\partial L}{\partial \upsilon^\nu} + L \frac{\partial^2 L}{\partial \upsilon^\mu \partial \upsilon^\nu},\tag{80}$$

one obtains

$$g\_{\mu\nu}(\mathbf{x}, w) = g\_{\lambda\sigma}(\mathbf{x}, v) \Lambda\_{\mu}^{\lambda}(\mathbf{x}) \Lambda\_{\nu}^{\sigma}(\mathbf{x}). \tag{81}$$

We conclude therefore, in view of this last equation and the definition of the Lorentz transformation given above, that the equality *gµν*(*x*, *w*) = *gµν*(*x*, *v*) holds true for any *w* and *v* that are connected to each other by a Lorentz transformation. Thus, setting *w* = *v* + *δv*, we obtain

$$\frac{\partial g\_{\mu\nu}(\mathbf{x},\boldsymbol{v})}{\partial \boldsymbol{v}^{\lambda}} = \lim\_{\delta \boldsymbol{v} \to 0} \left( \frac{\mathbf{g}\_{\mu\nu}(\mathbf{x},\boldsymbol{v} + \delta \boldsymbol{v}) - \mathbf{g}\_{\mu\nu}(\mathbf{x},\boldsymbol{v})}{\delta \boldsymbol{v}^{\lambda}} \right) = \mathbf{0}.\tag{82}$$

Thus, *L* must be of the form (*gµν*(*x*)*vµvν*)1/2. As we have seen, this result follows from the requirement of local Lorentz invariance. Such an assumption is the counterpart of the condition put by Helmholtz on a general metric space, in order to geometrically characterize Riemann spaces [23]. In this last case, local rotations played the role that is assigned to local Lorentz transformations in the physical case.

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

Carathéodory's approach to the calculus of variations appears to be an appropriate tool for uncovering some aspects of the quantum-classical relationship. Because it describes a whole field of extremals rather than a single one, Carathéodory's approach is, by its very nature, more akin to the quantal formulation. It remains still open how to introduce in this framework the second basic element of the quantal formulation, namely probability. By blending field and probability issues, it is likely that the ensuing result shed some light on questions concerning the quantum-classical correspondence. Here, by way of illustration of the capabilities of Carathéodory's approach, we have dealt with the two fundamental interactions of classical physics: electromagnetism and gravitation. We have seen that the London equations of superconductivity can be formally derived from the standard Lagrangian of a particle interacting with a prescribed electromagnetic field. The London equations have therefore not a distinctive quantum-mechanical origin, as it is often assumed. This does not mean, however, that we can explain superconductivity by recourse to classical physics alone. The conditions under which a system of charged particles behaves as described by the standard, classical Lagrangian, might be explainable only through quantum mechanics.

In the gravitational case, we recovered Weyl's results about the invariance of geodesics under some special transformation of the Christoffel symbols. Carathéodory's fundamental equations led us to formulate Weyl's result without having to resort to the tools of differential geometry. Furthermore, we have seen that the Lagrangian *L* = (*gµν*(*x*)*vµvν*)1/2 is a direct consequence of the assumption of local Lorentz invariance. The underlying principle that led us to state the appropriate questions was the principle of gauge invariance, something usually tied to a quantal approach.

In summary, Carathéodory's approach to variational calculus represents an alternative way to introduce some of the most basic principles of classical physics. It unifies different aspects that otherwise appear to be independent from one another, and it can help us in our quest for delimiting the quantum-classical correspondence.

### **Acknowledgements**

Partial financial support from DGI-PUCP and from the Science Department (PUCP) is gratefully acknowledged.

### **Author details**

22 Quantum Mechanics

means the inverse of <sup>∼</sup>

*L*(*xµ*, *wµ*) = *L*(*xµ*,

one obtains

obtain

A transformation in the tangent space, *<sup>v</sup>* <sup>→</sup> *<sup>w</sup>*, defined through *<sup>w</sup><sup>µ</sup>* <sup>=</sup> <sup>∼</sup>

*∂L*(*x*, *w*) *<sup>∂</sup>v<sup>µ</sup>* <sup>Λ</sup>*<sup>µ</sup>*

*∂*2*L*(*x*, *w*) *<sup>∂</sup>vµ∂v<sup>ν</sup>* <sup>Λ</sup>*<sup>µ</sup>*

2

*∂*2*L*<sup>2</sup> *<sup>∂</sup>vµ∂v<sup>ν</sup>* <sup>=</sup> *<sup>∂</sup><sup>L</sup>*

*<sup>g</sup>µν*(*x*, *<sup>w</sup>*) = *<sup>g</sup>λσ*(*x*, *<sup>v</sup>*)Λ*<sup>λ</sup>*

We conclude therefore, in view of this last equation and the definition of the Lorentz transformation given above, that the equality *gµν*(*x*, *w*) = *gµν*(*x*, *v*) holds true for any *w* and *v* that are connected to each other by a Lorentz transformation. Thus, setting *w* = *v* + *δv*, we

Thus, *L* must be of the form (*gµν*(*x*)*vµvν*)1/2. As we have seen, this result follows from the requirement of local Lorentz invariance. Such an assumption is the counterpart of the condition put by Helmholtz on a general metric space, in order to geometrically characterize Riemann spaces [23]. In this last case, local rotations played the role that is assigned to local

Carathéodory's approach to the calculus of variations appears to be an appropriate tool for uncovering some aspects of the quantum-classical relationship. Because it describes a whole field of extremals rather than a single one, Carathéodory's approach is, by its very nature, more akin to the quantal formulation. It remains still open how to introduce in this framework the second basic element of the quantal formulation, namely probability. By blending field and probability issues, it is likely that the ensuing result shed some light on questions concerning the quantum-classical correspondence. Here, by way of

transformation if it satisfies *<sup>g</sup>µν*(*x*, *<sup>v</sup>*) = *<sup>g</sup>λσ*(*x*, *<sup>v</sup>*)Λ*<sup>λ</sup>*

respect to *v*, we obtain the two following equations:

When these equations are substituted into the identity

*∂gµν*(*x*, *v*)

Lorentz transformations in the physical case.

**6. Summary and conclusions**

*<sup>g</sup>µν*(*x*, *<sup>v</sup>*) = <sup>1</sup>

*<sup>∂</sup>v<sup>λ</sup>* <sup>=</sup> lim *<sup>δ</sup>v*→<sup>0</sup>

Λ *µ*

∼ Λ *µ* Λ *µ*

*<sup>ν</sup>* (*x*) at any *fixed* point *<sup>x</sup>*. Here, <sup>Λ</sup>*<sup>µ</sup>*

*<sup>∂</sup>v<sup>ν</sup>* (78)

*<sup>∂</sup>vσ∂v<sup>τ</sup>* (79)

*<sup>∂</sup>vµ∂v<sup>ν</sup>* , (80)

*<sup>ν</sup>* (*x*). (81)

= 0. (82)

*<sup>ν</sup>* (*x*)Λ*<sup>σ</sup>*

*<sup>ν</sup>* <sup>=</sup> *<sup>∂</sup>L*(*x*, *<sup>v</sup>*)

*<sup>τ</sup>* <sup>=</sup> *<sup>∂</sup>*2*L*(*x*, *<sup>v</sup>*)

*∂L*

*µ*(*x*)Λ*<sup>σ</sup>*

 *<sup>g</sup>µν*(*x*, *<sup>v</sup>* + *<sup>δ</sup>v*) − *<sup>g</sup>µν*(*x*, *<sup>v</sup>*) *δv<sup>λ</sup>*

*<sup>∂</sup>v<sup>ν</sup>* <sup>+</sup> *<sup>L</sup> <sup>∂</sup>*2*<sup>L</sup>*

*∂v<sup>µ</sup>*

*σ*Λ*<sup>ν</sup>*

*<sup>ν</sup>* . Invariance of *L* under local Lorentz transformations means that

*<sup>ν</sup> vν*) = *L*(*xµ*, *vµ*). From this equality, by taking partial derivatives with

*<sup>ν</sup> vν*, is a local Lorentz

*ν*

Francisco De Zela

Departamento de Ciencias, Sección Física, Pontificia Universidad Católica del Perú, Lima, Peru

### **References**


[5] W. Farrell Edwards, "Classical Derivation of the London Equations", *Phys. Rev. Lett.* vol. 47, pp. 1863-1866, 1981.

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[23] D. Laugwitz, *Differential and Riemannian Geometry* , Academic Press, New York, 1965.

[25] H. Rund, *The differential geometry of Finsler spaces*, Springer, Berlin, 1959.

pp. 269-282, 1991.

