**3.1.2 Quantum problem of electron in magnetic field at bounded area**

The quasiclassical description of an electron in a magnetic field would not give the correct picture of the probability density distribution and the current density. It would not also describe the alternation of the energy spectrum when a perturbation does the classical

Statistical Mechanics That Takes into Account Angular

, 1; ! ! ! L *<sup>l</sup> nl x nl l n x <sup>r</sup> r rn* **<sup>r</sup>** , where L *<sup>l</sup>*

integer. The energy spectrum of the operator hˆ is

is necessary that

and when *<sup>c</sup> lUl*

expressed by formula:

Then the function

*r*

where L ; *<sup>l</sup> n* 

other perturbation. The polynomial L *<sup>l</sup>*

*l n* **<sup>r</sup>**

degenerate hypergeometric function has the form:

*r*

*n*

number *nr* 1 has the sign 1 *<sup>n</sup>* **<sup>r</sup>**

that are proportional to

 

energy spectrum of the operator <sup>0</sup> hˆ is

Momentum Conservation Law - Theory and Application 459

*<sup>c</sup> n* 1 2 where *n* is integer. There are two kinds of a degeneracy of energy levels of those Hamiltonians. The degeneracy of the first kind arise as result of the formulae: *n nl* <sup>0</sup> 2 1 *<sup>r</sup>* , and *nn l <sup>r</sup>* when 0 *l* . Every level is degenerated with multiplicity that equals to its number. The perturbation *U r* would be described as a power series without the linear term. Then the term that is proportional to <sup>2</sup> *<sup>r</sup>* in the spectrum of the Hamiltonian <sup>0</sup> hˆ should change the distance between levels without elimination of the degeneration. In the spectrum of the Hamiltonian hˆ the level would be split. The other terms of potential field series eliminate the degeneration in the first order of the perturbation theory. The second kind of the degeneracy is inherent only to the spectrum of the Hamiltonian hˆ in absence of any perturbation. Every level is degenerated with infinity multiplicity because when 0 *l* the energy value does not depend on *l* . At a bounded area the multiplicity would be limited but very large and would depend on the magnetic field. That will be shown below. This degeneration eliminates by any potential field. The modulo *l* would have many integer value,

That is essential because the explanations of many experimental effects rely on the discreteness of the Landau spectrum. In so doing ones do not study the model stability relative to the electrostatic interaction that is unavoidable perturbation. The other inconsistence of the common model is the large-scale negative total angular momentum of the ground state when

Let us study the degeneration eliminating by the boundary condition (33) in the absence of

<sup>2</sup>

Here j *l i*, is a null of the Bessel function J *<sup>l</sup> x* with number in the order of increasing *i* .

*n l x n l ln x A x T x*

*r r rn n n*

, 1 1 2 1 1! ,

proportional to <sup>1</sup> exp *<sup>r</sup> l n x x* **<sup>r</sup>** . Then the function (35) will have one more null at the large-

, 1; ! ! ! L ; , 1 1 ,

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

*n li*

Obviously, this function cannot satisfy to the boundary condition (33). When

*n r r r*

*A x x n l ln n*

*x* **<sup>r</sup>** is the polynomial, the nulls of which are less than

j , , ; 24 2 *<sup>r</sup>*

*l n*

*r l i*

all levels have the identical and large-scale multiplicity of degeneracy.

*nr* , where *nr* is integer or zero.

, if that's possible at some *l* , the spectrum would be quasicontinuous.

*<sup>n</sup> x* **<sup>r</sup>** has *n* simple zeroes, which, if *nr* 1 , are

. (36)

, and further the function tends to zero asymptotically.

**r r**

*nr*

*n li <sup>r</sup>* , ; by quantities

*r r* is

 

 

the

(37)

has *nr* 1 extremes that are decrease modulo. The extreme

*r*

, and *T x <sup>n</sup>***<sup>r</sup>** is the infinite series that at , ; *x n ln*

 

*l n*

*<sup>n</sup> x* **<sup>r</sup>** is the Laguerre polynomial. The

*n n ll c r* 22 1 ,

 

 

0 0 *n nl n c r* 22 1 2 *<sup>c</sup>* <sup>0</sup> , where *n*<sup>0</sup> is

motion nonperiodical. But that problem in quantum mechanics also is considered insufficiently. As suggested in the paper (Vagner et al., 2006) the density of the probability current of the wave function **r r** exp *i* is

$$\mathbf{j} = \frac{\hbar}{m} |\boldsymbol{\nu}|^2 \left( \nabla \Phi + \frac{e}{\hbar} \mathbf{A} \right). \tag{32}$$

The eigenfunctions *<sup>n</sup>* **r** cannot be chosen so that **j** 0 because any vector potential **A** cannot be equal to a gradient of a continuous function. Then it is necessary for stationarity of the states that the current lines to be closed in the area under consideration. The boundary condition best suited to the research of this problem is null of the function on the circumference that bounds the area:

$$
\psi\left(\circledast,\varphi\right) \equiv 0\tag{33}
$$

This condition retains the greatest possible symmetry. The current lines in this case are concentric circumferences. The density of the current can be zero only at separate circumferences. Therefore the magnetization is nonuniform. The magnitudes of the eigenfunctions should have the axial symmetry. The localization of the electron cannot coincide with any classical orbit because the uncertainties of values of the orbit centre coordinates (24) should satisfy to Heisenberg uncertainty relation.

The Hamiltonian is as follows:

$$\begin{split} \hat{\boldsymbol{\ell}} &= \frac{1}{2m} \Biggl[ \left( \hat{\boldsymbol{p}}\_{\times} - \frac{eHy}{2} \right)^{2} + \left( \hat{\boldsymbol{p}}\_{\times} + \frac{eHx}{2} \right)^{2} \Biggr] + \boldsymbol{U} \{ \boldsymbol{r} \} = \hat{\boldsymbol{\ell}}\_{\times} + \frac{o\boldsymbol{o}\_{c}}{2} \hat{\boldsymbol{\ell}}\_{\times} \, \boldsymbol{r} \\ & \qquad \hat{\boldsymbol{\ell}}\_{\circ} = \frac{1}{2m} \Bigl( \hat{\boldsymbol{p}}\_{\times}^{2} + \frac{\hat{\boldsymbol{l}}\_{\circ}^{2}}{r^{2}} \Bigr) + \frac{moo\_{c}^{2}r^{2}}{8} + \boldsymbol{U} \{ \boldsymbol{r} \} . \end{split} \tag{34}$$

The operators ˆ <sup>0</sup> h and <sup>ˆ</sup> *zl* commutate with each other and each of them with the Hamiltonian hˆ. The potential energy *U r* is created by the interaction with other electrons and with a neutralizing background. The nonuniform magnetization **M***r* should be neglected. If the potential energy *U r* also will be neglected the eigenfunctions of the Hamiltonians hˆ and ˆ <sup>0</sup> h will have the form:

$$\begin{split} \Psi &= \exp(il\rho)\nu\_{\parallel\mu}\left(r\right); \qquad l = 0, \pm 1, \pm 2, \cdots\\ \nu\_{\parallel\mu^{\alpha}}\left(r\right) &= \frac{A}{\lambda} \left(\frac{r}{\lambda\sqrt{2}}\right)^{\parallel l} \exp\left(-\frac{r^{2}}{4\lambda^{2}}\right) \Phi\left(\alpha\_{\prime}|l| + 1; \frac{r^{2}}{2\lambda^{2}}\right). \end{split} \tag{35}$$

Here *l* is eigenvalue of the operator of the angular momentum, *acx* , ; is the degenerate hypergeometric function, *A* is normalizing factor that depends on *l* and . The eigenvalues of energy are expressed by and *<sup>l</sup>* : for the Hamiltonian ˆ <sup>0</sup> h that is <sup>0</sup> *<sup>c</sup> l* 1 2 , and for hˆ that is *c c l l* 12 2 . The permissible values of are determined by the boundary condition. In the common theory (Landau, & Lifshitz, E.M., 1980b; Vagner et al., 2006) that is normability of the eigenfunctions at an infinity plane. Then it

motion nonperiodical. But that problem in quantum mechanics also is considered insufficiently. As suggested in the paper (Vagner et al., 2006) the density of the probability

> is

<sup>2</sup> *e*

**A** cannot be equal to a gradient of a continuous function. Then it is necessary for stationarity of the states that the current lines to be closed in the area under consideration. The boundary condition best suited to the research of this problem is null of the function on

