Quantum statistical mechanics of electron gas in magnetic field
Electron eigenstates in a magnetic field are considered. Density of the electrical current and an averaged
 magnetic moment are obtained. Density of states is investigated for two-dimensional electron in a circle that
 is bounded by the infinite potential barrier. The present study s...
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| Date: | 2006 |
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Інститут фізики конденсованих систем НАН України
2006
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| Cite this: | Quantum statistical mechanics of electron gas in magnetic field / I.M. Dubrovskii // Condensed Matter Physics. — 2006. — Т. 9, № 4(48). — С. 645–658. — Бібліогр.: 13 назв. — англ. |
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| author | Dubrovskii, I.M. |
| author_facet | Dubrovskii, I.M. |
| citation_txt | Quantum statistical mechanics of electron gas in magnetic field / I.M. Dubrovskii // Condensed Matter Physics. — 2006. — Т. 9, № 4(48). — С. 645–658. — Бібліогр.: 13 назв. — англ. |
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| description | Electron eigenstates in a magnetic field are considered. Density of the electrical current and an averaged
magnetic moment are obtained. Density of states is investigated for two-dimensional electron in a circle that
is bounded by the infinite potential barrier. The present study shows that the common quantum statistical
mechanics of electron gas in a magnetic field leads to incorrect results. The magnetic moment of electron
gas can be computed as the sum of averaged moments of the occupied states. The computations lead to the
results that differ from the ones obtained as the derivative of the thermodynamical potential with respect to
the magnetic field. Other contradictions in common statistical thermodynamics of electron gas in a magnetic
field are pointed out. The conclusion is done that these contradictions arise from using the incorrect statistical
operator. A new quantum function of distribution is derived from the basic principles, taking into account the
law of conservation of an angular momentum. These results are in accord with the theory that has been
obtained within the framework of classical statistical thermodynamics in the previous work.
Розглянуто власнi стани електрону в магнiтному полi. Одержано густину електричного струму i середнiй магнiтний момент. Дослiджено густину станiв для двовимiрного електрону в колi обмеженому
нескiнченним потенцiальним бар’єром. З використанням цих результатiв показано, що загальноприйнята квантова статистична механiка електронного газу в магнiтному полi приводить до помилкових результатiв. Магнiтний момент електронного газу може бути обчислений як сумма середнiх
моментiв заповнених станiв. Таке обчислення приводить до результату, що вiдрiзняється вiд того,
який одержується як похiдна термодинамiчного потенцiалу по магнiтному полю. Наведено й iншi
протирiччя у загальноприйнятiй статистичнiй термодинамiцi електронного газу в магнiтному полi.
Зроблено висновок, що цi протирiччя виникають внаслiдок використання неправильного статистичного оператора. Нова квантова функцiя розподiлу виведена з основних принципiв, беручи до уваги
закон збереження кутового моменту. Цi результати узгоджуються з теорiєю, яка була виведена у
рамках класичної статистичної термодинамiки у попереднiй роботi
|
| first_indexed | 2025-12-07T18:32:03Z |
| format | Article |
| fulltext |
Condensed Matter Physics 2006, Vol. 9, No 4(48), pp. 645–658
Quantum statistical mechanics of electron gas in
magnetic field
I.M.Dubrovskii∗
Institute for Metal Physics, 36 Academician Vernadsky Blvd., Kyiv–142, 03680, Ukraine
Received May 24, 2006
Electron eigenstates in a magnetic field are considered. Density of the electrical current and an averaged
magnetic moment are obtained. Density of states is investigated for two-dimensional electron in a circle that
is bounded by the infinite potential barrier. The present study shows that the common quantum statistical
mechanics of electron gas in a magnetic field leads to incorrect results. The magnetic moment of electron
gas can be computed as the sum of averaged moments of the occupied states. The computations lead to the
results that differ from the ones obtained as the derivative of the thermodynamical potential with respect to
the magnetic field. Other contradictions in common statistical thermodynamics of electron gas in a magnetic
field are pointed out. The conclusion is done that these contradictions arise from using the incorrect statistical
operator. A new quantum function of distribution is derived from the basic principles, taking into account the
law of conservation of an angular momentum. These results are in accord with the theory that has been
obtained within the framework of classical statistical thermodynamics in the previous work.
Key words: electron states, magnetic field, angular momentum, averaged magnetic moment, quantum
function of distribution, quantum statistical thermodynamics
PACS: 05.30.Ch, 75.20.-g
1. Introduction
It is generally agreed that the task mentioned in the title had been resolved by L.D. Landau
in work [1], the result of which is known as “the Landau diamagnetism”. The energy spectrum
discreteness was originally obtained in this work. The following consideration was based on quasi-
classical conceptions. It is assumed that an electron is localized in an orbit of classical circular
motion. The eigenvalue of the circle area is proportional to the energy eigenvalue. One from the orbit
center coordinates also has definite value in the eigenstate. An electron in an orbit creates a localized
magnetic moment. The orbit centers are homogeneously distributed over the area bound by infinite
potential barrier. These conceptions form the basis of numerous investigations concerning electron
theory of conductors (see [2–4], and the works cited there). The thermodynamical characteristics
of an electron gas in a magnetic field were also computed using these conceptions and the common
quantum function of distribution. The latter follows from the quantum statistical operator that
depends only on the Hamiltonian. Therefore the main term of thermodynamical potential does
not depend on a magnetic field, as it is observed in the classical Bohr – van Leeuwen theorem.
The averaged magnetic moment was computed as the derivative of thermodynamical potential
with respect to the magnetic field. It was obtained from a small term generated due to the energy
spectrum discreteness.
It was shown in work [5] that the classical Bohr – van Leeuwen theorem is incorrect because the
common density of distribution that depends only on the Hamiltonian is inappropriate in the case
of a charge particle gas in a magnetic field. In this case the density of distribution also depends
on the component of the full angular momentum that is parallel to the magnetic field. Then the
considered system should be inhomogeneous. The gas density should decrease with the distance
from the mass center. There are diamagnetic currents that envelop the mass center and generate
the diamagnetic moment.
∗E-mail: lodub@dildub.kiev.ua
c© I.M.Dubrovskii 645
I.M.Dubrovskii
The first purpose of the present study is to show that the quasi-classical conceptions and the
common quantum function of distribution lead to inconsistent results when applied to an electron
gas in a magnetic field. It is known that the wave functions localized near the classical orbits (the
coherent states described in work [6]) are not the eigenfunctions of the Hamiltonian. In the second
section of this paper the eigenstates of one-particle Hamiltonian are under investigation. It is shown
that these states give the probability current lines that envelop the zero of the coordinate system.