[24] F. De Zela, "Über mögliche Grenzen einer Gravitationstheorie", *Ann. der Physik*, vol. 48,

Charathéodory's "Royal Road" to the Calculus of Variations: A Possible Bridge Between Classical and Quantum Physics


24 Quantum Mechanics

1982.

1980.

1946.

New York, 1991.

London, 1966.

New York, 1975.

108, pp. 1175-1204, 1957.

*Physics*, Wiley, New York, 1984.

1597-1607, 1956.

pp. 212-221, 1961.

47, pp. 1863-1866, 1981.

*Rev. Lett.* vol. 49, pp. 416, 1982.

Equations"', *Phys. Rev. Lett.* vol. 49, p. 417, 1982.

– a review", *Am. J. Phys.*, vol. 80, pp. 164-169, 2011.

*Proc. Roy. Soc.(London)* vol. A149, pp. 71-88, 1935.

Interior", *Phys. Rev. Lett.*, vol. 100, pp. 075001-1-075001-4, 2008.

[5] W. Farrell Edwards, "Classical Derivation of the London Equations", *Phys. Rev. Lett.* vol.

[6] F. S. Henyey, "Distinction between a Perfect Conductor and a Superconductor", *Phys.*

[7] B. Segall, L. L. Foldy, and R. W. Brown, "Comment on 'Classical Derivation of the London

[8] J. B. Taylor, "A classical derivation of the Meissner effect?", *Nature* vol. 299, pp. 681-682,

[9] S. M. Mahajan, "Classical Perfect Diamagnetism: Expulsion of Current from the Plasma

[10] R. Balian, *From Microphysics to Macrophysics*, Vol. I, Springer-Verlag, Berlin, Heidelberg,

[11] H. Essén and M. C. N. Fiolhais, "Meissner effeect, diamagnetism, and classical physics

[12] H. Goldstein, *Classical Mechanics*, 2nd. Ed., Addison-Wesley, Reading, Massachusetts,

[13] H. Rund, *The Hamilton-Jacobi Theory in the Calculus of Variations*, D.van Nostrand Comp.,

[14] D. Lovelock and H. Rund, *Tensors, Differential Forms, and Variational Principles* Wiley,

[15] G. A. Bliss, *Lectures on the Calculus of Variations*, University of Chicago Press, Chicago,

[16] F. London and H. London, "The Electromagnetic Equations of the Supraconductor",

[17] J. Bardeen, L. N. Cooper, J. R. Schrieffer, "Theory of Superconductivity", *Phys. Rev.* vol.

[19] L. D. Landau, E. M. Lifshitz, *The Classical Theory of Fields*, Pergamon Press, Oxford, 1962.

[20] F. Halzen and A. D. Martin, *Quarks and Leptons: An Introductory Course in Modern Particle*

[21] R. Utiyama, "Invariant Theoretical Interpretation of Interaction", *Phys. Rev.* vol. 101, pp.

[22] T. W. B. Kibble, "Lorentz Invariance and the Gravitational Field", *J. Math. Phys.* vol. 2,

[18] J. D. Jackson, *Classical Electrodynamics*, 3rd edition, Wiley, New York, 1999.

**Chapter 4**

**Provisional chapter**

**The Improvement of the Heisenberg Uncertainty**

**The Improvement of the Heisenberg Uncertainty**

One of the fundamental cornerstone of quantum mechanics is the Heisenberg uncertainty principle. This principle is so fundamental to quantum theory that it is believed that if a single phenomenon that could violate it is found then the whole building of quantum mechanics will fall apart. However, since the formulation of the uncertainty principle until today there is not clear and universal agreement in its formulation or interpretation. Even Heisenberg was not clear about the exact meaning of *p*<sup>1</sup> and *x*<sup>1</sup> in their first formulation of

nor in the interpretation of the uncertainty principle. According to Heisenberg, in Eq. (1) *q*<sup>1</sup> represents "*the precision with which the value of q is know (q*<sup>1</sup> *is, say, the mean error of q), therefore here the wavelength of light. Let p*<sup>1</sup> *be the precision with which the value of p is determinable; that is, here, the discontinuous change of p in the Compton effect* [1]". He also thought the uncertainty principle in terms of disturbance produced on an observable when it is measured

The relevance of the uncertainty principle to Physics is that it introduced for the first time the indeterminacy in a physical theory, which mean the end of the era of *certainty* in Physics. That is to say, what uncertainty principle made evident was the peculiar characteristic of quantum theory of not being able to predict with certainty a property of a physical system; in words of Heisenberg: "*. . . canonically conjugate quantities can be determined simultaneously only with a characteristic indeterminacy. This indeterminacy is the real basis for the occurrence of*

> ©2012 Aguilar et al., 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 Aguilar et al.; 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 Aguilar et al.; 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.

*<sup>p</sup>*1*q*<sup>1</sup> ∼ *<sup>h</sup>*, (1)

L. M. Arévalo Aguilar, C. P. García Quijas and

C. P. García Quijas and Carlos Robledo-Sanchez

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

**Principle**

10.5772/54530

**1. Introduction**

the uncertainty relations [1]:

its canonical counterpart.

*statistical relations in quantum mechanics* [1]".

**Principle**

Carlos Robledo-Sanchez

L. M. Arévalo Aguilar,

http://dx.doi.org/10.5772/54530

**Provisional chapter**

### **The Improvement of the Heisenberg Uncertainty Principle The Improvement of the Heisenberg Uncertainty Principle**

L. M. Arévalo Aguilar, C. P. García Quijas and Carlos Robledo-Sanchez L. M. Arévalo Aguilar, C. P. García Quijas and Carlos Robledo-Sanchez

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/54530 10.5772/54530

### **1. Introduction**

One of the fundamental cornerstone of quantum mechanics is the Heisenberg uncertainty principle. This principle is so fundamental to quantum theory that it is believed that if a single phenomenon that could violate it is found then the whole building of quantum mechanics will fall apart. However, since the formulation of the uncertainty principle until today there is not clear and universal agreement in its formulation or interpretation. Even Heisenberg was not clear about the exact meaning of *p*<sup>1</sup> and *x*<sup>1</sup> in their first formulation of the uncertainty relations [1]:

$$p\_1 q\_1 \sim h,\tag{1}$$

nor in the interpretation of the uncertainty principle. According to Heisenberg, in Eq. (1) *q*<sup>1</sup> represents "*the precision with which the value of q is know (q*<sup>1</sup> *is, say, the mean error of q), therefore here the wavelength of light. Let p*<sup>1</sup> *be the precision with which the value of p is determinable; that is, here, the discontinuous change of p in the Compton effect* [1]". He also thought the uncertainty principle in terms of disturbance produced on an observable when it is measured its canonical counterpart.

The relevance of the uncertainty principle to Physics is that it introduced for the first time the indeterminacy in a physical theory, which mean the end of the era of *certainty* in Physics. That is to say, what uncertainty principle made evident was the peculiar characteristic of quantum theory of not being able to predict with certainty a property of a physical system; in words of Heisenberg: "*. . . canonically conjugate quantities can be determined simultaneously only with a characteristic indeterminacy. This indeterminacy is the real basis for the occurrence of statistical relations in quantum mechanics* [1]".

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 Aguilar et al.; 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 Aguilar et al.; 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 Aguilar et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative

Since now, you can perceive two different meanings of the Uncertainty Principle in the two quoted paragraphs above. In the first one, the uncertainty comes from a statistical property (according with Heisenberg, the mean error) of quantum theory; in the second meaning the uncertainty is a restriction to simultaneously measure two physical properties.

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. (3)

The Improvement of the Heisenberg Uncertainty Principle

<sup>2</sup> , (6)

, (7)

<sup>∆</sup>*B*<sup>ˆ</sup> |Ψ� = |*ψb*�. (4)

�*ψa*|*ψa*� �*ψb*|*ψb*� <sup>≥</sup> |�*ψa*|*ψb*�|<sup>2</sup> (5)

<sup>2</sup> <sup>=</sup> *<sup>δ</sup>A*<sup>2</sup> is the variance, the same for the operator *<sup>B</sup>*ˆ. From the

 2

= *A*ˆ*B*ˆ + *B*ˆ *A*ˆ, and *δA* and *δB* are the standard deviation. It is worth to notice

that the association of the standard deviation whit the uncertainty relations was not proposed by Heisenberg, it was Kennard and Robertson [2] who made this association. Although Heisenberg endorse it later. As it was stated above, Heisenberg associates *p*<sup>1</sup> and *q*<sup>1</sup> with the mean error, also in the same paper he associates these quantities with the widths of Gaussian

Some problems arises with the textbooks uncertainty relations: *i)* They are given in terms of the standard deviation, *ii)* They depend on the state of the system. Additionally, *iii)* They <sup>1</sup> There are others forms to obtain the uncertainty relations, this begin by defining an operator as *D*ˆ = ∆*A*ˆ + *λ*∆*B*ˆ and,

and the following quantum postulates:

= <sup>Ψ</sup>∗(*x*)*A*ˆΨ(*x*)*dx* =

Let them operate on an state |Ψ�, given:

Now, consider the following operators defined as 1:

Therefore, using the Schwarz inequality given in the Eq. (2),

 ∆*A*ˆ2

*δAδB* ≥

functions representing the quantum states of the system.