This condition retains the greatest possible symmetry. The current lines in this case are concentric circumferences. The density of the current can be zero only at separate circumferences. Therefore the magnetization is nonuniform. The magnitudes of the eigenfunctions should have the axial symmetry. The localization of the electron cannot coincide with any classical orbit because the uncertainties of values of the orbit centre

2 2

2 2

*r*

*m r*

hypergeometric function, *A* is normalizing factor that depends on *l* and

*c c l l* 12 2

 

*l*

<sup>ˆ</sup> 1 ˆ . 2 8

 

ˆ ˆ

*eHy eHx*

h h

2 22

*x y z*

*p p Ur l*

<sup>1</sup> <sup>ˆ</sup> ˆ ˆ - , 22 2 2

*z c*

hˆ. The potential energy *U r* is created by the interaction with other electrons and with a neutralizing background. The nonuniform magnetization **M***r* should be neglected. If the potential energy *U r* also will be neglected the eigenfunctions of the Hamiltonians hˆ and

2 2

*Ar r <sup>r</sup> r l*

Here *l* is eigenvalue of the operator of the angular momentum, *acx* , ; is the degenerate

determined by the boundary condition. In the common theory (Landau, & Lifshitz, E.M., 1980b; Vagner et al., 2006) that is normability of the eigenfunctions at an infinity plane. Then it

and *<sup>l</sup>* : for the Hamiltonian ˆ

exp ; 0, 1, 2,

*l*

*il r l*

*l mr p U r*

*<sup>n</sup>* **r** cannot be chosen so that **j** 0 because any vector potential

*zl* commutate with each other and each of them with the Hamiltonian

2 2

<sup>0</sup> h that is

 <sup>0</sup> *<sup>c</sup> l* 1 2 ,

exp , 1; .

. The permissible values of

 

2 4 2

. (32)

R, 0 (33)

*c*

(34)

(35)

. The eigenvalues

are

0

current of the wave function

The eigenfunctions

The Hamiltonian is as follows:

<sup>0</sup> h and <sup>ˆ</sup>

The operators ˆ

<sup>0</sup> h will have the form:

of energy are expressed by

and for hˆ that is

ˆ

the circumference that bounds the area:

**r r** exp *i*

*m* **<sup>j</sup> <sup>A</sup>**

> 

coordinates (24) should satisfy to Heisenberg uncertainty relation.

ˆ

h

0

*m*

*l*

 

is necessary that *nr* , where *nr* is integer or zero. , 1; ! ! ! L *<sup>l</sup> nl x nl l n x <sup>r</sup> r rn* **<sup>r</sup>** , where L *<sup>l</sup> <sup>n</sup> x* **<sup>r</sup>** is the Laguerre polynomial. The energy spectrum of the operator <sup>0</sup> hˆ is 0 0 *n nl n c r* 22 1 2 *<sup>c</sup>* <sup>0</sup> , where *n*<sup>0</sup> is integer. The energy spectrum of the operator hˆ is *n n ll c r* 22 1 , *<sup>c</sup> n* 1 2 where *n* is integer. There are two kinds of a degeneracy of energy levels of those Hamiltonians. The degeneracy of the first kind arise as result of the formulae: *n nl* <sup>0</sup> 2 1 *<sup>r</sup>* , and *nn l <sup>r</sup>* when 0 *l* . Every level is degenerated with multiplicity that equals to its number. The perturbation *U r* would be described as a power series without the linear term. Then the term that is proportional to <sup>2</sup> *<sup>r</sup>* in the spectrum of the Hamiltonian <sup>0</sup> hˆ should change the distance between levels without elimination of the degeneration. In the spectrum of the Hamiltonian hˆ the level would be split. The other terms of potential field series eliminate the degeneration in the first order of the perturbation theory. The second kind of the degeneracy is inherent only to the spectrum of the Hamiltonian hˆ in absence of any perturbation. Every level is degenerated with infinity multiplicity because when 0 *l* the energy value does not depend on *l* . At a bounded area the multiplicity would be limited but very large and would depend on the magnetic field. That will be shown below. This degeneration eliminates by any potential field. The modulo *l* would have many integer value, and when *<sup>c</sup> lUl* , if that's possible at some *l* , the spectrum would be quasicontinuous. That is essential because the explanations of many experimental effects rely on the discreteness of the Landau spectrum. In so doing ones do not study the model stability relative to the electrostatic interaction that is unavoidable perturbation. The other inconsistence of the common model is the large-scale negative total angular momentum of the ground state when all levels have the identical and large-scale multiplicity of degeneracy.

Let us study the degeneration eliminating by the boundary condition (33) in the absence of other perturbation. The polynomial L *<sup>l</sup> <sup>n</sup> x* **<sup>r</sup>** has *n* simple zeroes, which, if *nr* 1 , are expressed by formula:

$$\zeta'(n\_r, |l|; i) \approx \frac{j^2 \left(|l|, i\right)}{2|l| + 4n\_r + 2} \,. \tag{36}$$

Here j *l i*, is a null of the Bessel function J *<sup>l</sup> x* with number in the order of increasing *i* . Then the function *l n* **<sup>r</sup>** has *nr* 1 extremes that are decrease modulo. The extreme number *nr* 1 has the sign 1 *<sup>n</sup>* **<sup>r</sup>** , and further the function tends to zero asymptotically. Obviously, this function cannot satisfy to the boundary condition (33). When *nr* the degenerate hypergeometric function has the form:

$$\begin{split} \Phi\left(-n\_{r}-\gamma, \left|l\right|+1; \mathbf{x}\right) &= \left[ \left(n\_{r}\right)! \middle|l\right\rangle \left(\left|l\right|+n\_{r}\right)! \left\|\mathbf{L}\right\|\_{\mathbf{n}\_{\mathbf{r}}} \left(\gamma; \mathbf{x}\right) - \gamma A\_{\mathbf{n}\_{r}}\left(\gamma, \mathbf{x}\right) - \left(-1\right)^{\mathbf{n}} \gamma \left(1-\gamma\right) T\_{\mathbf{n}\_{\mathbf{r}}}\left(\mathbf{x}\right), \\ A\_{\mathbf{n}\_{\mathbf{r}}}\left(\gamma, \mathbf{x}\right) &= -\left(-\mathbf{x}\right)^{\mathbf{n}\_{r}+1} \left[ \Gamma\left(n\_{r}+1+\gamma\right) \Gamma\left(|l|+1\right) \right] \Gamma\left(|l|+n\_{r}+2\right) \Gamma\left(1+\gamma\right) \left(n\_{r}+1\right)! \right]^{-1}, \end{split} \tag{37}$$

where L ; *<sup>l</sup> n x* **<sup>r</sup>** is the polynomial, the nulls of which are less than *n li <sup>r</sup>* , ; by quantities that are proportional to , and *T x <sup>n</sup>***<sup>r</sup>** is the infinite series that at , ; *x n ln r r* is proportional to <sup>1</sup> exp *<sup>r</sup> l n x x* **<sup>r</sup>** . Then the function (35) will have one more null at the large-

Statistical Mechanics That Takes into Account Angular

j *l l* , 2 

where

about

continuous.

2 2 <sup>2</sup>

*n n <sup>r</sup>* 2 and

parameter of this expansion is 1 2

parabolic potential is equal to 2 2 *m*

2 and about <sup>0</sup> *R n*

averaged over the interval

**electrostatic interaction** 

believed that the magnetic band is finished when 2 2 *n n* <sup>0</sup> R 4

of the Hamiltonian hˆ in absence of the potential energy *U r* is:

spectrum 2 22 2 *m <sup>c</sup>* R .