The averaged value of the magnetic moment is always non-zero and negative in these states. The
energy spectrum is obtained with zero boundary condition on a large-radius circle. This could not be
done in work [1] because some theorems concerning zeros of the degenerate hypergeometric function
were obtained only in the forties. A complementary formulae about these zeros are obtained in
this work (section 5). In the third section it is shown that an electron gas in a magnetic field is a
non-homogeneous system. Then there is no sense passing to the thermodynamical limit and to the
densities of the extensive quantities. The magnetic moment of a finitesimal system is computed
by the conventional method with the application of the results obtained in the second section.
It is very different from the result of [1] and from the experimental data. Usually the averaged
magnetic moment was computed as the derivative of the averaged energy or the thermodynamical
potential. It is proved that this formula is incorrect in general case. Other contradictions in common
statistical thermodynamics of an electron gas in a magnetic field are also pointed out.
The second purpose of the present study is to derive the quantum function of distribution for
an electron gas in a magnetic field from the basics taking into account the conservation of the
angular momentum. This is done in section four. The averaged magnetic moment of the ideal gas
in a magnetic field without potential fields is computed. The qualitative investigation of the effect
of an external field is made.
2. Electron eigenstates in circle perpendicular to magnetic field
The probability current in a magnetic field is proportional to [∇Φ− (e/c~)A] where Φ(x, y, z)
is the phase of the wave function. A vector potential of the magnetic field A cannot be equal to
gradient of a single-valued function. Consequently, the eigenfunctions cannot be taken so that the
density of the probability current is identically equal to zero. Then every eigenstate must satisfy
the requirement of closure of all lines of the probability current within the considered volume. That
is why the choice of vector potential of a magnetic field is not arbitrary. In work [1] the vector
potential was chosen so that the lines of the probability current are parallel to the axis of the
Cartesian coordinate system that is perpendicular to the magnetic field. The choice of the vector
potential is appropriate for the case at which the considered system has a transport current. In
work [1] eigenstates were obtained by imposing the cyclic boundary condition upon the planes
perpendicular to this axis. This is the same as to obtain the approximate eigenfunctions in a
circular ring with a large radius and a small width. The computation of the average magnetic
moment can be incorrect in this approximation. An axially symmetrical vector potential is only
appropriate in the case of an isolated system. Then the probability current lines of the eigenstates
are circles. Such current cannot lead to alternation of the probability density with time.
The electron Hamiltonian with this vector potential is as follows:
Ĥ =
1
2m
[(
p̂x − eH
2c
y
)2
+
(
p̂y +
eH
2c
x
)2
]
+ u(|r|) = Ĥe + βHL̂z . (1)
Here
Ĥe =
1
2m
(
p̂x
2
+ p̂y
2
)
+
m
2
(
βH
~
)2
|r|2 + u(|r|), ~L̂z = xp̂y − yp̂x . (2)
The operators Ĥe and L̂z commutate with each other and each of them with the Hamiltonian.
For simplicity we shall restrict ourselves to the case of two degrees of freedom on a XY plane
perpendicular to the uniform magnetic field H = (0, 0,H). The charge of an electron is −e, the
mass of one is m, c is the speed of light in a vacuum, β is the Bohr magneton. The potential
646
Quantum statistical mechanics of electron gas in magnetic field
energy u(|r|) is created by the interaction with other electrons and with a neutralizing background
and involves the boundary barrier. In what follows it is assumed that the potential energy of the
Coulomb interaction can be described by the self-consistent field in the electron gas that is in the
equilibrium state. In this section we assume that the self-consistent field is exactly equal to zero,
and the boundary barrier is infinite just like in the conventional theory. These assumptions will be
discussed in section four.
The solutions of the time-independent two-dimensional Schrödinger equation for an electron in
a magnetic field are (see [7]):
ψε,l(r, ϕ) =
A(ε, l)
λ
exp(ilϕ) exp(− r2
2λ2
)
( r
λ
)|l|
Φ
(
− ε
2βH
+
l + |l| + 1
2
, |l| + 1;
r2
λ2
)
. (3)
We use the polar coordinates (r, ϕ), ε is the energy eigenvalue. The quantum number of an angular
momentum l is the eigenvalue of the operator L̂z. It takes any integer value. A normalizing factor is
denoted as A(ε, l)/λ. The square of the magnetic length is described by the formula λ2 = 2c~/|e|H.
The degenerate hypergeometric function (DHF) is written, as the convention, Φ(a, c; z). (In the
monograph [7] the number l has an opposite sign in the first parameter of the DHF, which is a
mistake.)
Let the wave functions be equal to zero beyond the circle with the radius R0. The boundary
condition ψ(R0, ϕ) ≡ 0 can be satisfied only if the first parameter of the DHF labelled as a is
negative a < 0. Then:
− ε
2βH
+
l + |l| + 1
2
= −N − γ(N, |l|), (4)
where N > 0 is integer, and 0 6 γ < 1. (On an infinite plane the functions can be normalized only
when γ = 0).
The eigenvalues of the operator
R̂2 = X̂c
2
+ Ŷc
2
=
λ2
2βH
(Ĥ − 2βHL̂z) (5)
give the spectrum of the square distance of the classical orbit center from the zero of the coordinate
system. The operators X̂c and Ŷc are the Cartesian coordinates of the classical orbit center. The last
equality in (5) is obtained when these operators are expressed as the functions of the coordinates
and the momenta of an electron (see [4,8]). All these operators commutate with Hamiltonian Ĥ.
The operators X̂c and Ŷc do not commutate with each other and [X̂c, Ŷc] = iλ2/2. Hence, if one
of the two coordinates of the orbit center is determined then the other one is undetermined. The
fundamental point of the problem investigated lies in the fact that an electron cannot be localized
in the determined classical orbit. From (5) it follows:
L̂z =
1
λ2
(
ρ̂2 − R̂2
)
, (6)
where ρ̂2 = (λ2/2βH)Ĥ is the operator of the square radius of the classical orbit. The spectrum of
the operator R̂2 differs from the spectrum of the operator Ĥ only by the same coefficient λ2/2βH.
The quantum number of the angular momentum can be positive only when ρ > R, i. e., for orbits
that envelop the zero of the coordinate system. The angular momentum has the same property in
classical mechanics.