 *A*ˆ, *B*ˆ <sup>2</sup> + *A*ˆ, *B*ˆ

 ∆*B*ˆ2 ≥ ∆*A*ˆ∆*B*ˆ 

notation).

 *A*ˆ 

we arrive to:

∆*A*ˆ2 =

*A*ˆ, *B*ˆ

then, requiring that

*A*ˆ <sup>2</sup> − *A*ˆ

*D*ˆ †*D*ˆ ≥ 0.

Eq. (6), it is not difficult to show that:

where

where

• The state of a quantum system is represented by a wave function Ψ(*x*, *t*) ( |Ψ�, in Dirac

• For every observable A there is a self-adjoint operator *A*ˆ, its expectation value is given by

∆*A*ˆ = *A*ˆ −

∆*B*ˆ = *B*ˆ −

∆*A*ˆ |Ψ� = |*ψa*�

*A*ˆ 

*B*ˆ 

Ψ|*A*ˆ|Ψ .

On the other hand, to elucidate the meaning of the time-energy uncertainty relation [1] *<sup>E</sup>*1*t*<sup>1</sup> ∼ *<sup>h</sup>* is quite difficult, for, contrary to the uncertainty relation given in Eq. (1), it is not possible to deduce it from the postulates of quantum mechanics, i. e. there is not an operator for time. In Heisenberg's paper the meaning of *t*<sup>1</sup> is the "*time during which the atoms are under the influence of the deflecting field*" and *E*<sup>1</sup> refers to the accuracy in the energy measurement. Heisenberg concludes that "*a precise determination of energy can only be obtained at the cost of a corresponding uncertainty in the time* [1]".

In this Chapter of the book, we will review the evolution of the Uncertainty Principle since its inception by Heisenberg until their application to measure entanglement. We will review some problems (usually untouched by quantum mechanic's textbooks) that the usual interpretation of the Uncertainty Principle have in terms of standard deviations and its dependence of the wave function. Also, we will review the efforts made to clarify the meaning of the Uncertainty Principle using uncertainty relations.

### **2. The relation between the Heisenberg Uncertainty Principle and the Uncertainty Relations**

The uncertainty principle is one of the fundamental issues in which quantum theory differs from the classical theories, then since its formulation has attracted considerable attention, even from areas normally outside the scientific development. This has lead to create misunderstandings about the content of the principle. Thus, it is important to mention that when we say that there is a lower limit on irreducible uncertainty in the result of a measurement, what we mean is that the uncertainty is not due to experimental errors or to inaccuracies in the laboratory. Instead, the restriction attributed to the uncertainty principle is fundamental and inherent to the theory and is based on theoretical considerations in which it is assumed that all observations are ideal and perfectly accurate.

A reading of the original Heisenber's paper shows that he writes (i. e. believes) in some pharagraps that the indeterminacies comes from the observational procedures. For, in his original paper, Heisenberg stated [1] that the concepts of classical mechanics could be used analogously in quantum mechanics to describe a mechanical system, however, the use of such concepts are affected by an indeterminacy originated *purely* by the observational procedures used to determine *simultaneously* two canonically conjugate variables. This could be contrasted with the called *Statistical Interpretation* where it is tough that the wave function represents and ensemble of identical prepared system and, therefore, the indeterminacy comes form an intrinsic indeterminacy of the physical properties.

Usually, the uncertainty principle is stated in terms of uncertainty relations. One of the first way to obtain this indeterminacy relation is due to Robertson [2]. Here, instead, we use the textbooks approach to deduce the uncertainty relations from the quantum postulates [3, 4]. This approach uses both the Schwarz inequality

$$
\langle \langle \phi | \phi \rangle \langle \phi | \varphi \rangle \ge \left| \langle \phi | \varphi \rangle \right|^2 \,, \tag{2}
$$

and the following quantum postulates:

2 Quantum Mechanics

*corresponding uncertainty in the time* [1]".

**Uncertainty Relations**

Since now, you can perceive two different meanings of the Uncertainty Principle in the two quoted paragraphs above. In the first one, the uncertainty comes from a statistical property (according with Heisenberg, the mean error) of quantum theory; in the second meaning the

On the other hand, to elucidate the meaning of the time-energy uncertainty relation [1] *<sup>E</sup>*1*t*<sup>1</sup> ∼ *<sup>h</sup>* is quite difficult, for, contrary to the uncertainty relation given in Eq. (1), it is not possible to deduce it from the postulates of quantum mechanics, i. e. there is not an operator for time. In Heisenberg's paper the meaning of *t*<sup>1</sup> is the "*time during which the atoms are under the influence of the deflecting field*" and *E*<sup>1</sup> refers to the accuracy in the energy measurement. Heisenberg concludes that "*a precise determination of energy can only be obtained at the cost of a*

In this Chapter of the book, we will review the evolution of the Uncertainty Principle since its inception by Heisenberg until their application to measure entanglement. We will review some problems (usually untouched by quantum mechanic's textbooks) that the usual interpretation of the Uncertainty Principle have in terms of standard deviations and its dependence of the wave function. Also, we will review the efforts made to clarify the

**2. The relation between the Heisenberg Uncertainty Principle and the**

The uncertainty principle is one of the fundamental issues in which quantum theory differs from the classical theories, then since its formulation has attracted considerable attention, even from areas normally outside the scientific development. This has lead to create misunderstandings about the content of the principle. Thus, it is important to mention that when we say that there is a lower limit on irreducible uncertainty in the result of a measurement, what we mean is that the uncertainty is not due to experimental errors or to inaccuracies in the laboratory. Instead, the restriction attributed to the uncertainty principle is fundamental and inherent to the theory and is based on theoretical considerations in which

A reading of the original Heisenber's paper shows that he writes (i. e. believes) in some pharagraps that the indeterminacies comes from the observational procedures. For, in his original paper, Heisenberg stated [1] that the concepts of classical mechanics could be used analogously in quantum mechanics to describe a mechanical system, however, the use of such concepts are affected by an indeterminacy originated *purely* by the observational procedures used to determine *simultaneously* two canonically conjugate variables. This could be contrasted with the called *Statistical Interpretation* where it is tough that the wave function represents and ensemble of identical prepared system and, therefore, the indeterminacy

Usually, the uncertainty principle is stated in terms of uncertainty relations. One of the first way to obtain this indeterminacy relation is due to Robertson [2]. Here, instead, we use the textbooks approach to deduce the uncertainty relations from the quantum postulates [3, 4].

�*φ*|*φ*� �*ϕ*|*ϕ*� ≥ |�*φ*|*ϕ*�|<sup>2</sup> , (2)

uncertainty is a restriction to simultaneously measure two physical properties.

meaning of the Uncertainty Principle using uncertainty relations.

it is assumed that all observations are ideal and perfectly accurate.

comes form an intrinsic indeterminacy of the physical properties.