 

Momentum Conservation Law - Theory and Application 461

values of energy not the greatest but other nulls, which are described by formula (36), satisfy the boundary condition. With the approximate formula for the nulls of the Bessel function

condition should be satisfied only not the greatest but other nulls of the degenerate

0 0 <sup>0</sup> <sup>2</sup> 2 2 <sup>0</sup> <sup>j</sup> , , <sup>j</sup> , 2 2 <sup>2</sup> *c c n n ll l*

eigenvalues energy in a circular potential well with reflecting boundaries (see formula (19)). The spectrum becomes quasicontinuous as distances between the nearest levels are proportional to <sup>2</sup> R . The density of states does not depend on energy, as well as for twodimensional gas of free particles. The function (35) would be expanded in series over the

coincide with free electron wave functions. This form of the spectrum would be illustrated by the classical consideration. If the formula (34) would be considered as the classical Hamiltonian, the ultimate energy for which the classical accessible area is determined by the

with the energy of the transitive area and with energy of the transition to quasicontinuous

These results can be described as energy spectrum breakdown into two bands. A spectrum lower part is denoted as a magnetic band, and the upper one will be denoted as a conduction band. Bands are not separated by a gap or sharp boundary, but far from transitive area the density of states and wave functions differ substantially. Fine structure of the density of states in the lower part of magnetic band represents the narrow zonule separated by gaps. The total width of the allowed zonule and gap is equal to

Number of states in an interval with number *n*<sup>0</sup> is equal to *n*<sup>0</sup> . In the transitive area gaps disappear and in the conduction band the spectrum is quasicontinuous. The maximum of magnitude of a wave function in the magnetic band is localized within a ring of width

> *<sup>c</sup>* <sup>2</sup> , is <sup>2</sup> 2 4 *n*0 0

transitive area this grows is decelerated, and in the conduction band the density of states does not depend on energy and is equal to 2 2 *m*R 2 (without spin consideration). It is

would be expected from the quasiclassical consideration. Then the density of states will be

The quantum-mechanical average value of the magnetic moment in the ordinary eigenstate

**3.2 Statistical mechanics of electron gas in uniform magnetic field with regard for** 

*<sup>c</sup>* R 8 . That within a factor

 R

Bessel functions (Erdélyi, 1953) and the first term is proportional to J 2 *<sup>l</sup> r m*

 

is not described by the formula (39). They coincide with

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

*n*<sup>0</sup> 8 2 R

*m*

. Hence, the wave functions in this approximation also

2 in radius. The density of states in the magnetic band,

*c c* and grows with energy. In the

 *<sup>b</sup>* or 2 2 <sup>0</sup> *m*

R

  *nr* is

the boundary

(40)

. The

> *<sup>c</sup>* 2 .

 *c b* R 8 as

that is close to 1 coincides

the second in magnitude null of the function (35) with

*nn n* 0 0 82 8 <sup>0</sup> . When it will be so that 2 22

 

hypergeometric function. Corresponding values of energy look like:

> *n*0

scale value *x* . This null *X* tends to infinity when tends to zero, and it would be shown that

$$\gamma = \frac{X^{2n\_r + \|l\| + 1}}{(|l| + n\_r)!(n\_r)!} \exp(-X) \, , \tag{38}$$

when 1 . When 1 , *X n ln r r* 1, ; 1 . Then the boundary condition (33) would be satisfied when 2 2 R 2 *X* . (In the mathematical handbook (Erdélyi, 1953) it is written that the function , 1; *l x* at *nr* has *nr* 1 nulls, but all these nulls are determined by the formula that is like to the formula (36). Then the boundary condition would be satisfied at arbitrarily large values 2 2 R 2 only when energy has also largescale value). The value of the null *n ln r r* 1, ; 1 increases when the value of *l* increases. The maximal value 1*l* , at which the inequality 2 2 <sup>1</sup> 2 1, ; 1 R *n ln r r* is fulfilled, determines the number of the eigenstates of the Hamiltonian hˆ that have 1 (ordinary states). It can be shown that when 1 *<sup>r</sup> l n* then 2 2 <sup>1</sup> 2 12 *<sup>r</sup> l n* R R . The quantum number of the energy *n* is equal to *nr* because when 0 *l* the energy depends on *l* only by *n l*, . This number of the ordinary states is consistent with the estimate that was obtained in classic mechanics theory in section 3.1.1 and in the work (Landau, 1930). Those ordinary states are quasi degenerated. The multiplicity of this degeneration is proportional to the magnetic field. Then the main term in the thermodynamical potential, as it was computed by Landau (Landau, 1930), should not depend on the magnetic field, and the Bohr – van Leeuwen theorem is vindicated. When <sup>1</sup> *l l* and 0 *l* the other nulls that are described by the formula (36) would satisfy the boundary condition. Those are the near-boundary states, and in this case are not restrictions on the values of besides 0 1 . There are *n* near-boundary states.

In the section 3.2 it will be shown that for statistical mechanics of the electron gas in the magnetic field the Hamiltonian ˆ <sup>0</sup> h has fundamental importance. Let as study the spectrum of this Hamiltonian with boundary condition (33). Then

$$\begin{split} \mathcal{L}\_{0}\left(n\_{0},n\_{r}\right) &= \frac{\hbar\alpha\_{c}}{2} \Big[n\_{0} + 2\gamma \left(n\_{0},n\_{r}\right)\Big], \quad n\_{0} = 2n\_{r} + |l| + 1, \\ \gamma\left(n\_{0},n\_{r}\right) &= \frac{\left(\circledast^{2}/2\lambda^{2}\right)^{n\_{0}}}{\left(n\_{0} - n\_{r} - 1\right)!(n\_{r})!} \exp\left(-\circledast^{2}/2\lambda^{2}\right). \end{split} \tag{39}$$

The degenerate levels are transformed in zonule. It follows from formula (39) that the zonule upper edge is determined by minimum value of *nn n* <sup>0</sup> *r r* 1! ! that is roughly <sup>2</sup> *n*<sup>0</sup> 2 ! . In the vicinity of *n n <sup>r</sup>* <sup>0</sup> 2 a shift of energy most slowly varies with *nr* . It means that a density of states is the highest in the vicinity of the zonule upper edge. The zonule lower edge is shifted by <sup>2</sup> min max 0 *n n* 2! ! 0 max . The zonule width max 0 *n* increases with *n*0 if 2 2 R 2 2 *n*<sup>0</sup> . If max 0 *n* 1 2 then number of states on each interval of energy values of width *<sup>c</sup>* 2 is equal to *n*<sup>0</sup> , (the spin will be taken into account subsequently). For max 0 *n* 1 2 zonules overlap, gaps in the spectrum disappear, and the number of states in the interval becomes less than *n*<sup>0</sup> , i.e., grows of the density of states is decelerated with energy increase. That is transitive area of energy. For higher

 2 1 exp ! !

*<sup>X</sup> <sup>X</sup>*

*n l*

**r**

 

*ln n*

*nr*

determined by the formula that is like to the formula (36). Then the boundary condition

increases. The maximal value 1*l* , at which the inequality 2 2

fulfilled, determines the number of the eigenstates of the Hamiltonian hˆ that have 1

(ordinary states). It can be shown that when 1 *<sup>r</sup> l n* then 2 2

The quantum number of the energy *n* is equal to *nr* because when 0 *l* the energy

estimate that was obtained in classic mechanics theory in section 3.1.1 and in the work (Landau, 1930). Those ordinary states are quasi degenerated. The multiplicity of this degeneration is proportional to the magnetic field. Then the main term in the thermodynamical potential, as it was computed by Landau (Landau, 1930), should not depend on the magnetic field, and the Bohr – van Leeuwen theorem is vindicated. When <sup>1</sup> *l l* and 0 *l* the other nulls that are described by the formula (36) would satisfy the boundary condition. Those are the near-boundary states, and in this case are not restrictions

In the section 3.2 it will be shown that for statistical mechanics of the electron gas in the

. There are *n* near-boundary states.

, 2 , , 2 1, 2

 **0**

would be satisfied at arbitrarily large values 2 2 R 2

 besides 0 1 

of this Hamiltonian with boundary condition (33). Then

0

max 0 *n* increases with *n*0 if 2 2 R 2 2

each interval of energy values of width

account subsequently). For

*n n*

*n*<sup>0</sup> 2 ! . In the vicinity of *n n <sup>r</sup>* <sup>0</sup> 2 a shift of energy

zonule lower edge is shifted by <sup>2</sup>

*r*

*c*

0 0 00 0

2 2

 

*nn n*

2 , exp 2 . 1! !

*r r*

The degenerate levels are transformed in zonule. It follows from formula (39) that the zonule upper edge is determined by minimum value of *nn n* <sup>0</sup> *r r* 1! ! that is roughly

means that a density of states is the highest in the vicinity of the zonule upper edge. The

and the number of states in the interval becomes less than *n*<sup>0</sup> , i.e., grows of the density of states is decelerated with energy increase. That is transitive area of energy. For higher

*n*<sup>0</sup> . If

*r r r n*

*nn n nn n n l*

<sup>R</sup>

0

 

 1 , *X n ln* 

, 1; *l x* at

*r r*

*n l*, . This number of the ordinary states is consistent with the

<sup>0</sup> h has fundamental importance. Let as study the spectrum

2 2

R

max 0 *n* 1 2 zonules overlap, gaps in the spectrum disappear,

min max 0 *n n* 2! ! 0 max

 

 . The zonule width

*<sup>c</sup>* 2 is equal to *n*<sup>0</sup> , (the spin will be taken into

tends to zero, and it would be shown

has *nr* 1 nulls, but all these nulls are

only when energy has also large-

<sup>1</sup> 2 1, ; 1 R

<sup>1</sup> 2 12 *<sup>r</sup> l n* R R 

*n ln r r* is

(39)

most slowly varies with *nr* . It

max 0 *n* 1 2 then number of states on

> .