In a state ψε,l(r, ϕ) only the ϕ-component of the density of the electrical current is non-zero
and is described by formula:
jϕ = − e~
mr
|ψε,l(r, ϕ)|2
[
l +
r2
λ2
]
. (7)
The current lines are concentric circles. Their center and the symmetry point of the probability
density is zero of coordinate system. This symmetry does not depend on the distance of the classical
orbit center from zero of coordinate system.
647
I.M.Dubrovskii
We introduce the new quantum numbers n and k as was done in works [4] and [8].
N =
n+ k − |n− k|
2
= min(n, k), l = n− k; n = N +
l + |l|
2
, k = N − l − |l|
2
. (8)
The numbers n, k take only integer non-negative values. They are the number of energy and the
number of the orbit center distance from the origin of coordinates.
εn,k = 2βH[n+ 1/2 + γ(n, k)], ρ2
n,k = λ2[n+ 1/2 + γ(n, k)],
R2
n,k = λ2[k + 1/2 + γ(n, k)]. (9)
Here ρn,k is the radius of the orbit of a classical particle that has the same energy. The radial
coordinate of the center of this orbit is Rn,k. The function γ(n, k) is obtained by transformation
γ(N, |l|) → γ(n, k) with the formulae (8). On the infinite plane the number n entirely determines
the eigenvalue of the energy, and the number k does the same for the eigenvalue of the operator R̂2
(see (5)). Functions (3) are eigenfunctions for each of these operators. Consequently each of these
quantum numbers is degenerated. The function γ(n, k) describes the removal of the degeneracy by
the insertion of the boundary barrier. The other pairs of the quantum numbers will be also used
in what follows.
The operator of the orbital magnetic moment is:
m̂z = − e
2c
(x̂̇y − ŷ̇x) = −β
(
L̂z +
r2
λ2
)
. (10)
Using (6) we can see that the formula (10) is in accord with (7). It is necessary to use the formula
(see [9]):
zΦ(a, c; z) = aΦ(a+ 1, c; z) − (2a− c)Φ(a, c; z) + (a− c)Φ(a− 1, c; z) (11)
for computing the averaged value of the magnetic moment (10) in the state ψn,l(r, ϕ). Then we
obtain:
〈mz〉n,l = −εn,l
H
. (12)
This formula is in accord with the time average of the magnetic moment of a classical charged
particle in a magnetic field.
Its result accidentally agrees with the result of the commonly used formula
〈mz〉n,l = −∂εn,l/∂H (13)
only in the absence of the boundary barrier. The formula (10) can be obtained as the result of the
symbolical differentiation
m̂z = −
(
∂Ĥ
∂H
)
symb
, (14)
where the operator (1) is differentiated as a customary function. This differentiation is symbolical
because the real derivation of an operator with respect to a parameter is the derivation of its
matrix. It is generally believed that the average value of the operator (∂Ĥ/∂H)symb obtained by
a symbolical differentiation is identically equal to the derivation of the operator Ĥ eigenvalue.
This inference is not always true. If there is a potential energy u(r), the formula (10) does not
change. A potential energy can be such that it alters only the energy eigenvalue. This alteration
can non-linearly depend on the magnetic field as it is in the case of the boundary barrier. In doing
so the view of the eigenfunctions does not alter. As this takes place the formulae (12) and (13)
should lead to the distinct results. The consequence of this mistake for statistical thermodynamics
will be considered in section three.
The DHF has real positive zeros only when its first parameter a is negative a < 0. The number
of these zeros is equal to the smallest integer that is greater than, or equal to (−a). Let us take
a = −N − γ. The DHF zeros are denoted as ξj(N + γ, c), where j is the zero’s number when the
648
Quantum statistical mechanics of electron gas in magnetic field
zeros are arranged in the increasing order. If j 6 N , the zero ξj(N +γ, c) decreases with increasing
γ in the interval [ζ
(c−1)
N+1,j , ζ
(c−1)
N,j ], where ζ
(c−1)
N,j is the j numbered zero of the associated Laguerre
polynomial L
(c−1)
N (z). It is shown in section five that the largest zero of the function Φ(−N−γ, c; z)
also depends on γ monotonously: ξN+1(N+γ, c) → ∞ when γ → 0, and ξN+1(N+γ, c) → ζ
(c−1)
N+1,N+1
when γ → 1.
We consider the distribution of the eigenvalues in the interval of the energy values [βH(2n +
1), βH(2n + 3)]. The number of states in this interval is equal to the number of such k values
at which the wave functions with fixed n can satisfy the boundary condition. The eigenvalues
are classified by the zero of DHF Φ(−N − γ, |l| + 1; z) which coincides with the boundary value
z0 = R2
0/λ
2. From the beginning we investigate the spectrum when n� z0. The maximum distance
of the orbit center from the origin of coordinates in this case can be close to R0. Therefore, the
maximum k is much greater than n. It is evident from (8) that then N = n, |l| = k − n � N .
In section five it is shown that when
√
k � n, the minimum zero and the maximum zero of the
Laguerre polynomial L
(k−n)
n (z) are described in the first order by the formulae:
ζ
(k−n)
n,1 u k −
√
kn(n− 1)
2
, ζ(k−n)
n,n u k +
√
kn(n− 1)
2
. (15)
All other zeros in this approximation are equal in the order of polynomial: ζ
(k−n)
n,j u k. The Laguerre
polynomial zeros increase when k increases and n is fixed. If at some k value we have z0 > ζ
(k−n)
n+1,n+1,
the boundary condition can be satisfied by the greater zero of the DHF Φ(−n − γ, k − n + 1; z)
with the definite value γ(n, k). It is possible, if k is less than or equal to quantity Dn, to obtain
from the equation:
ζ
(Dn−n)
n+1,n+1 =
R2
0
λ2
. (16)
Using (15) we obtain:
Dn = z0 −
√
n(n+ 1)
2
z0 . (17)
When R0 v 1 cm and H v 104 E then z0 v 8 · 1010. These numbers determine the values n for
which we can make such an approximation.
There are two other intervals of the values k > Dn where the boundary condition can be
satisfied by other zeros of the DHF Φ(−n − γ, k − n + 1; z). In this approximation all Laguerre
polynomial intermediate zeros coincide, i. e. they differ in the quantities that are congruent with
unity. Therefore, we can find the value γ(n, k) for any k from the interval [z0−
√
n(n− 1)z0/2 , z0]
so that intermediate zero of the DHF is equal to z0. If the minimum zero of the DHF satisfies
the boundary condition, the values k should be within the interval [z0 +
√
n(n− 1)z0/2 , z0 +√
n(n+ 1)z0/2]. The number of the states in the energy interval under consideration is equal in
sum of the numbers of the k values over all the intervals of the possible values. This is equal to
D0 = z0, i. e. degeneration of the Landau levels that was obtained in work [1]. Hence the density of
the states obtained by averaging over the interval 2βH in work [1] is the same as in the present work.