This approach uses both the Schwarz inequality


Now, consider the following operators defined as 1:

$$
\Delta \hat{A} = \hat{A} - \langle \hat{A} \rangle$$

$$
\Delta \mathcal{B} = \mathcal{B} - \langle \mathcal{B} \rangle.\tag{3}$$

Let them operate on an state |Ψ�, given:

$$
\begin{aligned}
\Delta\hat{A}\left|\Psi\right\rangle &= \left|\psi\_a\right\rangle \\
\Delta\hat{B}\left|\Psi\right\rangle &= \left|\psi\_b\right\rangle.
\end{aligned}
\tag{4}
$$

Therefore, using the Schwarz inequality given in the Eq. (2),

$$\left| \left< \psi\_a \middle| \psi\_a \right> \left< \psi\_b \middle| \psi\_b \right> \right> \geq \left| \left< \psi\_a \middle| \psi\_b \right> \right|^2 \tag{5}$$

we arrive to:

$$
\left\langle \Delta \hat{A}^2 \right\rangle \left\langle \Delta \mathcal{B}^2 \right\rangle \geq \left| \left\langle \Delta \hat{A} \Delta \hat{B} \right\rangle \right|^2,\tag{6}
$$

where ∆*A*ˆ2 = *A*ˆ <sup>2</sup> − *A*ˆ <sup>2</sup> <sup>=</sup> *<sup>δ</sup>A*<sup>2</sup> is the variance, the same for the operator *<sup>B</sup>*ˆ. From the Eq. (6), it is not difficult to show that:

$$
\delta A \delta B \ge \sqrt{\left| \langle \{ [\hat{A}, \hat{B}] \rangle \rangle \right|^2 + \left| \langle \{ \{ \hat{A}, \hat{B} \} \rangle \rangle \right|^2}},\tag{7}
$$

where *A*ˆ, *B*ˆ = *A*ˆ*B*ˆ + *B*ˆ *A*ˆ, and *δA* and *δB* are the standard deviation. It is worth to notice that the association of the standard deviation whit the uncertainty relations was not proposed by Heisenberg, it was Kennard and Robertson [2] who made this association. Although Heisenberg endorse it later. As it was stated above, Heisenberg associates *p*<sup>1</sup> and *q*<sup>1</sup> with the mean error, also in the same paper he associates these quantities with the widths of Gaussian functions representing the quantum states of the system.

Some problems arises with the textbooks uncertainty relations: *i)* They are given in terms of the standard deviation, *ii)* They depend on the state of the system. Additionally, *iii)* They

<sup>1</sup> There are others forms to obtain the uncertainty relations, this begin by defining an operator as *D*ˆ = ∆*A*ˆ + *λ*∆*B*ˆ and, then, requiring that *D*ˆ †*D*ˆ ≥ 0.

does not represent the meaning of the impossibilities of simultaneous measurement of two observables, *iv)* They does not quantify the role of the disturbance in the state after the measurement process. Finally, *v)* They does not address the concept of complementarity. There have been proposed some criteria to solve this problems, we are going to review this proposals in the next sections.

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= *M*<sup>2</sup> (10)

The Improvement of the Heisenberg Uncertainty Principle

<sup>2</sup> . (12)

, (13)

equations [7]:

 *<sup>x</sup>*0+*W*/2 *<sup>x</sup>*0−*W*/2 <sup>|</sup>*ψ*(*x*)<sup>|</sup>

)*ψ*(*x*′ <sup>−</sup> *<sup>w</sup>*)*dx*′

These quantities, i. e. *W* and *w*, provides a better characterization of the spread of the possible values of *x*ˆ and *p*ˆ, in particular there is not any divergence in these numbers. Based in these definitions Hilgevoord and Uffink give the following uncertainty relations, that they propose as a substitute to the uncertainty relation given by Kennard (∆*x*∆*p* ≥ 1/2), [7]:

*<sup>w</sup>φW<sup>ψ</sup>* <sup>≥</sup> arccos *<sup>M</sup>* <sup>+</sup> <sup>1</sup> <sup>−</sup> *<sup>N</sup>*

*<sup>w</sup>ψW<sup>φ</sup>* <sup>≥</sup> arccos *<sup>M</sup>* <sup>+</sup> <sup>1</sup> <sup>−</sup> *<sup>N</sup>*

In the quantum literature, there are many defined Entropic Uncertainty Relations. Mostly, they are based in terms of Shannon entropy [8, 9], although in last ten years there has been extension to other forms of entropy, like Renyi entropy [10]. In reference [11] there is a recent

One of the important result in this area was the one found by Deutsch [8]. What Deutsch pursuit was a quantitative expression of the Heisenberg uncertainty principle, he notice that the customary generalization has the drawback that the lower limit depends on the quantum

> 1 4 *A*ˆ, *B*ˆ

Deutsch stress that the right hand side of the Equ. (12) does not has a lower bound but is a function of the state |*ψ*�, even it vanishes for some choices of |*ψ*�. So, in search of a quantity that could represent the uncertainty principle Deutsch propose some elementary properties, like for example that the lower limit must vanishes if the observables have an eigenstate in common. Based in this considerations he proposed the following entropic uncertainty

where *SA*<sup>ˆ</sup> <sup>=</sup> <sup>−</sup> <sup>∑</sup>*<sup>a</sup>* |�*a*|*ψ*�|<sup>2</sup> *Ln* |�*a*|*ψ*�|<sup>2</sup> and *SB*<sup>ˆ</sup> <sup>=</sup> <sup>−</sup> <sup>∑</sup>*<sup>b</sup>* |�*b*|*ψ*�|<sup>2</sup> *Ln* |�*b*|*ψ*�|<sup>2</sup> are the Shanon

<sup>1</sup> <sup>+</sup> *sup*{|�*a*|*b*�|}

∆*A*∆*B* ≥

*SA*<sup>ˆ</sup> <sup>+</sup> *SB*<sup>ˆ</sup> <sup>≥</sup> <sup>2</sup>*Ln* <sup>2</sup>

entropy, and |*a*� and |*b*� are, respectively, the eigenstates of *A*ˆ and *B*ˆ.

these uncertainty relations works well for the single-slit and double-slit experiments.

 *<sup>ψ</sup>*∗(*x*′

**3.2. Entropic Uncertainty Relations**

review of this research area.

state, that is:

relation:

<sup>2</sup> *dx* = *N*

 

*N*

*N*

2

#### **3. Reformulations to the uncertainty principle**

In this section we will review some proposed solutions to the problems stated in the last paragraph of the previous section.

#### **3.1. The dependence on the standard deviation**

The principal criticism to the dependence of the uncertainty relation on the standard deviation comes from J. Hilgevoord and J. M. B. Uffink [5, 6]. Their argument is based on two reason, first, they argue that the standard deviation is an appropriate measure of the error of a measurement because errors usually follow a Gaussian distribution, and the standard deviation is an appropriate measurement of the spread of a Gaussian; however, this is not true for a general distribution. Secondly, they gave as a principal counter argument the fact that even for simple phenomenon as the single slit the standard deviation of momentum diverges. Their approach is inside the thinking that the uncertainty relations are the measure of the spread of the probability distribution, i. e. it is believed that ∆*x* and ∆*p* represents the probability distribution of the possibles properties of the system. In short, it represents the spread of values (of *x*ˆ or *p*ˆ) that are intrinsic in the physical system that are available to appear after a measurement.

The principal counter argument with regard to the standard deviations comes from the single-slit experiment. In this case, it is supposed that the state of an income beam of particles is represented by plane waves. This plane wave represents a particle of precise momentum *p*0. Then, the plane wave arrives at the single-slit and is diffracted by it. Therefore, the wave function at the screen, according to Hilgevoord and Uffink, is:

$$\psi(\mathbf{x}) = \begin{cases} (2a)^{-1/2} \text{, if } |\mathbf{x}| \le a; \\ 0, & \text{if } |\mathbf{x}| < a. \end{cases}$$

and

$$\phi(p) = (a/\pi)^{1/2} \frac{\sin ap}{ap}.\tag{8}$$

Now, the problem with the standard deviation, as defined in quantum mechanics, in this case is that it diverges:

$$
\Delta p = \left\langle \not p^2 \right\rangle - \left\langle \not p \right\rangle^2 \to \infty. \tag{9}
$$

Therefore, these authors defined, instead of the standard deviation, the overall width (*Wψ*) and the mean peak width of *ψ* as the smaller *W* and *w* that satisfies the following equations [7]:

4 Quantum Mechanics

proposals in the next sections.

paragraph of the previous section.

appear after a measurement.

case is that it diverges:

and

**3. Reformulations to the uncertainty principle**

**3.1. The dependence on the standard deviation**

function at the screen, according to Hilgevoord and Uffink, is:

*ψ*(*x*) =

∆*p* = *p*ˆ 2 − �*p*ˆ�

does not represent the meaning of the impossibilities of simultaneous measurement of two observables, *iv)* They does not quantify the role of the disturbance in the state after the measurement process. Finally, *v)* They does not address the concept of complementarity. There have been proposed some criteria to solve this problems, we are going to review this

In this section we will review some proposed solutions to the problems stated in the last

The principal criticism to the dependence of the uncertainty relation on the standard deviation comes from J. Hilgevoord and J. M. B. Uffink [5, 6]. Their argument is based on two reason, first, they argue that the standard deviation is an appropriate measure of the error of a measurement because errors usually follow a Gaussian distribution, and the standard deviation is an appropriate measurement of the spread of a Gaussian; however, this is not true for a general distribution. Secondly, they gave as a principal counter argument the fact that even for simple phenomenon as the single slit the standard deviation of momentum diverges. Their approach is inside the thinking that the uncertainty relations are the measure of the spread of the probability distribution, i. e. it is believed that ∆*x* and ∆*p* represents the probability distribution of the possibles properties of the system. In short, it represents the spread of values (of *x*ˆ or *p*ˆ) that are intrinsic in the physical system that are available to

The principal counter argument with regard to the standard deviations comes from the single-slit experiment. In this case, it is supposed that the state of an income beam of particles is represented by plane waves. This plane wave represents a particle of precise momentum *p*0. Then, the plane wave arrives at the single-slit and is diffracted by it. Therefore, the wave

> (2*a*)<sup>−</sup>1/2, if |*x*| ≤ *<sup>a</sup>*; 0, if |*x*| < *a*.