, (38)

*r r* 1, ; 1 . Then the boundary condition (33)

*n ln r r* 1, ; 1 increases when the value of *l*

 

*X* . (In the mathematical handbook (Erdélyi, 1953) it is

scale value *x* . This null *X* tends to infinity when

that

when 

1 . When

depends on *l* only by

on the values of

<sup>2</sup>

magnetic field the Hamiltonian ˆ

would be satisfied when 2 2 R 2

written that the function

scale value). The value of the null

values of energy not the greatest but other nulls, which are described by formula (36), satisfy the boundary condition. With the approximate formula for the nulls of the Bessel function j *l l* , 2 the second in magnitude null of the function (35) with *nr* is 2 2 <sup>2</sup> *nn n* 0 0 82 8 <sup>0</sup> . When it will be so that 2 22 *n*<sup>0</sup> 8 2 R the boundary condition should be satisfied only not the greatest but other nulls of the degenerate hypergeometric function. Corresponding values of energy look like:

$$
\varepsilon\_0 \left( n\_0 \right) = \frac{\hbar \alpha\_c}{2} \left( n\_0 + 2\gamma' \right) = \frac{\hbar \alpha\_c}{2} \left( \frac{\circledast \mathbb{1}^2}{\lambda^2} \right)^{-1} \mathbf{j}^2 \left( \left| l \right|, \nu \right) = \varepsilon\_0 \left( \left| l \right|, \nu \right) = \frac{\hbar^2}{2m \mathbb{R}^2} \mathbf{j}^2 \left( \left| l \right|, \nu \right) \tag{40}
$$

where *n n <sup>r</sup>* 2 and is not described by the formula (39). They coincide with eigenvalues energy in a circular potential well with reflecting boundaries (see formula (19)). The spectrum becomes quasicontinuous as distances between the nearest levels are proportional to <sup>2</sup> R . The density of states does not depend on energy, as well as for twodimensional gas of free particles. The function (35) would be expanded in series over the Bessel functions (Erdélyi, 1953) and the first term is proportional to J 2 *<sup>l</sup> r m* . The parameter of this expansion is 1 2 *n*0 . Hence, the wave functions in this approximation also coincide with free electron wave functions. This form of the spectrum would be illustrated by the classical consideration. If the formula (34) would be considered as the classical Hamiltonian, the ultimate energy for which the classical accessible area is determined by the parabolic potential is equal to 2 2 *m<sup>c</sup>* R 8 . That within a factor that is close to 1 coincides with the energy of the transitive area and with energy of the transition to quasicontinuous spectrum 2 22 2 *m <sup>c</sup>* R .

These results can be described as energy spectrum breakdown into two bands. A spectrum lower part is denoted as a magnetic band, and the upper one will be denoted as a conduction band. Bands are not separated by a gap or sharp boundary, but far from transitive area the density of states and wave functions differ substantially. Fine structure of the density of states in the lower part of magnetic band represents the narrow zonule separated by gaps. The total width of the allowed zonule and gap is equal to *<sup>c</sup>* 2 . Number of states in an interval with number *n*<sup>0</sup> is equal to *n*<sup>0</sup> . In the transitive area gaps disappear and in the conduction band the spectrum is quasicontinuous. The maximum of magnitude of a wave function in the magnetic band is localized within a ring of width about 2 and about <sup>0</sup> *R n* 2 in radius. The density of states in the magnetic band, averaged over the interval *<sup>c</sup>* <sup>2</sup> , is <sup>2</sup> 2 4 *n*0 0 *c c* and grows with energy. In the transitive area this grows is decelerated, and in the conduction band the density of states does not depend on energy and is equal to 2 2 *m*R 2 (without spin consideration). It is believed that the magnetic band is finished when 2 2 *n n* <sup>0</sup> R 4 *<sup>b</sup>* or 2 2 <sup>0</sup> *m c b* R 8 as would be expected from the quasiclassical consideration. Then the density of states will be continuous.

## **3.2 Statistical mechanics of electron gas in uniform magnetic field with regard for electrostatic interaction**

The quantum-mechanical average value of the magnetic moment in the ordinary eigenstate of the Hamiltonian hˆ in absence of the potential energy *U r* is:

Statistical Mechanics That Takes into Account Angular

To do this let us multiply the function <sup>1</sup>

where *fE fE* , 

the thermostat phase space leads to the formula:

 

*B*

*B*

number. The saddle-point <sup>1</sup>

*k T*

0 0

and the result is:

0

 from integral at <sup>1</sup>

integrate over *E* between 0 and . Then the integrations over E and

HE E

 *t aa t*

> *a i th th*

0 *B k T*

2

0

into (44). Then the non-normalized statistical operator is obtained:

<sup>2</sup> <sup>1</sup> 0

exp d .

*z*

canonical ensemble is considered, then the statistical operator of particle number

 <sup>ˆ</sup>*N B* exp *kT a a*

Momentum Conservation Law - Theory and Application 463

by using the formula (8). Let us generalize the Krutkov method (Krutkov, 1933; Zubarev, 1974) for this case. We calculate the Laplace transformation with respect the total energy *E* .

> 2 *<sup>E</sup> I*

 <sup>2</sup> 0

L L is Laplace transform of a function *f E* . The integration over

 *IE th <sup>E</sup> aa i* 

*h* 

<sup>1</sup> <sup>2</sup> 0 1 exp , exp d *<sup>N</sup> th th th*

where 1 is the phase space of a thermostat particle and *hth* is its Hamiltonian and *Nth* is the number of thermostat particles. As result of inverse Laplace transformation we obtain:

 <sup>1</sup> 1 2 0

*a i th*

This integration can be performed by the saddle-point method because *Nth* is a large-scale

exponent that depend on variables-operators of the quantum subsystem can be factored

0

*l aa i*

Here the first multiplier is the common formula of the statistical operator for the quantum Gibbs distribute (Landau, & Lifshitz, E.M., 1980a; Zubarev, 1974). The second multiplier would be computed by the Darwin – Fowler method (Fowler, & Guggenheim, 1939) like as in the formula (16) and describes the conservation of the particle number. If the grand

 

<sup>1</sup>

would be obtained from this multiplier by the Krutkov method. The last multiplier in formula (49) cannot be computed by those methods because it does not have any large-scale parameter. This multiplier imposes constraints on ensembles that the total angular momentum equal to zero. If Hamiltonian and an operator that should be averaged have the

ˆ exp exp d

*kT a a N a a i*

**th**

exp d d

 

 

is determined by the first exponent, and the second

 

**th** (47)

d d 2 exp ln exp d

as well as in the classical case.The result should be substituted

 

(50)

*<sup>E</sup> <sup>E</sup> N t a a N*

E EE (48)

 

1

(formulae (44 - 45) by exp

 

 

 

  *E* and

can be performed,

E (46)

 

(49)

$$
\mu\_z = -\left(\hat{\varepsilon}\varepsilon\left(n,l\right)\right|\hat{\varepsilon}\text{H}\right) = -\left\{\left(\hbar\epsilon/m\right)\left(n+\varphi\right) + \left(\hbar\epsilon\text{H}/m\right)\left(\hat{\varepsilon}\gamma/\hat{\varepsilon}\text{H}\right)\right\}\tag{41}
$$

It is a negative quantity because the positive term that proportional to *H* is small. In the paper (Landau, 1930) were taken into account only ordinary states. Then the quantumstatistical average value of the gas magnetic moment must be:

$$\mathbb{S}\mathbb{X} = -\sum\_{nl} \frac{\partial \varepsilon\left(n, l\right)}{\partial H} \ln\left(n, l\right) = -\sum\_{nl} \frac{\partial \varepsilon\left(n, l\right)}{\partial H} \left\{ D\left(H\right) \left[1 + \exp\left(\frac{\varepsilon\left(n, l\right) - \mu}{k\_B T}\right)\right]^{-1} \right\} < 0 \tag{42}$$

Here n*n l*, is the average occupation number of the state *n l*, , *D H* is the multiplicity of degeneracy that in this case does not depend on the state energy and depend on the magnetic field. But in this work the magnetic moment of the gas was computed as

$$\begin{split} \mathcal{R} &= -\frac{\partial \mathcal{Q}}{\partial H} = k\_{\mathcal{B}} T \frac{\partial}{\partial H} \sum\_{nl} \left\{ D(H) \ln \left[ 1 + \exp \left( \frac{\mu - \varepsilon \langle n, l \rangle}{k\_{\mathcal{B}} T} \right) \right] \right\} = \\ &k\_{\mathcal{B}} T \sum\_{nl} \left\{ \frac{\partial D(H)}{\partial H} \ln \left[ 1 + \exp \left( \frac{\mu - \varepsilon \langle n, l \rangle}{k\_{\mathcal{B}} T} \right) \right] \right\} - \sum\_{nl} \frac{\partial \varepsilon \langle n, l \rangle}{\partial H} \ln \langle n, l \rangle. \end{split} \tag{43}$$

When this result is compared with the formula (42) it is apparent that the thermodynamical potential is determined incorrectly.

