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<ノート>多体問題とグリーン関数との関係の研究--グリーン関数と多体問題(15)量子統計力学(7)

邦夫 橋爪

Published Dec 1, 2010• 0 citations
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Abstract

[Synopsis]In this paper, next subject is discussed. § 40. Magnetic properties of an imperfect electron gas. We discuss the effects of interparticle interaction affecting the magnetic properties of an electron gas. Two electrons prefer to be in an antisymmetric orbital wave function to minimize the repulsive energy. But in fermions, total wave function of two electrons with spin-1/2 must be antisymmetric. Therefore an antysimmetric orbital wave function requires a symmetric spin wave function. Thus the repulsive interaction would align the spins of the electrons. And the effect of a repulsive interaction enhances the paramagnetism. We here demonstrate this effect by considering an imperfect electron gas at very low temperature.

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〈ノート〉多体問題とグリーン関数との関係の研究--グリーン関数と多体問題(18) 量子統計力学(10)

橋爪 邦夫

[Synopsis] 0The alignment of the electron spin with the external magnetic field give rise to paramagnetism. Whereas, the orbital motions of the electrons give rise to diamagnetism. In this paper we consider the idealized problem of a free spinless electron gas in an external magnetic field H=H^^z . A free electron moves in a circular orbital quantized normal to the uniform magnetic field with the cycrotron frequency ω=<—eμ_0H>/m. The energy eigenvalues in two-dimesional electron system E_j =hω_0(j+1/2)=βH(2j+1) (j=0,1,2,・・・) are the Landau energy levels. β is a Bohr magneton. g=<—eμ_0H>/hL^2 is the degeneracy of a Landau level . In the low-temperature limit of electron in Landau levels, the electrons have a tendency to occupy the lowest available levels. As the magnetic field is decrease, each Landau level can accommodate fewer electrons because the degeneracy g is decreased. Consequently, some electrons will be forced to jump up to a higher level. This cause the oscillation of the low-temperature magnetic susceptibility X_m as a inverse function of the magnetic field 1/H. The magnetization M also oscillates as a function of the magnetic field. We call these phenomena the de Haas-Van Alphen effect.

近畿大学工学部研究報告•Dec 1, 2011•0 citations

〈ノート〉多体問題とグリーン関数との関係の研究--グリーン関数と多体問題(22)量子統計力学(14)

邦夫 橋爪

[Synopsis] The energy Hamiltonian, given by Heisenberg in 1928, to discuss the physical properties and the atomics tructure of ferromagnetic substance is as follows: H=—2Σ__ J_ S_i・S_j.-gβΣ__iS_i・H. A spinv ariable S_i is associated with each lattice site. In the Ising model of J_ =J_ =0 and J_ ≠0, proposed by E. Ising in Zeit. Phys. 31 253(1925), the above Hamiltonian is simplified as follows : H=-2Σ__ J_ S_ S_ -gβHΣ__iS_ ・S_ =±1/2. The system is an array of N fixed points calld lattice sites that form an n-dimensional periodic lattice. The energy of the system in the configuration specified by {S_i} is as follows : E{S_i}=εΣ__ S_iS_j-βHΣ__iS. S_i=+1 denotes that the ith site has spin ip and S_i=-1 detotes that the ith site has spin down. The partition function is Z(H, T)=Σ__ Σ__ ・・・Σ__ e^<- / >=e^<<1/k_BT>N(1/2γε-βH)>Σ^^N_ e^<-2・1/ (εγ-βH)N_+>Σ^1__ g(N_+,N_++)e^<4・1/ εN_++>. Next the Ising model can be made to simulate the system of a lattice gas and a binary alloy of other than a ferromagnetic substance by a change of names.

•Dec 1, 2011•0 citations

The Explanation of Oppositely Aligned Electron Pairs in Molecules

Shutao Wang

The symmetric properties of wave functions of bosons and fermons were proposed by the principles of quantum mechanics.The calculation of the expectation value of the square of the separation distance between two particles confirms that bosons with symmetric wave functions under exchange tend to be attractive,while fermions with antisymmetric wave functions under exchange have some repulsive properties.Based on this point of view,the electron wave function is product of position wave function and spin wave function,the position wave functions of oppositely aligned electrons must be symmetric and attraction appears,resulting in electron pairs.