> > *ap* . (8)

<sup>2</sup> → ∞. (9)

*<sup>φ</sup>*(*p*)=(*a*/*π*)1/2 sin *ap*

Now, the problem with the standard deviation, as defined in quantum mechanics, in this

Therefore, these authors defined, instead of the standard deviation, the overall width (*Wψ*) and the mean peak width of *ψ* as the smaller *W* and *w* that satisfies the following

$$\int\_{\mathbf{x}\_{0}-\mathbf{W}/2}^{\mathbf{x}\_{0}+\mathbf{W}/2} |\psi(\mathbf{x})|^{2} \, d\mathbf{x} = N$$

$$\left| \int \psi^{\*}(\mathbf{x}')\psi(\mathbf{x}'-\mathbf{w})d\mathbf{x}' \right|^{2} = M^{2} \tag{10}$$

These quantities, i. e. *W* and *w*, provides a better characterization of the spread of the possible values of *x*ˆ and *p*ˆ, in particular there is not any divergence in these numbers. Based in these definitions Hilgevoord and Uffink give the following uncertainty relations, that they propose as a substitute to the uncertainty relation given by Kennard (∆*x*∆*p* ≥ 1/2), [7]:

$$w\_{\Phi}W\_{\Psi} \ge \arccos\left(\frac{M+1-N}{N}\right)$$

$$w\_{\Psi}W\_{\Phi} \ge \arccos\left(\frac{M+1-N}{N}\right) \tag{11}$$

these uncertainty relations works well for the single-slit and double-slit experiments.

#### **3.2. Entropic Uncertainty Relations**

In the quantum literature, there are many defined Entropic Uncertainty Relations. Mostly, they are based in terms of Shannon entropy [8, 9], although in last ten years there has been extension to other forms of entropy, like Renyi entropy [10]. In reference [11] there is a recent review of this research area.

One of the important result in this area was the one found by Deutsch [8]. What Deutsch pursuit was a quantitative expression of the Heisenberg uncertainty principle, he notice that the customary generalization has the drawback that the lower limit depends on the quantum state, that is:

$$
\Delta A \Delta B \ge \frac{1}{4} \left| \langle \left[ \hat{A}, \hat{B} \right] \rangle \right|^2. \tag{12}
$$

Deutsch stress that the right hand side of the Equ. (12) does not has a lower bound but is a function of the state |*ψ*�, even it vanishes for some choices of |*ψ*�. So, in search of a quantity that could represent the uncertainty principle Deutsch propose some elementary properties, like for example that the lower limit must vanishes if the observables have an eigenstate in common. Based in this considerations he proposed the following entropic uncertainty relation:

$$S\_{\mathcal{A}} + S\_{\mathcal{B}} \ge 2Ln \left( \frac{2}{1 + \sup \{ |\, \langle a | b \rangle| \}} \right),\tag{13}$$

where *SA*<sup>ˆ</sup> <sup>=</sup> <sup>−</sup> <sup>∑</sup>*<sup>a</sup>* |�*a*|*ψ*�|<sup>2</sup> *Ln* |�*a*|*ψ*�|<sup>2</sup> and *SB*<sup>ˆ</sup> <sup>=</sup> <sup>−</sup> <sup>∑</sup>*<sup>b</sup>* |�*b*|*ψ*�|<sup>2</sup> *Ln* |�*b*|*ψ*�|<sup>2</sup> are the Shanon entropy, and |*a*� and |*b*� are, respectively, the eigenstates of *A*ˆ and *B*ˆ.

The next step in this line of research, was quite soon given by Hossein Partovi [12], he points out that the above uncertainty relation does not take into account the measurement process. Then, considering that the measuring device realizes a partitioning of the spectrum of the observable and the assignation of their corresponding probabilities, he proposes the following definition of entropy [12]:

$$S\_A = -\sum\_i p\_i \ln \left\{ p\_i \right\}. \tag{14}$$

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<sup>1</sup> ⊗ <sup>ˆ</sup>*I*<sup>2</sup>

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The Improvement of the Heisenberg Uncertainty Principle

*and momentum*." So, this sub-research area is concerned with the simultaneous measurement

One of the first work in this approach was that of Arthurs and Kelly [16], they analyze this problems as follows: First, they realize that as the problem is the measurement of two observables, then it is required two devices to perform the measurement. That is, the system is coupled to two devises. Then, they consider that as the two meter position commutes then it is possible to perform two simultaneous measurements of them. Therefore, the simultaneous measurement of the two meters constitutes a simultaneous measurement of two non-commuting observables of the system. As the two meters interacts with the

> *q*ˆ*P*ˆ *<sup>x</sup>* + *p*ˆ*P*ˆ *y*

where *q*ˆ and *p*ˆ correspond to the position and momentum of the quantum system,

Gaussian function as the initial wave function of the meters they arrive at the following

Therefore, the uncertainty relation of the simultaneous measurement of *q*ˆ and *p*ˆ is greater (by a factor of two) than the uncertainty relations based on the probability distribution of the

The next step in this approach was given by Arthurs and Goodman [17]. In this case, the approach is as follow: To perform a measurement, the system observables, *C*ˆ = *C*ˆ

and *<sup>D</sup>*<sup>ˆ</sup> = *<sup>D</sup>*<sup>ˆ</sup> <sup>1</sup> ⊗ <sup>ˆ</sup>*I*2, must be coupled to a measuring apparatus which is represented by the

meter operators then there must exist an uncertainty relations for these operators that puts a limit to the available information. Based in this consideration, they prove what they call a

*<sup>N</sup>*<sup>ˆ</sup> *<sup>R</sup>* = *<sup>R</sup>*<sup>ˆ</sup> − *GRC*ˆ(0),

where *C*ˆ(0) and *D*ˆ (0) are the system observables and *R*ˆ and *S*ˆ are the tracking apparatus observables, the latter obey the commutation rule [*R*ˆ, *S*ˆ] = 0. Also, it is required that the correlation between the system observables and the meter has, on average, a perfect match,

Using the previous condition, i. e. Equ (20), it is possible to show that the noise operator is uncorrelated with all system operators like *C*ˆ and *D*ˆ . Using all the previous properties of

generalized uncertainty relation. To prove it they defined a a noise operator by

*y* are the momentum of the two independent meters. Using two

*σxσ<sup>p</sup>* ≥ 1. (18)

<sup>1</sup> ⊗ <sup>ˆ</sup>*I*2. Then, if we consider that there is access only to the

*<sup>N</sup>*<sup>ˆ</sup> *<sup>S</sup>* = *<sup>S</sup>*<sup>ˆ</sup> − *GSD*<sup>ˆ</sup> (0) (19)

= 0. (20)

*H*ˆ*int* = *K*

uncertainty relation for the simultaneous measurement of two observables:

quantum system, they consider the following Hamiltonian:

two observables, the topic of the previous two sub-sections.