The same consideration can be done when n v z0. The greater zero of the Laguerre polynomial
in this case is determined by the formula (see [9]):
ζ
(c−1)
N+1,N+1 u
π2
4
(N + 1 + c/2 − 3/4)2
N + 1 + c/2
u
π2
8
(n+ k). (18)
Then we got from the formula (16): Dn = (8z0/π
2) − n. The number of the states for which the
greater zero satisfies the boundary condition is much smaller than D0 = z0. When n > (8z0/π
2)
these states are absent.
It is shown in section five that
γ w
z2N+c
0
(c+N − 1)!N !
exp (−z0) =
zk+n+1
0
k!n!
exp (−z0), (19)
649
I.M.Dubrovskii
if the boundary condition is satisfied by the greater zero of DHF. Therefore, the shift of the energy
value 2βHγ and its variation with a variation of k cannot be observed for the greater part of the
k values that are less than Dn. This means that the density of the states has very narrow peaks at
the energy values equal in the position of the Landau level. The number of the states in this peak
is Dn instead of D0 = z0 as it was asserted in work [1]. When the k value is close to Dn the γ(n, k)
value is close to unity. But the number of these states is small compared to Dn when n � Dn. It
is also shown in section five that the values of γ(n, k) can be supposed to be distributed over the
interval [0, 1) uniformly if the boundary condition is satisfied by any of the other zeros of DHF.
These states do not make a contribution to the peak of the density of states. All zeros of the DHF
except the maximum zero can be approximately obtained by the formula (see [9]):
ξi =
1
2c− 4a
j2c−1,i , (20)
if −a+c/2 � 1 and |a| > c/2. Here jc−1,i is the zero with number i of the first type Bessel function
with the index c − 1. Then from the boundary condition and from the formulae (3) and (20) we
obtain:
ε =
~2
2m
j2|l|,i
R2
0
+ lβH. (21)
It is interesting to note that the first term is the energy of a free electron with the angular
momentum ~l in a circular potential box, and the second term is the energy of its magnetic
moment in a magnetic field. Hence the energy spectrum for these states is quasi-continuous and
its density of the states is like the one for a free electron.
The main result of this research is the density of the state being broken up into the narrow peaks
and the background that is obtained within the framework of the unified classification of the states.
This successfully allows us to take into account a background contribution in the computation of
the thermodynamical functions. Hereafter we shall see this contribution to be substantial.
It is widely believed that we can use classical mechanics and quasi-classical quantization in
considering the electron gas in a metal in a magnetic field (see, for example, [2–4,10,11]). The
periodical motion of a classical particle can be described as distributions of probability to find a
particle in a given point or as a current density. From the above it is seen that distributions of
probability density and the density of probability current for an electron are very different from
the classical ones. Among other things, a particle in a quasi-classical state also alternates its energy
due to the collision with a boundary. Then the number of states on the energy interval of length
2βH is not equal to πR2
0m/2π~2 = D0/2βH as it should be.
3. Contradictions of common statistical thermodynamics of electron gas in
magnetic field
First of all we compute the magnetic moment by means of a common method using some results
of the second section. A change to the thermodynamical limit and densities of the extensive quan-
tities is correct only for systems that can be named spatial-additive. The determinative property
of these systems is that they can be mentally divided into parts, where each of them should be
similar to a complete system. It was shown above that the electron gas in a magnetic field is not a
spatial-additive system as long as the probability current lines envelope the origin of coordinates.
Then we should compute a thermodynamical potential of the limited system taking into account
the effect of the boundary.
650
Quantum statistical mechanics of electron gas in magnetic field
The computation of the thermodynamical potential of a three-dimensional electron gas in a
magnetic field (see [2]) leads to a formula:
Ω = − 2h
π~
√
2m
∑
n
(
D12∑
k=0
φnk +
D0∑
k=D11
φnk +
D22∑
k=D21
φnk
)
, φnk = φ(xn + 2βHγ(n, k)),
φ(x) =
∫ ∞
x
(ε− x)1/2f0(ε)dε, xn = 2βH
(
n+
1
2
)
,
Dqr = D0 + (−1)q
√
D0
2
n[n+ (−1)r]. (22)
Here h is the height of the circle cylinder that contains the gas, and f0(ε) is the Fermi function.
The integration is done over energy. We made the change of variables with the formula pz(ε, n) =
±
√
2m[ε− βH(2n+ 1). The summation over k is made using the results of the second section. Let
us compute the formula (22) by expanding φnk in terms of γ(n, k) in the first order. We suppose
that γ(n, k) = γ̄ w 1/2 on the both intervals of the k values where the boundary states are located.
In the first sum in (22) we suppose γ(n, k) = 0. Then we obtain:
Ω = − 2h
π~
√
2m
∑
n
[
φ(xn)D0 + 2βHγ̄
(
∂φ
∂x
)
x=xn
√
D0
2
n(n+ 1)
]
. (23)
The first term leads to the formula of thermodynamical potential that is obtained in work [1].
The second term describes the contribution of boundary states. We can take the sum in this term
changing over integration since the dependence upon the magnetic field does not disappear in
so doing. Then we obtain the ratio between the magnetic moment of work [1] and the magnetic
moment created by the conversion of the part of Landau level states to the background. It is in the
region of −(βH/ζF)5/2(R0pF/~) where ζF is the Fermi energy and pF is the Fermi momentum. We
can see that the magnetic moment created by the background can be larger than the one in the
conventional theory by a factor of two or three hundred. Therefore the agreement of the results in
the conventional theory with experimental data is caused by the theoretical mistakes.
Let us note some other contradictions that demonstrate the incorrectness of the conventional
theory. In the second section it was shown that the averaged magnetic moment is negative in any
eigenstate (see formula (12)). The averaged magnetic moment of ideal gas should be equal to
Mz =
∑
n,l
Nn,l〈mz〉n,l , (24)
where
Nn,l = [1 + exp (εn,l − µ)/kBT )]
−1
. (25)
Evidently Mz cannot be zero when it is computed by change of summation into integration. The
same paradox in classical thermodynamics was described in work [5]. This is more obvious in
quantum thermodynamics because there is the formula (24). The mistake of the conventional
theory is in the formula (see [12]) 〈
∂Ĥg
∂H
〉
=
∂Ωg
∂H
, (26)
where Ĥg is the gas Hamiltonian that is the sum of the one-particle Hamiltonians. Firstly
〈(∂Ĥ/∂H)symb〉n,l 6= ∂εn,l/∂H generally. Secondly
∑
n,l Nn,l(∂εn,l/∂H) 6= ∂Ωg/∂H. Let us prove
the last assertion.
