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Journal of physics. A, Proceedings of the Physical Society. General

A note on ring integrals and the Fermi function

G J Papadopoulos, A V Jones

Published Jul 1, 1970• 1 citation
Publisher

Abstract

A method which proceeds in a natural fashion via the symmetry properties of the wave functions of a many-body fermion (boson) system of independent identical particles, is employed for the derivation of the Fermi (Bose) function. The analysis makes use of the grand partition function expressed in terms of ring integrals. Spin considerations are included in the case of electrons in a constant uniform magnetic field.

DOI
10.1088/0305-4470/3/4/004

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A method which proceeds in a natural fashion via the symmetry properties of the wave functions of a many-body fermion (boson) system of independent identical particles, is employed for the derivation of the Fermi (Bose) function. The analysis makes use of the grand partition function expressed in terms of ring integrals. Spin considerations are included in the case of electrons in a constant uniform magnetic field.
The methods that have been described in previous chapters can be used to study general quantum Bose systems. They are based in a direct way on the introduction of a generating function of symmetric wave functions of bosons. In the case of fermion systems, however, one faces the problem that fermion wave functions, or fermion correlation functions (or Green functions) are antisymmetric with respect to the exchange of a fermion pair. Thus, the construction of generating functions requires the introduction of an antisymmetric or Grassmann algebra of ‘classical functions’. It is then possible to generalize to Grassmann algebras the notions of derivatives and integrals, yielding quite parallel formalisms for bosons and fermions, in particular, to define a path integral for fermion systems, analogous to the holomorphic path integral for bosons. This chapter discusses differentiation and integration in Grassmann algebras, Gaussian integrals and perturbative expansion, partition function, and quantum Fermi gas.
In classical physics, the question whether there are subsets of identical particles is not particularly interesting. After all, whether the particles are identical or not, we can always track each of them at any time. As we shall see, in quantum mechanics identical particles are indistinguishable because we cannot track trajectories, and this fact induces some important symmetry properties on the total wave function describing the system of particles under consideration. Generally speaking, a symmetry is an effect of an invariance, that is, the system does not change if one effects certain transformations (see Ch. 8). In this case, the system is not modified under exchange of two identical particles. Moreover, such a symmetry yields a most interesting connection between spin and statistics. After a short introduction to the significance of statistics in quantum mechanics (in Sec. 7.1), we define the permutation operator and discuss the symmetry properties of the wave function of N identical particles (Sec. 7.2). In Sec. 7.3 we show that quantum particles, differently from classical particles (that obey the Maxwell–Boltzmann statistics), may be subject to two different statistics, namely the Fermi–Dirac statistics for fermions and the Bose–Einstein statistics for bosons. In Sec. 7.4 we treat the so-called exchange interaction and, finally, in Sec. 7.5 we consider some applications of the two statistics. Statistics and quantum mechanics Identical particles are those particles whose Hamiltonian is symmetric under exchange of one particle with the other.
By expanding the grand partition function, we obtain the expression for partition function of two dimensional free particles which is seen to be of the same form as that of one dimensional harmonic oscillators. For such systems it is shown that the difference in statistics does not appear in thermal behavior. When the energy levels of the particles are modified as ε n 1 , n 2 = Δ + b ( n 1 2 + n 2 2 ), Δ >0, the Bose condensation occurs and, if we set Δ =2 b , the condensation temperature qualitatively agrees with that of Osborne. The speciffic heat of free Bosons in a narrow box is calculated as the function of temperature, its width d being taken 7.1 Å and 14.2 Å. The results show that the maximum of the specific heat occurs at higher temperatures than that of the case d =∞, and its height becomes lower. The weak excitation of a Fermi system can be treated as excitons which obey Bose statistics. And we can show, by the method analogous to that of Ward and Wilks and Dingle, that the sound wave in a Fermi gas should be considered as the second sound, or the propagation of fluctuation in exciton distribution. Its propagation velocity is shown to become p0/3mp_{0}/\sqrt{3}mp0​/3​m, where p 0 is the momentum at the Fermi surface.
This chapter develops the basic equations that will be used to analyse the Fermi-Dirac and Bose-Einstein gases. The representation of many-particle states in terms of products of single-particle states is presented. The reasons for using the quantum grand canonical ensemble are given, and a general expression for the grand canonical partition function is derived. The essential equations for fermions, bosons, and distinguishable particles are developed, and the basic strategy for using them to solve problems is given.
Abstract Two concepts, of fundamental importance to quantum mechanics, will be discussed in this chapter. The first is that of symmetry. Even though the concept of symmetry is familiar in a wide range of natural sciences, the way symmetry constrains the consequences of quantum mechanics is rather subtle and, at the same time, far-reaching. The second is the symmetry property of the states under the exchange of two identical particles. Identical particles with a half-integer spin (fermions) obey the Fermi–Dirac statistics: the wave functions are totally antisymmetric under the exchange of the particles. Identical particles with an integer spin (bosons) are instead subject to the Bose–Einstein statistics. Their wave functions are totally symmetric.