*Tr ρ*ˆ*N*ˆ *<sup>R</sup>*,*<sup>S</sup>* = *R*ˆ − *GR C*ˆ(0) 

*<sup>x</sup>* and *P*ˆ

operators *<sup>R</sup>*<sup>ˆ</sup> = <sup>ˆ</sup>*I*<sup>1</sup> ⊗ *<sup>R</sup>*<sup>ˆ</sup> <sup>2</sup> and *<sup>C</sup>*<sup>ˆ</sup> = *<sup>C</sup>*<sup>ˆ</sup>

that is:

of two observables.

respectively, and *P*ˆ

where *pi* = *ψ <sup>π</sup>*<sup>ˆ</sup> *<sup>A</sup> i ψ* / �*ψ*|*ψ*� and *π*ˆ *<sup>A</sup> <sup>i</sup>* is the projection onto the subspaces spanned by the states corresponding to the partition induced by the measuring apparatus [12]. In this case, *pi* gives the probability of obtaining the outcome of a measurement in a subset of the partition realized by the measuring apparatus. In this approach, the whole spectrum correspond to the observable *A*ˆ but its partitioning correspond to the measuring device. Using these considerations Hossein Patrovi proses the following lower bound for the uncertainty relation:

$$S\_{\vec{A}} + S\_{\vec{B}} \ge 2Ln \left( \frac{2}{1 + \sup\_{i} \{ ||\mathfrak{H}\_{i}^{A} + \mathfrak{H}\_{j}^{B}|| \}} \right). \tag{15}$$

In the special case where the partition realized by the measuring device includes only one point of the spectrum of *A*ˆ, i. e. *π*ˆ *<sup>A</sup> <sup>i</sup>* <sup>=</sup> <sup>|</sup>*ai*� �*ai*<sup>|</sup> and *<sup>B</sup>*ˆ, i. e. *<sup>π</sup>*<sup>ˆ</sup> *<sup>B</sup> <sup>j</sup>* = *bj bj* , then Equ. (15) reduces to Equ. (13). Finally, it is worth to mention that the Patrovi's formulation requires a formulation of the details of the measuring devices, specifically, the kind of partition that induces (or could be used) in the spectrum of the observable.

There were two additional improvement on the lower bound of the entropic uncertainty relations defined above. The first one was due to Bialynicki-Birula who presented, based in his earlier wok [9], a lower bound for the angle-angular momentum pair [13] *S<sup>φ</sup>* + *SLz* ≥ −*ln*(∆*φ*/2*π*) and an improved lower bound for the position-momentum pair *S<sup>x</sup>* + *S<sup>p</sup>* ≥ 1 − *ln*(2) − *ln*(*γ*), where *γ* = ∆*x*∆*p*/*h*. The second one was proposed by Maasen and Uffink [14] who demonstrated, based on a previous work of Kraus [15], that

$$S^A + S^B \ge -2\ln(\varepsilon),\tag{16}$$

where *c* = *maxjk aj*|*bk* .

#### **3.3. Simultaneous measurement**

Whereas in the previous two subsection we treated the face of the Uncertainty Principle that is related with the probability distribution of observables of a given wave function, in this subsection we talk a bout a second version of The Uncertainty Principle. This version is related with the fact that it is not possible to determine simultaneously, with precision, two canonically conjugate observable and usually called *joint measurement*. This is stated, generally, as: "*It is impossible to measure simultaneously two observables like, for example, position* *and momentum*." So, this sub-research area is concerned with the simultaneous measurement of two observables.

6 Quantum Mechanics

where *pi* =

where *c* = *maxjk*

following definition of entropy [12]:

point of the spectrum of *A*ˆ, i. e. *π*ˆ *<sup>A</sup>*

 *aj*|*bk* .

**3.3. Simultaneous measurement**

*ψ <sup>π</sup>*<sup>ˆ</sup> *<sup>A</sup> i ψ* 

The next step in this line of research, was quite soon given by Hossein Partovi [12], he points out that the above uncertainty relation does not take into account the measurement process. Then, considering that the measuring device realizes a partitioning of the spectrum of the observable and the assignation of their corresponding probabilities, he proposes the

*SA* <sup>=</sup> <sup>−</sup> ∑

/ �*ψ*|*ψ*� and *π*ˆ *<sup>A</sup>*

*SA*<sup>ˆ</sup> <sup>+</sup> *SB*<sup>ˆ</sup> <sup>≥</sup> <sup>2</sup>*Ln*

induces (or could be used) in the spectrum of the observable.

[14] who demonstrated, based on a previous work of Kraus [15], that

*i*

states corresponding to the partition induced by the measuring apparatus [12]. In this case, *pi* gives the probability of obtaining the outcome of a measurement in a subset of the partition realized by the measuring apparatus. In this approach, the whole spectrum correspond to the observable *A*ˆ but its partitioning correspond to the measuring device. Using these considerations Hossein Patrovi proses the following lower bound for the uncertainty relation:

2

*<sup>i</sup>* <sup>+</sup> *<sup>π</sup>*<sup>ˆ</sup> *<sup>B</sup> <sup>j</sup>* ||}

<sup>1</sup> + *supij*{||*π*<sup>ˆ</sup> *<sup>A</sup>*

*<sup>i</sup>* <sup>=</sup> <sup>|</sup>*ai*� �*ai*<sup>|</sup> and *<sup>B</sup>*ˆ, i. e. *<sup>π</sup>*<sup>ˆ</sup> *<sup>B</sup>*

In the special case where the partition realized by the measuring device includes only one

reduces to Equ. (13). Finally, it is worth to mention that the Patrovi's formulation requires a formulation of the details of the measuring devices, specifically, the kind of partition that

There were two additional improvement on the lower bound of the entropic uncertainty relations defined above. The first one was due to Bialynicki-Birula who presented, based in his earlier wok [9], a lower bound for the angle-angular momentum pair [13] *S<sup>φ</sup>* + *SLz* ≥ −*ln*(∆*φ*/2*π*) and an improved lower bound for the position-momentum pair *S<sup>x</sup>* + *S<sup>p</sup>* ≥ 1 − *ln*(2) − *ln*(*γ*), where *γ* = ∆*x*∆*p*/*h*. The second one was proposed by Maasen and Uffink

Whereas in the previous two subsection we treated the face of the Uncertainty Principle that is related with the probability distribution of observables of a given wave function, in this subsection we talk a bout a second version of The Uncertainty Principle. This version is related with the fact that it is not possible to determine simultaneously, with precision, two canonically conjugate observable and usually called *joint measurement*. This is stated, generally, as: "*It is impossible to measure simultaneously two observables like, for example, position*

*piln* {*pi*} . (14)

*<sup>i</sup>* is the projection onto the subspaces spanned by the

*<sup>j</sup>* = *bj bj* 

*S<sup>A</sup>* + *S<sup>B</sup>* ≥ −2*ln*(*c*), (16)

. (15)

, then Equ. (15)

One of the first work in this approach was that of Arthurs and Kelly [16], they analyze this problems as follows: First, they realize that as the problem is the measurement of two observables, then it is required two devices to perform the measurement. That is, the system is coupled to two devises. Then, they consider that as the two meter position commutes then it is possible to perform two simultaneous measurements of them. Therefore, the simultaneous measurement of the two meters constitutes a simultaneous measurement of two non-commuting observables of the system. As the two meters interacts with the quantum system, they consider the following Hamiltonian:

$$
\hat{H}\_{\text{int}} = K \left( \hbar \hat{P}\_{\text{x}} + \hbar \hat{P}\_{\text{y}} \right) \tag{17}
$$

where *q*ˆ and *p*ˆ correspond to the position and momentum of the quantum system, respectively, and *P*ˆ *<sup>x</sup>* and *P*ˆ *y* are the momentum of the two independent meters. Using two Gaussian function as the initial wave function of the meters they arrive at the following uncertainty relation for the simultaneous measurement of two observables:

$$
\sigma\_{\mathfrak{X}} \sigma\_{\mathfrak{p}} \ge 1. \tag{18}
$$

Therefore, the uncertainty relation of the simultaneous measurement of *q*ˆ and *p*ˆ is greater (by a factor of two) than the uncertainty relations based on the probability distribution of the two observables, the topic of the previous two sub-sections.

The next step in this approach was given by Arthurs and Goodman [17]. In this case, the approach is as follow: To perform a measurement, the system observables, *C*ˆ = *C*ˆ <sup>1</sup> ⊗ <sup>ˆ</sup>*I*<sup>2</sup> and *<sup>D</sup>*<sup>ˆ</sup> = *<sup>D</sup>*<sup>ˆ</sup> <sup>1</sup> ⊗ <sup>ˆ</sup>*I*2, must be coupled to a measuring apparatus which is represented by the operators *<sup>R</sup>*<sup>ˆ</sup> = <sup>ˆ</sup>*I*<sup>1</sup> ⊗ *<sup>R</sup>*<sup>ˆ</sup> <sup>2</sup> and *<sup>C</sup>*<sup>ˆ</sup> = *<sup>C</sup>*<sup>ˆ</sup> <sup>1</sup> ⊗ <sup>ˆ</sup>*I*2. Then, if we consider that there is access only to the meter operators then there must exist an uncertainty relations for these operators that puts a limit to the available information. Based in this consideration, they prove what they call a generalized uncertainty relation. To prove it they defined a a noise operator by

$$\begin{aligned} \hat{N}\_R &= \hat{R} - G\_R \hat{C}(0), \\ \hat{N}\_S &= \hat{S} - G\_S \hat{D}(0) \end{aligned} \tag{19}$$

where *C*ˆ(0) and *D*ˆ (0) are the system observables and *R*ˆ and *S*ˆ are the tracking apparatus observables, the latter obey the commutation rule [*R*ˆ, *S*ˆ] = 0. Also, it is required that the correlation between the system observables and the meter has, on average, a perfect match, that is:

$$\operatorname{Tr}\left(\mathbb{j}\hat{\mathsf{N}}\_{\mathsf{R},\mathsf{S}}\right) = \left\langle \hat{\mathsf{R}} \right\rangle - \operatorname{G}\_{\mathsf{R}}\left\langle \hat{\mathsf{C}}(\mathsf{0}) \right\rangle = \mathbf{0}.\tag{20}$$