Ωg(µ, T ) =
∑
n,l
Ω(µ, T ; εn,l) =
∑
n
Dn(H)Ω(µ, T ;n), (27)
where Dn(H) is the degeneracy of the Landau energy levels in the conventional theory. The
assertion is proved by taking the derivative with respect to H from (27). ∂Ω(µ, T ; εn,l)/∂H =
651
I.M.Dubrovskii
Nn,l(∂εn,l/∂H) but the degeneracy of energy levels also depend on H. Through this dependence
the gas thermodynamical potential does not depend on the magnetic field when it is computed by
change of summation into integration.
Let us compute the averaged magnetic moment by means of a formulae (24) using (25) and (12).
Mz = − 1
H
∑
n,l
Nn,lεn,l = −Eg
H
=
T 2
H
(
∂
∂T
Ωg
T
)
. (28)
This formula can’t be correct because Ωg does not depend on the magnetic field. We should obtain
the magnetic moment that is inversely proportional to the magnetic field. It pinpoints that the
Fermi function of distribution (25) is not applied in the case of an electron gas in a magnetic field.
It is in accord with the conclusion of the work [5].
4. Principles of quantum statistical mechanics of electron gas in magnetic
field
It was shown in work [5] that the density of distribution of the classical statistical ensemble
for a charged particle gas in a magnetic field depends on the component of the system’s angular
momentum that is parallel to the magnetic field. The angular momentum also plays an important
part in the quantum statistical mechanics in a magnetic field. In the second section we showed that
the equilibrium state wave functions should be also the eigenfunctions of the angular momentum.
All one-particle eigenstate wave functions are also the angular momentum eigenfunctions. They
exponentially decrease at great distance from the origin of coordinate. The remoteness of this
asymptotic domain for the one-particle state depends on |l|. It is conditioned by the quadratic
potential term of the operator Ĥe and cannot be abandoned by any potential u(r). Formula (4)
demonstrates that if the one-particle energy eigenvalue εN,l is fixed, the possible positive values of
the angular momentum number can have only values l 6 ε/2βH. Therefore, if the energy and the
angular momentum of the electron gas are fixed, the negative values of the one-particle angular
momentum number should be also limited. Consequently both the degeneracy of the gas energy
value and the phase space domain volume in classical statistical thermodynamics should be finite.
We can make the same conclusions that the magnetic field should be the mandatory parameter
instead of the volume and the electron gas density in the magnetic field is non-homogeneous. It
is common knowledge that a magnetic field confines a two-dimensional plasma. That is another
reason why the electron gas in a magnetic field cannot be considered a spatial-additive system.
In the case under investigation we should derive the function of distribution from the ground
principles. To compute the degeneracy of a state of a two-dimensional electron gas in a magnetic
field without a potential field, the generating function can be taken as follows:
F =
∞∏
n=0
−∞∏
l=n
(
1 + xy2n+1z−l
)
. (29)
Here the unity of energy is taken as βH. This formula is the extension of the corresponding
formula from [13]. The degeneracy of a gas macroscopic state C(N ;E;L) is equal to the coefficient
of the term xNyE/βHz−L when the function F (x, y, z) is represented as expansion in the powers
of arguments. Here N is a number of electrons, E is a fixed gas energy and L is a fixed angular
momentum regarding the mass center measured by a unit ~. This degeneracy can be obtained by
computing the integrals regarding the closed paths using the steepest descent method (see [13]).
Let us consider a composite system made up of an electron gas in a magnetic field and other
subsystem (thermostat), a particle number of which is considerably greater than the one of the
electron gas. Quantities of the electron gas are marked by index 1, quantities of the thermostat
are marked by index 2, and quantities of the all system are not marked by any index. The energy
of the full system is conserved. The electron gas angular momentum is equal to zero. We suppose
652
Quantum statistical mechanics of electron gas in magnetic field
that the magnetic field does not effect the thermostat. Then the probability that the electron gas
has the energy E1 is:
w(E1) =
C2(N2;E − E1)
C(N1, N2;E;L1 = 0)
. (30)
In this equation we suppose that the energies are expressed by integers in the unity ∆ε that is
independent of the magnetic field. We need to compute four integrals regarding closed pathes in the
planes of complex variables x1, x2, y and z for C(N1, N2;E;L1 = 0) evaluation. The corresponding
saddle points ξ1, ξ2, θ and ζ should satisfy the equations:
∞∑
n1=0
−∞∑
l=n1
(−l)ξ1θ2n1+1
1 ζ−l
1 + ξ1θ
2n1+1
1 ζ−l
= 1,
∞∑
n1=0
−∞∑
l=n1
ξ1θ
2n1+1
1 ζ−l
1 + ξ1θ
2n1+1
1 ζ−l
= N1; (31)
βH
∆ε
∞∑
n1=0
−∞∑
l=n1
(2n1 + 1)ξ1θ
2n1+1
1 ζ−l
1 + ξ1θ
2n1+1
1 ζ−l
+
∞∑
n2=0
n2ξ2θ
n2
1 + ξ2θn2
= (E − E1) + E1 ,
∞∑
n2=0
ξ2θ
n2
1 + ξ2θn2
= N2 , θ1 = θβH/∆ε. (32)
Let us use N1 � N2 as it is commonly performed. We neglect the first term in the left member of
the first equation from (32) in the zeroth order. We also neglect the last term in the right member
of this equation. Following those approximations, the equations (32) relate only to the thermostat.
They coincide with the equations for the saddle points of the numerator of (30). We determine θ
from these equations. As in [13] θ = exp(−∆ε/kBT ) . In the first approximation the first equation
(32) is the equation for the electron gas:
∞∑
n1=0
−∞∑
l=n1
(2n1 + 1)ξ1θ
2n1+1
1 ζ−l
1 + ξ1θ
2n1+1
1 ζ−l
=
∆εE1
βH
. (33)
Let us add this equation to the first equation (31) and take into account that E1/βH � 1.