Let us obtain the density of distribution for an electron gas that is at equilibrium with thermostat, which is described by classical mechanics. The conservation of the zero value of the angular momentum also will be taken into account. The characteristic function of this system is:

$$\begin{split} \rho = \left(2\pi\right)^{-3} I\_{\mathbb{E}} \int\_{0}^{2\pi} \int\_{0}^{\infty} \exp\left[\left(N - \sum\_{\mathbf{v}} a\_{\mathbf{v}}^{+} a\_{\mathbf{v}}\right) \left(i\phi + \chi\right) + \left(\sum\_{\mathbf{v}} l\_{\mathbb{z}}\left(\Psi\right) a\_{\mathbf{v}}^{+} a\_{\mathbf{v}}\right) \left(i\boldsymbol{\omega} + \eta\right)\right] \mathrm{d}\boldsymbol{\phi} \mathrm{d}\boldsymbol{a} \\\ I\_{\mathbb{E}} = 2\pi \int\_{0}^{\kappa} \rho\_{1}\left(\boldsymbol{\delta}\right) \rho\_{2}\left(\boldsymbol{E} - \boldsymbol{\delta}\right) \mathrm{d}\boldsymbol{\delta} = \\\ \left. \int\_{0}^{\kappa} \delta \left[\boldsymbol{\delta}\mathbb{U}\_{\mathbb{H}} - \left(\boldsymbol{E} - \boldsymbol{\delta}\right)\right] \prod\_{0}^{2\pi} \exp\left[\left(\boldsymbol{\delta} - \sum\_{\mathbf{v}} \left[\boldsymbol{\varepsilon}\_{0}\left(\boldsymbol{\Psi}\right) + \left(\hbar a\_{\boldsymbol{\epsilon}}\left/\boldsymbol{2}\right) l\_{\boldsymbol{\epsilon}}\left(\boldsymbol{\Psi}\right)\right] a\_{\mathbf{v}}^{+} a\_{\mathbf{v}}\right) \left(i\boldsymbol{\tau} + \boldsymbol{\mathcal{G}}\right)\right] \mathrm{d}\boldsymbol{\tau} \mathrm{d}\boldsymbol{\delta}\right]. \end{split} \tag{44}$$

This formula is obtained on a basis of the properties of characteristic functions that was described in the formulae (6 – 8), and the quantum characteristic function (see formula (15)). Here *E* is a total energy, E is the energy of the electron gas, H*th* is the Hamiltonian of the thermostat that is a summatory function of the classical variables. Values of energy are expressed by the dimensionless positive integers, and the angular momentum is measured in . The expression of the Hamiltonian hˆ as the sum 0 <sup>ˆ</sup> <sup>2</sup> ˆ *c z* h *l* is taken into account. Then the second equality in (44) would be rewritten as:

$$I\_E = \int\_0^\varepsilon \delta \left[ \,^\varepsilon \mathbb{K}\_\hbar - \left( E - \mathcal{G} \right) \right] \prod\_0^2 \exp \left[ \left( \mathcal{S} - \sum\_\mathbf{v} \varepsilon\_0 \left( \Psi \right) a\_\mathbf{v}^+ a\_\mathbf{v} \right) \left( i\tau + \mathcal{G} \right) \right] \mathrm{d}\tau \mathrm{d}\mathcal{G} \tag{45}$$

*<sup>z</sup> n l H e m n eH m H* ,

the paper (Landau, 1930) were taken into account only ordinary states. Then the quantum-

Here n*n l*, is the average occupation number of the state *n l*, , *D H* is the multiplicity of degeneracy that in this case does not depend on the state energy and depend on the magnetic field. But in this work the magnetic moment of the gas was computed as

*nl nl B n l n l n l n l D H H H k T*

*nl B nl*

 <sup>1</sup> ,, , , 1 exp <sup>0</sup>

<sup>M</sup> <sup>n</sup> (42)

, ln 1 exp

 

 

 

 

 

HE E <sup>E</sup> (45)

   

*l* is taken into account.

(44)

*nl B*

*n l kT D H H H k T*

 

*D H n l n l k T n l H k T H*

When this result is compared with the formula (42) it is apparent that the thermodynamical

Let us obtain the density of distribution for an electron gas that is at equilibrium with thermostat, which is described by classical mechanics. The conservation of the zero value of the angular momentum also will be taken into account. The characteristic function of this

*I N aa i l aa i*

1 2

 

0

0

*th c z*

in . The expression of the Hamiltonian hˆ as the sum 0 <sup>ˆ</sup> <sup>2</sup> ˆ *c z* h

*E z*

*I E*

2d

*E*

*E l aa i*

= 2 exp d d

 

This formula is obtained on a basis of the properties of characteristic functions that was described in the formulae (6 – 8), and the quantum characteristic function (see formula (15)). Here *E* is a total energy, E is the energy of the electron gas, H*th* is the Hamiltonian of the thermostat that is a summatory function of the classical variables. Values of energy are expressed by the dimensionless positive integers, and the angular momentum is measured

HE E E

E EE

 

 <sup>2</sup> 0

*IE th E* exp *aa i* d d

 

exp 2 dd .

<sup>n</sup>

 

, , ln 1 exp , .

It is a negative quantity because the positive term that proportional to

 

(41)

  *H* is small. In

(43)

*B*

potential is determined incorrectly.

2 2 <sup>3</sup>

 

0 0

0 0

2

Then the second equality in (44) would be rewritten as:

0 0

system is:

 M

 

statistical average value of the gas magnetic moment must be:

*B*

by using the formula (8). Let us generalize the Krutkov method (Krutkov, 1933; Zubarev, 1974) for this case. We calculate the Laplace transformation with respect the total energy *E* . To do this let us multiply the function <sup>1</sup> 2 *<sup>E</sup> I* (formulae (44 - 45) by exp*E* and integrate over *E* between 0 and . Then the integrations over E and can be performed, and the result is:

$$I\_E = \int\_0^\varepsilon \mathcal{S} \left[ \mathcal{K}\_\hbar - \left( E - \mathcal{S} \right) \right] \prod\_0^{2x} \exp \left[ \left( \mathcal{S} - \sum\_\mathbf{q} \varepsilon\_0 \left( \Psi \right) a\_\mathbf{q}^+ a\_\mathbf{q} \right) \left( i\tau + \mathcal{G} \right) \right] \mathbf{d} \,\tau \mathbf{d} \,\mathbb{S} \tag{46}$$

where *fE fE* , L L is Laplace transform of a function *f E* . The integration over the thermostat phase space leads to the formula:

$$\left(2\pi\right)^{-1}\left[t\_{\hbar}\left(\mathcal{G}\right)\right]^{\mathbb{N}\_{\mathbf{n}}}\exp\left(-\mathcal{G}\sum\_{\mathbf{v}}\varepsilon\_{0}\left(\Psi^{\prime}\right)a\_{\mathbf{v}}^{+}a\_{\mathbf{v}}\right), \ t\_{\hbar}\left(\mathcal{G}\right) = \int\_{\Gamma\_{1}}\exp\left(-\mathcal{H}t\_{\hbar}\right)d\Gamma\_{1} \tag{47}$$