Using the previous condition, i. e. Equ (20), it is possible to show that the noise operator is uncorrelated with all system operators like *C*ˆ and *D*ˆ . Using all the previous properties of the system, meter and noise operators they arrive to the following generalized Heisenberg uncertainty relation:

$$
\sigma\_{\vec{\xi}} \sigma\_{\eta} \ge \left| \text{Tr} \left( \not\!\left[ \vec{\mathbb{C}} , \not\!D \right] \right) \right| , \tag{21}
$$

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. (24)

(25)

*M*ˆ *out*, *B*ˆ *out* = 0, Ozawa was able to show the

The Improvement of the Heisenberg Uncertainty Principle

. (26)

Using this operators, and considering that

2  *N*(*A*ˆ), *<sup>B</sup>*ˆ*in* +

*<sup>ǫ</sup>*(*A*) =

*<sup>η</sup>*(*B*) =

of entanglement in a quantum state could be experimentally tested.

proposed in references [8, 14]) to test the entanglement:

This uncertainty relation has been recently experimentally tested, see reference [20]

**4. Entanglement determination using entropic uncertainty relations**

Nowadays entanglement is considered as an useful resource to make non-clasical task. As a resource it is convenient to have adequate measures to quantify how much entanglement are in a given entangled state. However, until recently the most known proposed measures have the unwanted fact of being difficult to apply in experimental settings. Therefore, it was necessary to find out new ways of entanglement determination that enable that the amount

Recently there has been much research to proposed new entanglement determination based, mostly, in uncertainty relations. In this case, the entropic uncertainty relations helps to realize this task. Recently, Berta et. al. [21] have proposed a new uncertainty relation (based on that

to propose this equation Berta et. al. consider that the system, with observables S and R, is entangled with a memori, with observable B, so in equation *S*(*R*, *B*) is the von Neumann entropy and gives the uncertainty about the measurement of R given information stored in a quantum memory, B. The term S(A|B) quantifies the amount of entanglement between the

particle and the memory. This relation was experimentally tested in reference [22].

*S*(*R*|*B*) + *S*(*S*|*B*)*log*2*c* + *S*(*A*|*B*) (27)

experimental variable *M*ˆ *out* from the theoretical variable *A*ˆ*in*:

*A*ˆ*in*, *<sup>D</sup>*(*B*ˆ)

*<sup>M</sup>*<sup>ˆ</sup> *out* <sup>−</sup> *<sup>A</sup>*ˆ*in*1/2

*<sup>B</sup>*<sup>ˆ</sup> *out* <sup>−</sup> *<sup>B</sup>*ˆ*in*1/2

where the noise *ǫ*(*A*) was defined by Ozawa as the root-mean-square deviation of the

and the disturbance *η*(*B*) on observable *B*ˆ is the change in the observable caused by the

 ≥ 1 2 *ψ A*ˆ, *B*ˆ *ψ* 

following uncertainty relation [19]:

*<sup>ǫ</sup>*(*A*)*η*(*B*) + <sup>1</sup>

measurement process:

where *ρ*ˆ is the state of the system, *σξ* and *ση* are, respectively the standard deviation of the normalized operators *ξ* = *R*ˆ /*GR* and *η* = *R*ˆ /*GR*. This uncertainty relation is four times the corresponding uncertainty relation for *C*ˆ and *D*ˆ . Notice that in the left hand side of the Eq. (21) there is information of the meter operator whereas in the right hand side there is information of the system operators and that we have access only to the meter system. In reference [18] there was published an experimental verification of this uncertainty relation.

#### **3.4. Disturbance due to measurement**

The disturbance produced on an observable due to the measurement of another observable is, perhaps, the face of the uncertainty principled most talked about but the least studied. This comes from the fact that in quantum mechanics any measurement introduces an unforeseeable disturbance in the measured quantum system. It was only recently that there have been some research and understanding of this effect.

Originally, the idea that the measuring process disturb observables comes from Heisenberg's analysis of the observation of an electron by means of a microscope. This kind of uncertainty principle is written down, to use recent terminology, as [19]:

$$
\epsilon(\mathfrak{x})\eta(p) \ge \frac{1}{2} \left| \langle \psi \left| [\mathfrak{x}, \mathfrak{f}] \right| \psi \rangle \right| . \tag{22}
$$

where *ǫ*(*x*) is the noise in the measurement in position and *η*(*p*) is the disturbance caused by the apparatus [19]. Using a general description of measurement Ozawa demonstrated that the uncertainty relation for disturbance and noise given by the Eq. (22) does not accurately represent the disturbance process. He has show that this kind of uncertainty relation includes additional terms not present in Eq. (22). In the measurement process, the quantum system interacts with a measuring device. He considers that this devices measures observable *A precisely* if its experimental probabilty distribution coincides with the theoretical probability distribution of the observable. In the measurement process, when the interaction have been turned off, the device is subject to a measurement of an observable *M*. Then, *A*ˆ*in* = *A*ˆ ⊗ ˆ*I* is the input observable, *A*ˆ *out* = *U*ˆ † *A*ˆ ⊗ ˆ*I U*ˆ is the observable after the mesaurement, *M*ˆ *in* = ˆ*I* ⊗ *M*ˆ is the device observable when the interaction begin, *M*ˆ *out* = *U*ˆ †(ˆ*I* ⊗ *M*ˆ )*U*ˆ and *U*ˆ is the unitary time evolution operator

To show that the original uncertainty relation need additional terms, he introduces the following noise *N*(*A*ˆ) and disturbance *D*(*B*ˆ) operators:

$$N(\hat{A}) = \hat{M}^{out} - \hat{A}^{in},$$

$$D(\mathcal{B}) = \mathcal{B}^{out} - \mathcal{B}^{in}.\tag{23}$$

Using this operators, and considering that *M*ˆ *out*, *B*ˆ *out* = 0, Ozawa was able to show the following uncertainty relation [19]:

8 Quantum Mechanics

uncertainty relation:

**3.4. Disturbance due to measurement**

the input observable, *A*ˆ *out* = *U*ˆ †

unitary time evolution operator

have been some research and understanding of this effect.

principle is written down, to use recent terminology, as [19]:

*ǫ*(*x*)*η*(*p*) ≥

*A*ˆ ⊗ ˆ*I* 

following noise *N*(*A*ˆ) and disturbance *D*(*B*ˆ) operators:

the system, meter and noise operators they arrive to the following generalized Heisenberg

*Tr ρ*ˆ *C*ˆ, *D*ˆ 

where *ρ*ˆ is the state of the system, *σξ* and *ση* are, respectively the standard deviation of the normalized operators *ξ* = *R*ˆ /*GR* and *η* = *R*ˆ /*GR*. This uncertainty relation is four times the corresponding uncertainty relation for *C*ˆ and *D*ˆ . Notice that in the left hand side of the Eq. (21) there is information of the meter operator whereas in the right hand side there is information of the system operators and that we have access only to the meter system. In reference [18] there was published an experimental verification of this uncertainty relation.