Journal of Dezhou University•Jan 1, 2011•0 citations

Bosons and Fermions

William J. Mullin

Bosons have symmetric wave function under particle exchange, while fermions have antisymmetric wave functions. This leads to differences in behavior of the two kinds of particles even in the gaseous state. Strong repulsive forces between atoms at close range are due to the Pauli principle that forbids two electrons to be in the same energy or position state. A repulsive Pauli pressure operates similarly in white dwarf stars. Bosons undergo Bose–Einstein condensation because of Bose statistics. The wave function symmetry of particles with spin can affect the magnetic states and the behavior of transport properties, for example, viscosity and thermal conductivity. The symmetry properties of bosons at a beam splitter are demonstrated in the Hong–Ou–Mandel effect or in a Michelson interferometer via the Hanbury Brown-Twiss effect. Researchers see boson bunching and fermion antibunching in experiments considered.

Oxford University Press eBooks•Mar 16, 2017•0 citations

Short-ranged correlations and the ferromagnetic electron gas

A. K. Rajagopal, John C. Kimball, Mahadev Banerjee

A discussion of the various properties of pair correlation functions of a magnetic electron gas is presented. The cusp conditions obeyed by the many-electron wave functions for a pair of electrons with parallel and antiparallel spins in close proximity are expressed as derivative conditions on the corresponding parallel-and antiparallel-spin pair correlation functions. In particular, it is shown that in a fully ferromagnetic gas the parallel spin correlations determine the correlation energy whereas in the paramagnetic case, this function is of less importance in comparison to the antiparallel-spin correlation function. In the high-momentum limit, it is found that the momentum distribution of electrons is dominated by antiparallel-spin correlations in the partially magnetic case whereas in the fully ferromagnetic case, it is dominated by parallel-spin correlations but is smaller by a factor of ${q}^{\ensuremath{-}2}$. A calculation of the parallel-spin correlation function for the ferromagnetic gas in perturbation theory is presented. A model for the exchange correlation energy of the system which takes into account the exact conditions given here is proposed. Corresponding results for the two-dimensional electron gas are also stated.

Physical review. B, Condensed matter•Sep 1, 1978•117 citations

Thermodynamic properties of a weakly interacting Fermi gas in a strong magnetic field

Men Fu-Dian, Wang Bing-Fu, He Xiao-Gang +1 more

Based on the pseudopotential method and the local-density approximation, the thermodynamic properties of a weakly interacting Fermi gas in a strong magnetic filed are studied, the integrated analytical expressions of thermodynamic quantities of the system are derived, and the effects of magnetic field as well as interparticle interactions on the thermodynamic properties of the system are analyzed. It is shown that at both high and low temperatures, magnetic field may adjust the effects of interacting. At low temperatures, magnetic field can lower the chemical potential, total energy and heat capacity of the system compared with the situation of Fermi gas in the absence of the magnetic field. The repulsive interactions may increase the chemical potential, but reduce the total energy and heat capacity of the system compared with the situation of non-interacting Fermi gas. At high temperatures, magnetic field as well as repulsive interactions can reduce the total energy and increase heat capacity of the system, moreover, strong magnetic field may change the effects of interaction on the total energy and the heat capacity of the system.