where 1 is the phase space of a thermostat particle and *hth* is its Hamiltonian and *Nth* is the number of thermostat particles. As result of inverse Laplace transformation we obtain:

$$\int\limits\_{\Gamma\_{\mathsf{H}}} \int\limits\_{0}^{\eta} \rho\_{1}(\mathcal{E}) \rho\_{2}(E-\mathcal{E}) \mathrm{d}\mathbf{\mathcal{E}} \mathrm{d}\Gamma\_{\mathsf{H}} = \left(2\pi\right)^{-1} \int\limits\_{\mathsf{a}\to\mathsf{o}} \exp\left[N\_{\mathsf{H}}\left(\mathcal{G}\frac{E}{N\_{\mathsf{H}}} + \ln t(\mathcal{G})\right)\right] \exp\left[-\mathcal{G}\sum\_{\mathsf{v}} \varepsilon\_{0}\left(\Psi\right)a\_{\mathsf{v}}^{+}a\_{\mathsf{v}}\right] \mathrm{d}\mathcal{B}\left(48\right) = \left(2\pi\right)^{-1} \int\limits\_{\mathsf{a}\to\mathsf{o}} \exp\left[N\_{\mathsf{H}}\left(\mathcal{G}\frac{E}{N\_{\mathsf{H}}} + \ln t(\mathcal{G})\right)\right] \mathrm{d}\mathcal{B}\left(48\right)$$

This integration can be performed by the saddle-point method because *Nth* is a large-scale number. The saddle-point <sup>1</sup> 0 *B k T* is determined by the first exponent, and the second exponent that depend on variables-operators of the quantum subsystem can be factored from integral at <sup>1</sup> *B k T* as well as in the classical case.The result should be substituted into (44). Then the non-normalized statistical operator is obtained:

$$\begin{split} \hat{\rho} = \exp\Big[ - \left( k\_{\text{s}} T \right)^{-1} \sum\_{\text{v}} \varepsilon\_{0} \left( \Psi \right) a\_{\text{v}}^{+} a\_{\text{v}} \Big] \Big| \int\_{0}^{2\pi} \exp\Big[ \left( N - \sum\_{\text{v}} a\_{\text{v}}^{+} a\_{\text{v}} \right) \left( i\phi + \chi \right) \Big] \mathrm{d}\phi \, \times \\ \int\_{0}^{2\pi} \exp\Big[ \left( \sum\_{\text{v}} l\_{z} \left( \Psi \right) a\_{\text{v}}^{+} a\_{\text{v}} \right) \left( i\alpha + \eta \right) \Big] \mathrm{d}\alpha \, . \end{split} \tag{49}$$

Here the first multiplier is the common formula of the statistical operator for the quantum Gibbs distribute (Landau, & Lifshitz, E.M., 1980a; Zubarev, 1974). The second multiplier would be computed by the Darwin – Fowler method (Fowler, & Guggenheim, 1939) like as in the formula (16) and describes the conservation of the particle number. If the grand canonical ensemble is considered, then the statistical operator of particle number

$$\hat{\rho}\_{\mathcal{N}} = \exp\left[ -\mu \left( k\_{\mathcal{B}} T \right)^{-1} \sum\_{\Psi} a\_{\Psi}^{+} a\_{\Psi} \right] \tag{50}$$

would be obtained from this multiplier by the Krutkov method. The last multiplier in formula (49) cannot be computed by those methods because it does not have any large-scale parameter. This multiplier imposes constraints on ensembles that the total angular momentum equal to zero. If Hamiltonian and an operator that should be averaged have the

Statistical Mechanics That Takes into Account Angular

the chemical potential or the Fermi energy is obtained:

 

conveniently by new frequency 0 4 3 *r c*

commonly decrease the frequency

<sup>ˆ</sup> <sup>ˆ</sup> <sup>2</sup> 2 2 *<sup>N</sup>*

*<sup>i</sup> <sup>i</sup> <sup>l</sup>* , and <sup>ˆ</sup> ˆ *<sup>N</sup>*

alter. Hence it should be considered separately.

orbital motion. Then the energy of the magnetic band is:

The number of states in the magnetic band equals to:

1

*mb r n l n*

*nb*

The distance between the neighboring spikes in *Emb* is <sup>2</sup>

2

 

factor <sup>4</sup>

of electrons <sup>2</sup> *N*

R 

ˆ 0 *<sup>N</sup>*

and as result we obtain:

where 2 2 *a*0 0 4

where

Momentum Conservation Law - Theory and Application 465

0 0 0 0

 

> 

RR R R (55)

<sup>0</sup> 43 8 *<sup>c</sup> a mr* R 

*a* R . Then one-particle effective Hamiltonian

h (57)

or another way the effective charge *<sup>r</sup> e e* by

0 4 4 2

R (58)

*r b n* . If *Nb* is less than the number

*m*

2 2 2 32 *N n n n ea H b bb b* 1 <sup>0</sup> 12 R (59)

0 the energy of the ground state with a glance the formula (56) is:

, which has meaning of

. It is expressed

4 4 1 , 1 , <sup>12</sup> <sup>3</sup> <sup>6</sup> <sup>3</sup>

0 00 0 12 12 *mm a m a c c*

0 is the Fermi energy of two-dimensional electron gas in absence of a magnetic

 

2 22

*l mr*

<sup>0</sup> 4 3 10 *a* R . This factor does not depend on the electron density if this density

*<sup>r</sup> <sup>i</sup>* M M h *<sup>H</sup>* . The energy magnetic splitting over spin does not

 R R , (56)

*c c a a*

 

 R R

2 2 2 2

e *me* is Bohr radius. The Lagrange multiplier

*ma ma*

2 2

22 2

2 2 <sup>ˆ</sup> <sup>1</sup> <sup>ˆ</sup> 2 8 ˆ *z r*

 

The Coulomb interaction and the electron density inhomogeneous that are neglected

satisfies to the condition of the degeneration. The operator of the total magnetic moment

*<sup>r</sup> i ri <sup>i</sup>* M *e m l e Hr* . When the quantum-statistical average is computed,

The electron gas always interacts with electromagnetic field that in the ordinary circumstance has zero temperature. This leads to the fact that the gas passes to the ground state by the spontaneous photon irradiation. The role of statistical mechanics is that it imposes a constraint on a value of the angular momentum in the ground state. The energy of the ground state with zero angular momentum would be computed by the spectrum of the one-particle states that was described in the previous subsection. The energy levels, the degeneracy multiplicity, and the boundary of the magnetic band are determined only the

3 42

*n ea E nn <sup>H</sup>*

The quantity *nb* is discrete; therefore the formulae (58 – 59) describe a smoothed function.

3 108

*r b*

*p m r* 

*r r*

*c r* 

 

field. The residual potential energy has the form 2 2

 

will have the form that was study in the previous subsection:

commutative term that is proportional to the total angular momentum operator, this term should be eliminated when the averaging is performed. That is the reason for the change to 0 in the Hamiltonian <sup>0</sup> *a a* in the formula (48).

The model that will be considered below is described by the Hamiltonian:

$$\hat{\mathbf{x}}\hat{\mathbf{X}} = \sum\_{i=1}^{N} \hat{\boldsymbol{\xi}}\_{0i}; \qquad \hat{\boldsymbol{\xi}}\_{\circ} = \frac{\mathbf{1}}{2m} \left( \hat{p}\_r^2 + \frac{\hat{l}\_z^2}{r^2} \right) + \frac{m o \rho\_c^2 r^2}{8} + \mathcal{U}(r) \tag{51}$$

Here *U r* is potential of the self-consisted electric field that describes interaction of an electron with other electrons and neutralizing background that have the form of circle with radius R .