(Hereafter we shall denote the electron gas energy ∆εE1 as E1). If we assume ζ = θ1 in the
obtained equation, then summation indexes n1 and l will get only as the linear combination:
2n1 + 1 − l = n+ k + 1 = t. (34)
Therefore, we can change to the summation over t. The quantum numbers n and l or k (see (8))
completely determine the electron state. Then the quantum number t is t-fold degenerated. We
obtain the equations for finding the saddle points of the electron gas:
∞∑
t=1
t
tξ1θ
t
1
1 + ξ1θt
1
=
E1
βH
,
∞∑
t=1
t
ξ1θ
t
1
1 + ξ1θt
1
= N1 . (35)
These equations coincide with the equation set that defines the function of distribution of the
effective gas of the N1 particles which are described by Hamiltonian Ĥe (see (2)). The saddle point
ξ1 determines the chemical potential µ:
ξ1 = exp
µ
kBT
, βHt = τ, Nt =
t
1 + exp (τ − µ)/kBT
. (36)
The chemical potential µ(T,N1) is determined by the second equation (35). When T → 0 then
µ0 = βH
√
2N1. If the spin degeneracy is taken into account, then µ0 = βH
√
N1.
We can clarify these results in the following manner. The Hamiltonian of electron gas can
be presented as the sum of the Hamiltonian of effective particle gas and the operator that is
proportional to a full angular momentum of this gas like this is done in formula (1) for one
653
I.M.Dubrovskii
electron. These operators are commutative. Therefore, the energy eigenvalues of the electron gas
can be presented as the sum of the corresponding eigenvalues. We consider only the electron gas
states in which the full angular momentum is equal to zero. Therefore, in this case the statistical
thermodynamics should be determined by the Hamiltonian of effective gas. This consequence of
the angular momentum conservation law was not taken into account in the conventional theory.
The average magnetic momentum of the electron gas in the magnetic field is obtained from the
formulae (12), (24) and (35):
Mz = −β
∞∑
t=1
t
tξ1θ
t
1
1 + ξ1θt
1
= −E1
H
. (37)
This formula is applicable when the wave functions have the view like in (3). Derivation of the
function of distribution in the formulae (29)–(36) is founded on the supposition that the Hamil-
tonian of the system can be represented as a sum of the one-particle Hamiltonians. To do this it
is necessary to describe the particle interaction by the self-consistent field. The sum of this field
with the neutralizing background field should not be zero because the electron gas density is non-
homogeneous. This potential field should modify the view of the eigenfunctions. It differs from
those in the formula (3).
For a qualitative analysis we assume that the potential energy is as follows:
u(r) =
u1 > 0, when 0 < r < R1 ,
0, when R1 < r < R0 ,
u0 � u1 , when R0 < r.
(38)
The investigation of operator Ĥe spectrum can be carried out in the same way as in section two.
Density of states also has got very narrow peaks at the effective energy values τ = βHt and the
uniform background. Quantum numbers (t, l) define the assignment of state to the peak or to
the background. This can be illustrated by quantization of classical orbits that are obtained by
Hamiltonian Ĥe. If u(r) ≡ 0, these orbits are ellipses with a center in the origin of coordinates.
The orbit with the energy τ and the angular momentum χ has the axes, the squares of which are
ρ2
± =
2
mω2
(
2τ ±
√
4τ2 − ω2χ2
)
. (39)
By quantization of the adiabatic invariants we obtain χ = ~l, τ = βHt. The axes of the corre-
sponding orbit are obtained by formula:
ρ2
± = λ2
(
t±
√
t2 − l2
)
. (40)
Let us determine the potential by the formula (38). If R1 < ρ−(t, |l|) 6 ρ+(t, |l|) < R0, the
orbit and its energy remain unchanged. These states fall into the peaks of the density of states.
If one or both of these inequalities do not hold, the quantum condition for the radial adiabatic
invariant should be a transcendental equation for τ(t, |l|;H). It can be shown that in this case
τ(t, |l|;H) = βHf(t, |l|) > βHt. These states fall into the background of the density of states.
The states can be depicted by points with integer coordinates at the (t, l)-plane. All states
fall into the sector bounded by the bisectors of the first and fourth quadrants. In figure 1 the
upper half of the symmetric diagram is shown with R2
0/λ
2 = 50, R2
1/λ
2 = 5. The states with
ρ+(t, |l|) < R0 and R1 < ρ−(t, |l|) are positioned on the triangle ABC that is constructed both
by the segment BC of the parabola R2
0/λ
2 = t +
√
t2 − l2 and the segment AB of the parabola
R2
1/λ
2 = t−
√
t2 − l2. The states positioned in this domain have the effective energy values βHt.
Effective energy values of other states are above βHt. The states with τ < βH
√
N1 are completed
at zero temperature. When a magnetic field increases, the parabolas move to the right. Therefore,
the energy of electron gas should nonlinearly vary with the magnetic field. In this model a magnetic
moment is proportional to the energy of electron gas.
654
Quantum statistical mechanics of electron gas in magnetic field
Figure 1. The domains of quantum numbers t and l in which the effective energy is determined
by different formulae.
5. Supplement. Some details about features of zeroes of Laguerre polyno-
mial and degenerate hypergeometric function
Let us consider the Laguerre polynomial Lb
ν(z) = Φ(ν, b+ 1; z) where ν > 2, b > 0 are integers.
Thus we have established that its maximum and minimum zeroes are described by the formulae
ζb
ν,ν u b+
√
bν(ν − 1)
2
, ζb
ν,1 u b−
√
bν(ν − 1)
2
, (41)
in a first approximation when ν2 � b. Other zeroes are approximately equal in b, that is order of
polynomial.
According to the definition of Laguerre polynomial
Lb
ν(z) =
ν∑
k=0
(−z)kν!b!
(ν − k)!k!
1
(b+ k)!
. (42)
Let us introduce [(b+ k)!](−1) u [b!]−1b−k(1 − b−1
∑k
j=0 j). In (42) we denote zb−1 = y and make
some identical transforms:
Lb
ν(z) u
ν∑
k=0
(−y)kν!
(ν − k)!k!
1 − b−1
k∑
j=0
j
=
[
(1 − y)ν − 1
2b
ν∑
k=0
(−y)kν!k(k + 1)
(ν − k)!k!
]
=
[
(1 − y)ν − 1
2b
d
dy
y2 d
dy
ν∑
k=0
(−y)kν!
(ν − k)!k!