Acta Physica Sinica•Jan 1, 2011•2 citations
[Synopsis]In this paper, next subject is discussed. § 40. Magnetic properties of an imperfect electron gas. We discuss the effects of interparticle interaction affecting the magnetic properties of an electron gas. Two electrons prefer to be in an antisymmetric orbital wave function to minimize the repulsive energy. But in fermions, total wave function of two electrons with spin-1/2 must be antisymmetric. Therefore an antysimmetric orbital wave function requires a symmetric spin wave function. Thus the repulsive interaction would align the spins of the electrons. And the effect of a repulsive interaction enhances the paramagnetism. We here demonstrate this effect by considering an imperfect electron gas at very low temperature.
[Synopsis] 0The alignment of the electron spin with the external magnetic field give rise to paramagnetism. Whereas, the orbital motions of the electrons give rise to diamagnetism. In this paper we consider the idealized problem of a free spinless electron gas in an external magnetic field H=H^^z . A free electron moves in a circular orbital quantized normal to the uniform magnetic field with the cycrotron frequency ω=<—eμ_0H>/m. The energy eigenvalues in two-dimesional electron system E_j =hω_0(j+1/2)=βH(2j+1) (j=0,1,2,・・・) are the Landau energy levels. β is a Bohr magneton. g=<—eμ_0H>/hL^2 is the degeneracy of a Landau level . In the low-temperature limit of electron in Landau levels, the electrons have a tendency to occupy the lowest available levels. As the magnetic field is decrease, each Landau level can accommodate fewer electrons because the degeneracy g is decreased. Consequently, some electrons will be forced to jump up to a higher level. This cause the oscillation of the low-temperature magnetic susceptibility X_m as a inverse function of the magnetic field 1/H. The magnetization M also oscillates as a function of the magnetic field. We call these phenomena the de Haas-Van Alphen effect.
[Synopsis] The energy Hamiltonian, given by Heisenberg in 1928, to discuss the physical properties and the atomics tructure of ferromagnetic substance is as follows: H=—2Σ__ J_ S_i・S_j.-gβΣ__iS_i・H. A spinv ariable S_i is associated with each lattice site. In the Ising model of J_ =J_ =0 and J_ ≠0, proposed by E. Ising in Zeit. Phys. 31 253(1925), the above Hamiltonian is simplified as follows : H=-2Σ__ J_ S_ S_ -gβHΣ__iS_ ・S_ =±1/2. The system is an array of N fixed points calld lattice sites that form an n-dimensional periodic lattice. The energy of the system in the configuration specified by {S_i} is as follows : E{S_i}=εΣ__ S_iS_j-βHΣ__iS. S_i=+1 denotes that the ith site has spin ip and S_i=-1 detotes that the ith site has spin down. The partition function is Z(H, T)=Σ__ Σ__ ・・・Σ__ e^<- / >=e^<<1/k_BT>N(1/2γε-βH)>Σ^^N_ e^<-2・1/ (εγ-βH)N_+>Σ^1__ g(N_+,N_++)e^<4・1/ εN_++>. Next the Ising model can be made to simulate the system of a lattice gas and a binary alloy of other than a ferromagnetic substance by a change of names.
The symmetric properties of wave functions of bosons and fermons were proposed by the principles of quantum mechanics.The calculation of the expectation value of the square of the separation distance between two particles confirms that bosons with symmetric wave functions under exchange tend to be attractive,while fermions with antisymmetric wave functions under exchange have some repulsive properties.Based on this point of view,the electron wave function is product of position wave function and spin wave function,the position wave functions of oppositely aligned electrons must be symmetric and attraction appears,resulting in electron pairs.
Bosons have symmetric wave function under particle exchange, while fermions have antisymmetric wave functions. This leads to differences in behavior of the two kinds of particles even in the gaseous state. Strong repulsive forces between atoms at close range are due to the Pauli principle that forbids two electrons to be in the same energy or position state. A repulsive Pauli pressure operates similarly in white dwarf stars. Bosons undergo Bose–Einstein condensation because of Bose statistics. The wave function symmetry of particles with spin can affect the magnetic states and the behavior of transport properties, for example, viscosity and thermal conductivity. The symmetry properties of bosons at a beam splitter are demonstrated in the Hong–Ou–Mandel effect or in a Michelson interferometer via the Hanbury Brown-Twiss effect. Researchers see boson bunching and fermion antibunching in experiments considered.
A discussion of the various properties of pair correlation functions of a magnetic electron gas is presented. The cusp conditions obeyed by the many-electron wave functions for a pair of electrons with parallel and antiparallel spins in close proximity are expressed as derivative conditions on the corresponding parallel-and antiparallel-spin pair correlation functions. In particular, it is shown that in a fully ferromagnetic gas the parallel spin correlations determine the correlation energy whereas in the paramagnetic case, this function is of less importance in comparison to the antiparallel-spin correlation function. In the high-momentum limit, it is found that the momentum distribution of electrons is dominated by antiparallel-spin correlations in the partially magnetic case whereas in the fully ferromagnetic case, it is dominated by parallel-spin correlations but is smaller by a factor of ${q}^{\ensuremath{-}2}$. A calculation of the parallel-spin correlation function for the ferromagnetic gas in perturbation theory is presented. A model for the exchange correlation energy of the system which takes into account the exact conditions given here is proposed. Corresponding results for the two-dimensional electron gas are also stated.
Based on the pseudopotential method and the local-density approximation, the thermodynamic properties of a weakly interacting Fermi gas in a strong magnetic filed are studied, the integrated analytical expressions of thermodynamic quantities of the system are derived, and the effects of magnetic field as well as interparticle interactions on the thermodynamic properties of the system are analyzed. It is shown that at both high and low temperatures, magnetic field may adjust the effects of interacting. At low temperatures, magnetic field can lower the chemical potential, total energy and heat capacity of the system compared with the situation of Fermi gas in the absence of the magnetic field. The repulsive interactions may increase the chemical potential, but reduce the total energy and heat capacity of the system compared with the situation of non-interacting Fermi gas. At high temperatures, magnetic field as well as repulsive interactions can reduce the total energy and increase heat capacity of the system, moreover, strong magnetic field may change the effects of interaction on the total energy and the heat capacity of the system.