The electron density in the magnetic field should be distributed in such a way as to shield the external potential 2 2 *m r <sup>c</sup>* 8 . This shielding cannot be perfect. The current that would be generated by a residual potential would be compensated by the diffusion current that generated by an inhomogeneity of the electron density. Because the kinetic energy in the Hamiltonian (51) has the standard form, electrons density distribution and self-consistent potential can be calculated by the density functional method, described in the work (March, 1983). In this approximation, energy of the ground state of electron gas is presented in the form of functional of the gas density, i.e., number of particles per unit of area *r* :

$$\begin{split} \operatorname{E}\_{0}\left[\rho\left(\mathbf{r}\right)\right] &= \left(\pi\hbar^{2}/2m\right)\left[\rho^{2}\left(\mathbf{r}\right)\mathbf{dr} + \left(e^{2}/8\pi\mathbf{e}\_{0}\right)\right] \int\_{\mathbb{C}\times\mathbb{C}} \rho\left(\mathbf{r}\right)\rho\left(\mathbf{r}'\right) \left|\mathbf{r} - \mathbf{r}'\right|^{-1} \operatorname{d}\mathbf{r} \operatorname{d}\mathbf{r}' - \\ & \quad \left(e^{2}/4\pi\mathbf{e}\_{0}\right) \int\_{\mathbb{C}} \rho\left(\mathbf{r}\right) \left[u\left(\mathbf{r}\right) + u\_{0}\right] \operatorname{d}\mathbf{r} + \left(m\rho\_{c}^{2}/8\right) \int\_{\mathbb{C}} \rho\left(\mathbf{r}\right) r^{2} \operatorname{d}\mathbf{r} \ . \end{split} \tag{52}$$

Here 2 2 2*m* is density of kinetic energy of two-dimensional degenerated Fermi gas (each state is supposed to be twice degenerated), the second addend describes energy of electrons Coulomb interaction, e0 is vacuum inductivity, the third addend is energy of electrostatic interaction with the neutralizing background, the last addend described gas potential energy in the effective harmonic potential depending on magnetic field. The neutralizing background has the charge density <sup>0</sup> *eY r* R and generate the potential *eu***r** . This potential would be expressed by the elliptical integrals and is approximated by the parabola and a repulsing bound. The step function *Y x* 1 at 0 *x* and *Y x* 0 at *x* 0 . The constant 0 *eu* would be assigned so that the potential energy in the effective Hamiltonian would be zero at 0 *r* . All integrations are made over the area *C* , which is a circle of radius R . The functional *E*<sup>0</sup> **r** should be minimized under the following supplementary condition:

$$\int\_{\mathbb{C}} \rho(\mathbf{r}) \, \mathrm{d}\mathbf{r} = \pi \mathbb{R}^2 \rho\_0 \tag{53}$$

The quadratic term of the residual potential is of chief interest. Therefore the minimization would be performed in the quadratic approximation. The test function has the form:

$$
\rho(r) = \rho\_0 + a\mathbb{R}^{-2} - \mathcal{J}\mathbb{R}^{-4}r^2 \,, \tag{54}
$$

and as result we obtain:

464 Thermodynamics – Interaction Studies – Solids, Liquids and Gases

commutative term that is proportional to the total angular momentum operator, this term should be eliminated when the averaging is performed. That is the reason for the change

2

*m r*

Here *U r* is potential of the self-consisted electric field that describes interaction of an electron with other electrons and neutralizing background that have the form of circle with

The electron density in the magnetic field should be distributed in such a way as to shield

generated by a residual potential would be compensated by the diffusion current that generated by an inhomogeneity of the electron density. Because the kinetic energy in the Hamiltonian (51) has the standard form, electrons density distribution and self-consistent potential can be calculated by the density functional method, described in the work (March, 1983). In this approximation, energy of the ground state of electron gas is presented in the

22 2 1

 

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

2 d8 d d

**rr r rr**

2*m* is density of kinetic energy of two-dimensional degenerated Fermi gas

*c*

 

> 

2 22

*<sup>c</sup>* 8 . This shielding cannot be perfect. The current that would be

*r* :

(52)

R and generate the potential

**r** should be minimized under the following

**r r** <sup>R</sup> (53)

*r r* R R , (54)

<sup>H</sup> <sup>0</sup> h h (51)

*z c*

*l mr p U r*

in the formula (48).

0 2

*i r*

form of functional of the gas density, i.e., number of particles per unit of area

e

0 0

0 0

 

neutralizing background has the charge density <sup>0</sup> *eY r*

 

*E me*

 

circle of radius R . The functional *E*<sup>0</sup>

supplementary condition:

e

*e uu m r*

 4 d 8 d . *C C C*

2 2 2

*C C*

(each state is supposed to be twice degenerated), the second addend describes energy of electrons Coulomb interaction, e0 is vacuum inductivity, the third addend is energy of electrostatic interaction with the neutralizing background, the last addend described gas potential energy in the effective harmonic potential depending on magnetic field. The

*eu***r** . This potential would be expressed by the elliptical integrals and is approximated by the parabola and a repulsing bound. The step function *Y x* 1 at 0 *x* and *Y x* 0 at *x* 0 . The constant 0 *eu* would be assigned so that the potential energy in the effective Hamiltonian would be zero at 0 *r* . All integrations are made over the area *C* , which is a

> <sup>2</sup> <sup>0</sup> C d

The quadratic term of the residual potential is of chief interest. Therefore the minimization

2 42

 

> 

would be performed in the quadratic approximation. The test function has the form:

0

 

<sup>ˆ</sup> <sup>1</sup> <sup>ˆ</sup> ; ˆ 2 8

The model that will be considered below is described by the Hamiltonian:

to 0 

radius R .

Here 2 2 

the external potential 2 2 *m r*

in the Hamiltonian <sup>0</sup>

*a a*

ˆ ˆ *<sup>N</sup>*

1

*i*

$$\alpha = \frac{\left(m o\_c \Re^2\right)^2 a\_0}{12\pi \hbar^2 \mathcal{R}} \left(1 - \frac{4a\_0}{\Im \mathcal{R}}\right), \quad \mathcal{J} = \frac{\left(m o\_c \Re^2\right)^2 a\_0}{6\pi \hbar^2 \mathcal{R}} \left(1 - \frac{4a\_0}{\Im \mathcal{R}}\right). \tag{55}$$

where 2 2 *a*0 0 4e *me* is Bohr radius. The Lagrange multiplier , which has meaning of the chemical potential or the Fermi energy is obtained:

$$
\zeta = \left(\pi \hbar^2 \rho\_0 \not\!/ \text{m}\right) + m o\_c^2 \mathbb{R} a\_0 \not\!/ \text{12} = \zeta\_0 + m o\_c^2 \mathbb{R} a\_0 \not\!\!/ \text{12} \,\,\,\,\tag{56}
$$

where 0 is the Fermi energy of two-dimensional electron gas in absence of a magnetic field. The residual potential energy has the form 2 2 <sup>0</sup> 43 8 *<sup>c</sup> a mr* R . It is expressed conveniently by new frequency 0 4 3 *r c a* R . Then one-particle effective Hamiltonian will have the form that was study in the previous subsection:

$$\hat{\boldsymbol{\kappa}}\_r = \frac{1}{2m} \left( \hat{\boldsymbol{p}}\_r^2 + \frac{\hat{\boldsymbol{I}}\_z^2}{r^2} \right) + \frac{m o \rho\_r^2 r^2}{8} \tag{57}$$

The Coulomb interaction and the electron density inhomogeneous that are neglected commonly decrease the frequency *c r* or another way the effective charge *<sup>r</sup> e e* by factor <sup>4</sup> <sup>0</sup> 4 3 10 *a* R . This factor does not depend on the electron density if this density satisfies to the condition of the degeneration. The operator of the total magnetic moment <sup>ˆ</sup> <sup>ˆ</sup> <sup>2</sup> 2 2 *<sup>N</sup> <sup>r</sup> i ri <sup>i</sup>* M *e m l e Hr* . When the quantum-statistical average is computed, ˆ 0 *<sup>N</sup> <sup>i</sup> <sup>i</sup> <sup>l</sup>* , and <sup>ˆ</sup> ˆ *<sup>N</sup> <sup>r</sup> <sup>i</sup>* M M h *<sup>H</sup>* . The energy magnetic splitting over spin does not alter. Hence it should be considered separately.

The electron gas always interacts with electromagnetic field that in the ordinary circumstance has zero temperature. This leads to the fact that the gas passes to the ground state by the spontaneous photon irradiation. The role of statistical mechanics is that it imposes a constraint on a value of the angular momentum in the ground state. The energy of the ground state with zero angular momentum would be computed by the spectrum of the one-particle states that was described in the previous subsection. The energy levels, the degeneracy multiplicity, and the boundary of the magnetic band are determined only the orbital motion. Then the energy of the magnetic band is:

$$E\_{mb} = \hbar o\_r \sum\_{\nu=1}^{n\_b} \left( n + 2\gamma\_{\nu \parallel} \right) \eta \approx \frac{\hbar o\_r n\_b^3}{3} = \frac{e^4 a\_0^2}{108\hbar^2 m} \mathcal{R}^4 H^4 \tag{58}$$

The number of states in the magnetic band equals to:

$$N\_b = n\_b \left( n\_b + 1 \right) \approx n\_b^2 = \left( e^2 a\_o \Big/ 12 \,\text{h}^2 \right) \otimes {}^3H^2 \tag{59}$$

The quantity *nb* is discrete; therefore the formulae (58 – 59) describe a smoothed function. The distance between the neighboring spikes in *Emb* is <sup>2</sup> *r b n* . If *Nb* is less than the number of electrons <sup>2</sup> *N* R 0 the energy of the ground state with a glance the formula (56) is:

Statistical Mechanics That Takes into Account Angular

account the angular momentum conservation law.