]
=
[
(1 − y)ν − 1
2b
d
dy
y2 d
dy
(1 − y)ν
]
= (1 − y)ν−2
[
y2
(
1 − ν
2b
− ν2
2b
)
− 2y
(
1 − ν
2b
)
+ 1
]
. (43)
The obtained approximate expression for the Laguerre polynomial has zeroes that coincide with
(41)in the first order after multiplying by b. Thus, this proves the statement.
The distribution of states in the energy value interval 2βH is governed by the distributions of
the values γ(n, k) between zero and unity in each interval of the possible values k. Let us introduce
the DHF at the point z0 as:
Φ(−N − γ, c; z0) = PN (γ, z0) − γAN (γ, z0) − γ(1 − γ)TN (γ, z0), (44)
655
I.M.Dubrovskii
PN (γ, z0) =
N∑
i=0
(−N − γ)iz
i
0
(c)ii!
, AN (γ, z0) =
(−1)N (1 + γ)Nz
N+1
0
(c)N+1(N + 1)!
,
TN (γ, z0) =
(−1)N (1 + γ)Nz
N+2
0
(c)N+2(N + 2)!
∞∑
i=0
(2 − γ)iz
i
0
(c+N + 2)i(N + 3)i
, (45)
where
(b)i = b(b+ 1) · · · (b+ i− 1) = Γ(b+ i)/Γ(b), (b)0 = 1 (46)
is the Pochhammer symbol. It is evident from the DHF expansion that AN and TN have the same
sign. Then DHF can have zero at z0 only if PN has the same sign. The values of PN , AN and TN are
slightly varied when γ takes on the values between zero and unity. Let us neglect this dependence.
Then we put (44) equal to zero and solve this like equation for γ:
γ =
1
2
+
AN
2TN
− 1
2
√
1 +
2AN
TN
− 4PN
TN
+
(
AN
TN
)2
. (47)
Here the sign of the radical is chosen in order to get 0 < γ < 1.
We consider the problem of the maximum zero of DHF Φ(−N − γ, c; z). This zero cannot be
described by the formula (20) since it should vanish continuously when γ → 0. This is possible
only when ξN+1 → ∞ in so doing. When γ = 1 then ξN+1 = ζ
(c−1)
N+1,N+1. Therefore, when each
z0 > ζ
(c−1)
N+1,N+1 it is possible to take γ < 1 such that z0 is the greatest zero of the function
Φ(−N − γ, c; z). When z0 � 1 we can change the expression of TN (z0) as a power series (45) into
the first term of its asymptotic expansion. To derive this expansion we consider the identity:
Φ(−N − γ, c; z) ≡
N∑
k=0
(−N − γ)k
(c)k
zk
k!
+
(−N − γ)N+1
(c)N+1
zN+1
(N + 1)!
+
(−N − γ)N+2
(c)N+2
×
∫ z
0
dz1 · · ·
∫ zi−1
0
dzi · · ·
∫ zN+1
0
dzN+2Φ(2 − γ, c+N + 2; zN+2). (48)
This identity is easily proved by integrating the power series of function Φ(2− γ, c+ N+ 2; zN+ 2).
We correlate (48) with (44) and see that
TN (z0) =
(−1)NN !
(c)N+2
∫ z
0
dz1 · · ·
∫ zi−1
0
dzi · · ·
∫ zN+1
0
dzN+2Φ(2, c+N + 2; zN+2). (49)
We assume γ = 0. This can be done in computating the TN (z0). We express Φ(2, c+N + 2; z) by
integral representation (see ([9])):
Φ(2, c+N + 2; z) =
Γ(c+N + 2)
Γ(2)Γ(c+N)
∫ 1
0
exp (zu)u(1 − u)c+N−1du, (50)
and obtain:
TN (z0) =
(−1)NN !Γ(c)
Γ(c+N)
∫ 1
0
[
exp (zu) −
N+1∑
k=o
(zu)k
k!
]
u−N−1(1 − u)c+N−1du. (51)
This expression is convenient for TN numerical computation. In formula (49)the antiderivative of
each integration at inferior limit is equal to zero. Therefore the function Φ(2− γ, c+N + 2; z) can
be changed by its asymptotic approximation in this formula. This approximation is
Φ(2, c+N + 2; z) w Γ(c+N + 2)
exp (z)
zc+N
(
1 − c+N
z
)
. (52)
656
Quantum statistical mechanics of electron gas in magnetic field
For the integration in (49) we use the formula:
∫ x exp(t)
tn
dt =
exp(x)
xn
[
L∑
l=0
(n)l
xl
+ O
(
1
xL+1
)]
. (53)
Let us find the coefficient of the term exp(z0)/z
c+N+s
0 in the asymptotic expansion. To obtain this
factor we can pick the terms of expansion (53) that have l = mj in the integration over each zj in
the formula (49) so that
∑N+2
j=1 mj = σ. Here σ = s for the first item of (52) and σ = s− 1 for the
second one. The consequence of (46) is (b)n(b+n)m = (b)n+m. Therefore, the result of integration
does not depend on the set of numbers mj . It is proportional to (c+N)s/(c+N)s−σ. The number
of these items is equal to the number of the disposition of σ indistinguishable objects at N + 2
boxes. The net result is
TN (z0) = (−1)NN !(c− 1)!
exp(z0)
zc+N
0
[
S∑
s=0
(c+N)s
zs
0
(
s+N
s
)
+ O
(
1
xS+1
)]
. (54)
If we restrict ourselves to the first term of this sum we obtain:
TN w (−1)NN !(c− 1)!
exp (z0)
zc+N
0
. (55)
The greatest zero of the polynomial PN is smaller than z0. That is why this polynomial can be
changed by its last term. The formula (19) has been obtained. If the value of k is in other intervals
of the possible values, the equation (47) does not have a great parameter. It is evident from the
fact that zero of Φ(−N −γ, c; z) in this case is located between zeroes of the Laguerre polynomials
that are the function Φ(−N − γ, c; z) with γ = 0 and γ = 1. The quantity of TN does not take
part in the computation of these zeroes.
References
1. Landau L.D., Zs. phys., 1930, 64, 629.
2. Lifshits I.M., Azbel M.J., Kaganov M.I. Electron Theory for Metals. (in Russian) Nauka, Moscow, 1971.
3. Shoenberg D. Magnetic oscillations in metals. Cambridge University Press, 1984.
4. Dubrovskii I.M. Theory of Electron Phenomena in Deformed Cristal. (in Russian) RIO IMF, Kyiv,
1999.