The disturbance produced on an observable due to the measurement of another observable is, perhaps, the face of the uncertainty principled most talked about but the least studied. This comes from the fact that in quantum mechanics any measurement introduces an unforeseeable disturbance in the measured quantum system. It was only recently that there

Originally, the idea that the measuring process disturb observables comes from Heisenberg's analysis of the observation of an electron by means of a microscope. This kind of uncertainty

1

where *ǫ*(*x*) is the noise in the measurement in position and *η*(*p*) is the disturbance caused by the apparatus [19]. Using a general description of measurement Ozawa demonstrated that the uncertainty relation for disturbance and noise given by the Eq. (22) does not accurately represent the disturbance process. He has show that this kind of uncertainty relation includes additional terms not present in Eq. (22). In the measurement process, the quantum system interacts with a measuring device. He considers that this devices measures observable *A precisely* if its experimental probabilty distribution coincides with the theoretical probability distribution of the observable. In the measurement process, when the interaction have been turned off, the device is subject to a measurement of an observable *M*. Then, *A*ˆ*in* = *A*ˆ ⊗ ˆ*I* is

ˆ*I* ⊗ *M*ˆ is the device observable when the interaction begin, *M*ˆ *out* = *U*ˆ †(ˆ*I* ⊗ *M*ˆ )*U*ˆ and *U*ˆ is the

To show that the original uncertainty relation need additional terms, he introduces the

*N*(*A*ˆ) = *M*ˆ *out* − *A*ˆ*in*,

, (21)

<sup>2</sup> |�*<sup>ψ</sup>* <sup>|</sup>[*x*ˆ, *<sup>p</sup>*ˆ]<sup>|</sup> *<sup>ψ</sup>*�| , (22)

*U*ˆ is the observable after the mesaurement, *M*ˆ *in* =

*D*(*B*ˆ) = *B*ˆ *out* − *B*ˆ*in*. (23)

*σξση* <sup>≥</sup>

$$\left| \{ \epsilon(A)\eta(B) + \frac{1}{2} \left| \left\langle \left[ N(\hat{A}), \hat{B}^{\mathrm{in}} \right] \right\rangle \right. \right| + \left\langle \left[ \hat{A}^{\mathrm{in}}, D(\hat{B}) \right] \right\rangle \right| \geq \frac{1}{2} \left| \left\langle \psi \left| \left[ \hat{A}, \hat{B} \right] \right| \psi \right\rangle \right|. \tag{24}$$

where the noise *ǫ*(*A*) was defined by Ozawa as the root-mean-square deviation of the experimental variable *M*ˆ *out* from the theoretical variable *A*ˆ*in*:

$$\epsilon(A) = \left\langle \left( \hat{M}^{out} - \hat{A}^{in} \right)^{1/2} \right\rangle \tag{25}$$

and the disturbance *η*(*B*) on observable *B*ˆ is the change in the observable caused by the measurement process:

$$\eta(B) = \left\langle \left(\hat{\mathcal{B}}^{\rm out} - \hat{\mathcal{B}}^{\rm in}\right)^{1/2} \right\rangle. \tag{26}$$

This uncertainty relation has been recently experimentally tested, see reference [20]

#### **4. Entanglement determination using entropic uncertainty relations**

Nowadays entanglement is considered as an useful resource to make non-clasical task. As a resource it is convenient to have adequate measures to quantify how much entanglement are in a given entangled state. However, until recently the most known proposed measures have the unwanted fact of being difficult to apply in experimental settings. Therefore, it was necessary to find out new ways of entanglement determination that enable that the amount of entanglement in a quantum state could be experimentally tested.

Recently there has been much research to proposed new entanglement determination based, mostly, in uncertainty relations. In this case, the entropic uncertainty relations helps to realize this task. Recently, Berta et. al. [21] have proposed a new uncertainty relation (based on that proposed in references [8, 14]) to test the entanglement:

$$S(R|B) + S(S|B) \log 2c + S(A|B) \tag{27}$$

to propose this equation Berta et. al. consider that the system, with observables S and R, is entangled with a memori, with observable B, so in equation *S*(*R*, *B*) is the von Neumann entropy and gives the uncertainty about the measurement of R given information stored in a quantum memory, B. The term S(A|B) quantifies the amount of entanglement between the particle and the memory. This relation was experimentally tested in reference [22].

### **5. Conclusions**

In this chapter we review some of the most important improvements of the Heisenberg uncertainty relation. Although there are advances in their understanding and formulation, it remains yet as an open research area, specially in the quantification of entanglement.

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The Improvement of the Heisenberg Uncertainty Principle

[10] I. Bialynicki-Birula, Phys. Rev. A 74, 052102 (2006).

[12] M. Hossein Partovi, Phys. Rev. Lett. 50, 1883 (1983).

[14] H. Maasen and J. B. Uffink, Phys. Rev. Lett. 60, 1103 (1988).

[16] E. Arthurs and J. L. Kelly JR., Bell. Syst. Tech. J. 44, 725 (1965).

[18] A. Trifonov, G. Bjork and J. S ˙ oderholm, Phys. Rev. Lett. 86, 4423 (2001). ˙

[20] J. Erhart, S. Sponar, G. Sulyok, G. Badurek, M. Ozawa and Y. Hasegawa, Nature Physics

[21] M. Berta, M. Christandl, R. Colbeck, J. M. Renes and R. Renner, Phys. Nature Physics 6,

[22] Chuan-Feng Li, Jin-Shi Xu, Xiao-Ye Xu, Ke Li & Guang-Can Guo, Nature Physics 7, 752

[17] E. Arthurs, M. S. Goodman, Phys. Rev. Lett. 60, 2447 (1988).

[11] S. Wehner, New Jour. of Phys. 12, 025009 (2010).

[13] I. Bialynicki-Birula, Phys. Lett. A 103, 253 (1984).

[15] Krauss, Phys. Rev. D 35, 3070 (1987).

[19] M. Ozawa, Phys. Rev. A67, 042105 (2003).

8, 185Ð189 (2012).

659 (2010).

(2011).

### **Acknowledgements**

We thanks Consejo Nacional de Ciencia y Tecnologia (CONACYT). L. M. Arévalo Aguilar acknowledge the support from Vicerrectoria de Investigación y Posgrado VIEP-BUAP under grand ARAL-2012-I. P. C. Garcia Quijas acknowledges CONACYT for a posdoctoral scholarship at the Universidad Autonoma de Guadalajara.

### **Author details**

L. M. Arévalo Aguilar1,⋆, C. P. García Quijas<sup>2</sup> and Carlos Robledo-Sanchez<sup>3</sup>

<sup>⋆</sup> Address all correspondence to: olareva@yahoo.com.mx

1 Facultad de Ciencias Físico Matemáticas, Benemérita Universidad Autónoma de Puebla, Puebla, México

2 Departamento de F´sica, Universidad de Guadalajara, Guadalajara, Jalisco, México

3 Facultad de Ciencias Fisico Matemáticas, Benemérita Universidad Autónoma de Puebla, Puebla, México

### **References**


10 Quantum Mechanics

**5. Conclusions**

**Acknowledgements**

**Author details**

Puebla, México

Puebla, México

**References**

2004).

2009).

Merve et. al. Eds.

L. M. Arévalo Aguilar1,⋆,

In this chapter we review some of the most important improvements of the Heisenberg uncertainty relation. Although there are advances in their understanding and formulation, it remains yet as an open research area, specially in the quantification of entanglement.

We thanks Consejo Nacional de Ciencia y Tecnologia (CONACYT). L. M. Arévalo Aguilar acknowledge the support from Vicerrectoria de Investigación y Posgrado VIEP-BUAP under grand ARAL-2012-I. P. C. Garcia Quijas acknowledges CONACYT for a posdoctoral

1 Facultad de Ciencias Físico Matemáticas, Benemérita Universidad Autónoma de Puebla,

3 Facultad de Ciencias Fisico Matemáticas, Benemérita Universidad Autónoma de Puebla,

[1] W. Heisenberg, Zeitschrift fur Physik 43, 172 (1927). Translated in the book, Quantum

[3] D. Griffiths, *Introduction to Quantum Mechanics*, Addison-Wesley; 2nd edition (April 10,

[4] N. Zettili, *Quantum Mechanics: Concepts and Applications*, Wiley; 2 edition (March 4,

[6] J. Hilgevoord and J. M. M. Uffink, The mathematical expression of the uncertainty principle, in the book *Microphysical Reality and Quantum Formalism*, 91-114, A. van der

2 Departamento de F´sica, Universidad de Guadalajara, Guadalajara, Jalisco, México

scholarship at the Universidad Autonoma de Guadalajara.

C. P. García Quijas<sup>2</sup> and Carlos Robledo-Sanchez<sup>3</sup>

Measurement, Weeler and Zurek editors.

[5] J. Hilgevoord and J. M. M. Uffink, Eur. J. Phys. 6, 165 (1985).

[7] J. M. M. Uffink and J. Hilgevoord, Found. Phys. 15, 925 (1985).

[9] I. Bialynicki-Birula and J. Mycielski, Commun. math. Phys. 44, 129 (1975).

[2] H. P. Robertson, Phys. Rev. 46, 794 (1934).

[8] D. Deutsch, Phys. Rev. Lett. 50, 631 (1983).

<sup>⋆</sup> Address all correspondence to: olareva@yahoo.com.mx


**Section 2**

**The Schrödinger Equation**