U235, McGraw – Hill, New York.

Abrikosov, A.A. (1972). *Introduction to the Theory of Normal Metals,* Nauka, Moskow.

Cohen, K. (1951). *The Theory of Isotope Separation as Applied to the Large Scale Production of* 

Erdélyi, A. (1953). *Higher Transcendental Functions, based, in part, on notes left by Harry Bateman,* Vol. 1, Mc Graw-Hill book company, INC, New York Toronto London. Feynman, R.P., Leighton R.B., & Sands M. (1964). *The Feynman lectures on physics,* Vol. 2,

Fowler, R.H., & Guggenheim, E.A. (1939). *Statistical Thermodynamics,* Cambridge University

Khinchin, A.Y. (1949). *Mathematical Foundation of Statistical Mechanics,* Ed. Dover, NewYork. Khinchin, A.Y. (1960). *Mathematical Foundation of Quantum Statistics,* Ed. Dover, NewYork.

Addison-Wesley Publishing Company, Inc. Reading, Massachusetts. Palo Alto.

If 19 -2 

**4. Conclusion** 

**5. References** 

London

Press, London.

Momentum Conservation Law - Theory and Application 467

<sup>12</sup> *Hs* 5.15 10 T

The fundamental theory of statistical mechanics requires taking into account the law of the angular momentum conservation. The fulfilment of this requirement does not introduce any essential alterations into statistical thermodynamics, when the angular momentum of the system equal zero and the system Hamiltonian is a positive definite quadric form of all momenta. An equilibrium isolated system would have nonzero angular momentum only if an attraction of particles can resist centrifugal forces, as it is in nebulas. A gas can be in equilibrium with a rotating envelope that is a termospinstat. The condition of this equilibrium is the equality of the average value of sum of particles angular velocities to the angular velocity of the envelope. The Gibbs density of distribution and the thermodynamical functions are generalized for this case. If a system has the angular momentum equal to zero, the conservation of this value is important only when the Hamiltonian or/and an averaged quantity depend on the angular momentum. The problem of an electron gas in a uniform magnetic field is considered with taking into account the conservation of the zero value of the angular momentum. This consideration eliminates the paradoxical statement of the conventional theory that diamagnetic moment of the gas equals zero in classical as well as quantum physics (the Bohr – van Leeuwen theorem). The new formulae for the magnetic moment of the electron gas are obtained. It also leads to the effect of confinement of two-dimensional gas of charged particles by magnetic field. This results in effect of a non-uniform density of a gas, which decreases with distance from a center according both to classical as well as quantum theory. Then the model of noninteracting charged particles does not have areas of application. Many theories should be reconsidered, if they are founded on this model and on the statistical mechanics which does not take into

0

<sup>0</sup> 10 m , 0.1 m R , then *Hs* 5,15 T and 2 -1 2.92 10 J T M*<sup>s</sup>* .

*e a* 

<sup>1</sup> <sup>1</sup> <sup>2</sup> <sup>2</sup> 0 0 <sup>10</sup>

R R (64)

$$\begin{split} E\_0 &= E\_{mb} + \frac{m\mathfrak{R}^2}{\hbar^2} \int\_{\mathfrak{a}} \mathfrak{c} \mathbf{d} \mathcal{L} - \frac{m\mathfrak{R}^2}{\hbar^2} \left( \frac{e\hbar}{2m} H \right)^2 = \\ &\frac{m\mathfrak{R}^2 \zeta\_0^2}{2\hbar^2} + \frac{\pi \rho\_0 a\_0 e^2}{12m} \mathfrak{R}^3 H^2 - \frac{a\_0^2 e^4}{864\hbar^2 m} \mathfrak{R}^4 H^4 - \frac{e^2}{4m} \mathfrak{R}^2 H^2. \end{split} \tag{60}$$

Here the first term is the energy of the electron gas in the absence of a magnetic field. The second term that makes the main addend in the magnetic moment is the product of the Fermi energy of the electron gas <sup>0</sup> and the additional term in it, which depends on the magnetic field (formula (56)). The third term in the formula (60) is created by the addends *Emb* , the squared additional term of the Fermi energy, and the negative addend in the result of integration 22 2 2 *m <sup>b</sup>* R . The last term in this formula describes the energy lowering by the spin polarization. It always is smaller than the second term. The magnetic moment is:

$$\delta \mathcal{R} = -\frac{\partial \left(E\_0 - E\_p\right)}{\partial H} - \frac{E\_p}{H} = -\frac{\pi \rho\_0 a\_0 e^2}{6m} \mathcal{R}^3 H + \frac{a\_0^2 e^4}{216 \hbar^2 m} \mathcal{R}^4 H^3 + \frac{e^2}{4m} \mathcal{R}^2 H \tag{61}$$

where <sup>2</sup> *E eH m <sup>p</sup>* R 4 . The first term is the diamagnetic moment of the orbital motion in the states of the conduction band. It is proportional to the electron density and third power of the radius as distinct from the result of the Landau theory (see to formula (5)). This formula would be obtained by other way. Let us consider the density of electric current (formula (32)) in the ground state with zero angular momentum:

$$j\_{\rho} = -\frac{e\_r \hbar}{2m} \left\langle \sum\_{i=1}^{N} \left\{ \left[ \frac{\hat{l}\_i}{r\_i} + \frac{e\_r H r\_i}{2\hbar} \right] \delta \left( \mathbf{r}\_i - \mathbf{r} \right) + \delta \left( \mathbf{r}\_i - \mathbf{r} \right) \left[ \frac{\hat{l}\_i}{r\_i} + \frac{e\_r H r\_i}{2\hbar} \right] \right\} \right\rangle \tag{62}$$

The magnetic moment is:

$$\mathrm{r\u0t} = \pi \int\_0^\mathrm{s} j\_\rho(r) r^2 \mathrm{d}r = -\frac{e\_r \hbar}{m} \sum\_{nl} \mathrm{n}\left(n, l\right) l \pi \int\_0^\mathrm{s} \left|\nu\_{nl}\left(r\right)\right|^2 r \mathrm{d}r - \frac{\pi e\_r^2 H}{2m} \int\_0^\mathrm{s} \sum\_{nl} \left|\nu\_{nl}\left(r\right)\right|^2 \mathrm{n}\left(n, l\right) r^3 \mathrm{d}r \quad \text{(63)}$$

The first term equals to zero because <sup>2</sup> <sup>0</sup> 2 d1 *nl r rr* <sup>R</sup> and , 0 *nl* <sup>n</sup> *nll* . Let us change in the second addend <sup>2</sup> , *nl nl r nl r* n . Then the integration gives both the first and the second terms of the formula (61). If the sum , *nl* <sup>n</sup> *nll* will be computed in the common theory, the linear with respect to the magnetic field terms in the formula (63) will cancel.

The orbital diamagnetic susceptibility decreases with increasing of the magnetic field. This function also has a spikes that is caused by the addend 2 2 *E H mb* . When *Nb* will be equal to <sup>2</sup> *N* R 0 the diamagnetic susceptibility will be zero. When the magnetic field is stronger, the Fermi energy remains on the level *n N <sup>F</sup>* of the magnetic band and increases proportionally to magnetic field. The magnetic moment remains constant. The paramagnetic moment also does not increase because the density of states on the Fermi level decreases linearly. The value of the magnetic induction whereby the saturation of the magnetic moment will start is:

$$H\_s = \frac{\hbar}{c} \left(\frac{12\,\pi\rho\_0}{a\_o\,\text{\textdegree R}}\right)^{\text{\textdegree K}} = 5.15 \cdot 10^{-10} \left(\frac{\rho\_0}{\text{\textdegree R}}\right)^{\text{\textdegree K}}\,\text{T} \tag{64}$$

If 19 -2 <sup>0</sup> 10 m , 0.1 m R , then *Hs* 5,15 T and 2 -1 2.92 10 J T M*<sup>s</sup>* .