5. Dubrovskii I. M., Condensed Matter Physics, 2006, 9, No 1(45).
6. Feldman A., Kahn A. H., Phys. Rev. B, 1970, 1, 4584.
7. Landau L.D., Lifshits E.M. Quantum mechanics. Nonrelativistic theory. Pergamon Press, New York,
1980.
8. Dubrovskii I. M., Low Temperature Physics, 2002, 28, 845.
9. Higher Transcendental Functions. (Based, in part, on notes left by Harry Bateman and compiled by
the staff of the Bateman manuscript project, director Arthur Erdelyi) Vol. 1. Mc Graw-Hill book
company, INC, New York, Toronto, London, 1953.
10. Ginzburg Vl.L., Maksimov P.I., Fiz. Nizk. Temp., 1977, 3, 1285 (in Russian).
11. Gutnikov A.I., Feldman E.P., JETF, 1972, 63, 1054 (in Russian).
12. Landau L.D., Lifshits E.M. Statistical Physics, 3rd rev.,part I. Pergamon Press, New York, 1980
13. Fowler R.H. and Guggenheim E.A. Statistical Thermodynamics. Cambridge, 1939.
657
I.M.Dubrovskii
Квантова статистична мехнiка електронного газу в
магнiтному полi
I.М.Дубровський
Iнститут металлофiзики, бульв. Вернадського 36, Київ–142, 03680, Україна
Отримано 24 травня 2006 р.
Розглянуто власнi стани електрону в магнiтному полi. Одержано густину електричного струму i сере-
днiй магнiтний момент. Дослiджено густину станiв для двовимiрного електрону в колi обмеженому
нескiнченним потенцiальним бар’єром. З використанням цих результатiв показано, що загально-
прийнята квантова статистична механiка електронного газу в магнiтному полi приводить до помил-
кових результатiв. Магнiтний момент електронного газу може бути обчислений як сумма середнiх
моментiв заповнених станiв. Таке обчислення приводить до результату, що вiдрiзняється вiд того,
який одержується як похiдна термодинамiчного потенцiалу по магнiтному полю. Наведено й iншi
протирiччя у загальноприйнятiй статистичнiй термодинамiцi електронного газу в магнiтному полi.
Зроблено висновок, що цi протирiччя виникають внаслiдок використання неправильного статисти-
чного оператора. Нова квантова функцiя розподiлу виведена з основних принципiв, беручи до уваги
закон збереження кутового моменту. Цi результати узгоджуються з теорiєю, яка була виведена у
рамках класичної статистичної термодинамiки у попереднiй роботi.
Ключовi слова: електроннi стани, магнiтне поле, кутовий момент, середнiй магнiтний момент,
квантова функцiя розподiлу, квантова статистична термодинамiка
PACS: 05.30.Ch, 75.20.-g
658
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| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1607-324X |
| language | English |
| last_indexed | 2025-12-07T18:32:03Z |
| publishDate | 2006 |
| publisher | Інститут фізики конденсованих систем НАН України |
| record_format | dspace |
| spelling | Dubrovskii, I.M. 2017-06-14T08:05:15Z 2017-06-14T08:05:15Z 2006 Quantum statistical mechanics of electron gas in magnetic field / I.M. Dubrovskii // Condensed Matter Physics. — 2006. — Т. 9, № 4(48). — С. 645–658. — Бібліогр.: 13 назв. — англ. 1607-324X PACS: 05.30.Ch, 75.20.-g DOI:10.5488/CMP.9.4.645 https://nasplib.isofts.kiev.ua/handle/123456789/121376 Electron eigenstates in a magnetic field are considered. Density of the electrical current and an averaged
 magnetic moment are obtained. Density of states is investigated for two-dimensional electron in a circle that
 is bounded by the infinite potential barrier. The present study shows that the common quantum statistical
 mechanics of electron gas in a magnetic field leads to incorrect results. The magnetic moment of electron
 gas can be computed as the sum of averaged moments of the occupied states. The computations lead to the
 results that differ from the ones obtained as the derivative of the thermodynamical potential with respect to
 the magnetic field. Other contradictions in common statistical thermodynamics of electron gas in a magnetic
 field are pointed out. The conclusion is done that these contradictions arise from using the incorrect statistical
 operator. A new quantum function of distribution is derived from the basic principles, taking into account the
 law of conservation of an angular momentum. These results are in accord with the theory that has been
 obtained within the framework of classical statistical thermodynamics in the previous work. Розглянуто власнi стани електрону в магнiтному полi. Одержано густину електричного струму i середнiй магнiтний момент. Дослiджено густину станiв для двовимiрного електрону в колi обмеженому
 нескiнченним потенцiальним бар’єром. З використанням цих результатiв показано, що загальноприйнята квантова статистична механiка електронного газу в магнiтному полi приводить до помилкових результатiв. Магнiтний момент електронного газу може бути обчислений як сумма середнiх
 моментiв заповнених станiв. Таке обчислення приводить до результату, що вiдрiзняється вiд того,
 який одержується як похiдна термодинамiчного потенцiалу по магнiтному полю. Наведено й iншi
 протирiччя у загальноприйнятiй статистичнiй термодинамiцi електронного газу в магнiтному полi.
 Зроблено висновок, що цi протирiччя виникають внаслiдок використання неправильного статистичного оператора. Нова квантова функцiя розподiлу виведена з основних принципiв, беручи до уваги
 закон збереження кутового моменту. Цi результати узгоджуються з теорiєю, яка була виведена у
 рамках класичної статистичної термодинамiки у попереднiй роботi en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Quantum statistical mechanics of electron gas in magnetic field Квантова статистична мехнiка електронного газу в магнiтному полi Article published earlier |
| spellingShingle | Quantum statistical mechanics of electron gas in magnetic field Dubrovskii, I.M. |
| title | Quantum statistical mechanics of electron gas in magnetic field |
| title_alt | Квантова статистична мехнiка електронного газу в магнiтному полi |
| title_full | Quantum statistical mechanics of electron gas in magnetic field |
| title_fullStr | Quantum statistical mechanics of electron gas in magnetic field |
| title_full_unstemmed | Quantum statistical mechanics of electron gas in magnetic field |
| title_short | Quantum statistical mechanics of electron gas in magnetic field |
| title_sort | quantum statistical mechanics of electron gas in magnetic field |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/121376 |
| work_keys_str_mv | AT dubrovskiiim quantumstatisticalmechanicsofelectrongasinmagneticfield AT dubrovskiiim kvantovastatističnamehnikaelektronnogogazuvmagnitnomupoli |