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Journal of Physics A Mathematical and General

Classical and semiclassical theory for the exchange symmetry of identical particles

J G Leopold, Ian Colin Percival, D Richards

Published Mar 1, 1982• 17 citations
Publisher

Abstract

The primitive semiclassical (EBK) theory of bound states is applied to finite non-relativistic systems with a discrete symmetry, particularly systems of identical particles. The symmetry properties of invariant tori are obtained, and illustrated for simple models with a pair of identical particles, each with one freedom. Examples without and with interaction between the particles, and with vibration and rotation are all considered. Particular care has to be taken with interacting particles that rotate around a ring.

DOI
10.1088/0305-4470/15/3/019

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The primitive semiclassical (EBK) theory of bound states is applied to finite non-relativistic systems with a discrete symmetry, particularly systems of identical particles. The symmetry properties of invariant tori are obtained, and illustrated for simple models with a pair of identical particles, each with one freedom. Examples without and with interaction between the particles, and with vibration and rotation are all considered. Particular care has to be taken with interacting particles that rotate around a ring.
Despite the seminal connection between classical multiply periodic motion and Heisenberg matrix mechanics and the massive amount of work done on the associated problem of semiclassical Einstein-Brillouin-Keller (EBK) quantization of bound states, we show that there are, nevertheless, a number of previously unexploited aspects of this relationship that bear on the quantum-classical correspondence. In particular, we emphasize a quantum variational principle that implies the classical variational principle for invariant tori. We also expose the more indirect connection between commutation relations and quantization of action variables. In the special case of a one-dimensional system a different and succinct algebraic derivation of the WKB quantization rule for bound states is given. With the help of several standard models with one or two degrees of freedom, we then illustrate how the methods of Heisenberg matrix mechanics described in this paper may be used to obtain quantum solutions with a modest increase in effort compared to semiclassical calculations. We also describe and apply a method for obtaining leading quantum corrections to EBK results. Finally, we suggest several modified applications of EBK quantization. \textcopyright{} 1996 The American Physical Society.
Semiclassical periodic orbit theory is used in many branches of physics. However, most applications of the theory have been to systems that involve only single-particle dynamics. In this paper, we develop a semiclassical formalism to describe the density-of-states for two noninteracting particles. This includes accounting properly for particle exchange symmetry. As specific examples, we study two identical particles in a disk and in a cardioid. In each case, we demonstrate that the semiclassical formalism correctly reproduces the quantum densities.
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.
Systems of identical particles possessing non-local interactions are capable of exhibiting extra-classical properties beyond the characteristic quantum length scales. This letter derives the dynamics of such systems in the non-relativistic and degenerate limit, showing the effect of exchange symmetry and correlations on structure both in and out of equilibrium. Such descriptions may be crucial to understanding systems ranging from nuclei to dark matter. Correlated condensed solutions are also compared and contrasted to the more established approaches of standard mean field theory and classical field theory.
We discuss the rotational excitations of highly quantum many-body systems (for example, polyatomic molecules or microdroplets of helium). For a general system, the states Fί, where and ί is a rotationally invariant ground or vibrational state, are shown to be eigenfunctions of L 2 and Lz , with eigenvalues L(L+1)ħ 2 and Lħ (for even L). These wavefunctions preserve the translational invariance and the permutation and inversion symmetries of the N-particle state ί. For harmonic pair interactions, the f = 1 wavefunctions are shown to be exact eigenstates of the N-body hamiltonian. For large N, the states Fί(f=1) represent surface oscillations of the type first proposed by Bohr. An inequality for the rotational excitation energy is obtained variationally; it depends on two, three, and four-particle correlations. Other translationally invariant angular momentum eigenfunctions are also discussed.