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Progress of Theoretical Physics Supplement

Symmetry of the Ground-State Wave Function of Many Particle System

Yosuke Nagaoka, Tunemaru Usui

Published Jan 1, 1968• 3 citations
Publisher

Abstract

It is proved that, if the Hamiltonian of a system of spinless particles is symmetric with respect to interchange of the particles, the wave function with the lowest energy attainable without any restriction on its symmetry property, is symmetric with respect to interchange of the coordinates. It is also discussed what kind of conditions must be imposed upon the Hamiltonian for the theorem to be valid.

DOI
10.1143/ptps.e68.392

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It is proved that, if the Hamiltonian of a system of spinless particles is symmetric with respect to interchange of the particles, the wave function with the lowest energy attainable without any restriction on its symmetry property, is symmetric with respect to interchange of the coordinates. It is also discussed what kind of conditions must be imposed upon the Hamiltonian for the theorem to be valid.
It is demonstrated that in a system of NNN similar particles, the permissible wave functions are either symmetrical or antisymmetrical with respect to interchanges of the particles, if the following principles are assumed to be valid: (1) The interchange of two like particles produces no change in any measurable property of the system. In particular, if $\ensuremath{\psi}$ is to be a permissible wave function, $\ensuremath{\psi}\overline{\ensuremath{\psi}}$ must be unaltered by such an interchange. (2) All the wave functions obtained from a given permissible wave function by permuting the similar particles are also permissible wave functions for the same eigenvalue.
The symmetrization postulate (SP) states that wave functions are either completely symmetric or completely antisymmetric under permutations of identical particles. It is shown by one-dimensional counter-examples that SP is not demanded by the usual physical interpretation of the mathematical formalism of wave mechanics unless one makes use of further physical properties of real systems; the error in a standard proof of SP which ignores these properties is pointed out. It is then proved that SP is true for those systems of spinless particles which have the following properties: (a) probability densities are permutation-invariant, (b) allowable wave functions are continuous with continuous gradient, (c) the 3n3n3n-dimensional configuration space is connected, (d) the Hamiltonian is symmetric, and (e) the nodes of allowed wave functions have certain properties. The counterexamples show that SP is not a necessary property of those systems which do not have property (c). The proof is extended to particles with internal degrees of freedom (including spin) by noting that any observable commutes with every permutation and making use of a particular observable acting only on internal variables. Extraneous mathematical assumptions, such as that of the existence of a "complete" set of commuting observables, criticized by Messiah and Greenberg, are not employed. Some comments are made on the conventional nature of the concept of identity for similar particles; the equivalence between the usual formulation in which different species of similar particles are treated as distinct, and that in which they are regarded as identical particles in different internal states, is emphasized.
The Hamiltonian for a system of n particles that are identical, that is, particles that can be substituted for each other with no physical change, must be completely symmetric under any interchange of its arguments. Let Ψ(1, 2, …, n) be any solution of the Schrödinger equation, depending on the coordinates (spatial and spin) of the n identical particles 1 to n, and let P be any permutation of the n numbers 1 to n. Then PΨ(1, 2, …, n) ≡ Ψ(P1, P2, …, Pn) is a function which depends on the coordinates of particle Ψ in the same manner as the original function Ψ depended on the coordinates of particle i. Then using the symmetry of H, we can show that the operator P commutes with the Hamiltonian ( 2-1 ) H ( P Ψ ) = P ( H Ψ ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429493645/749bab20-4a79-4f5a-8c66-d9e9a36a87b9/content/eq89.tif"/>
Consider a pair of particles that may be either two fermions or two bosons. e Schrödinger equation for the system is written in terms of the Hamiltonian operator for the system as H( , ) ( , ) ( , ),1 2 1 2 1 2c c= E (12.1) where 1 represents the coordinates of particle 1 and 2 the coordinates of particle 2. e Hamiltonian (1, 2) must be symmetric under particle interchange because of the indistinguishability of the particles, giving H H( , ) ( , ).1 2 2 1= (12.2) e pair wave function χ(1, 2) may be either symmetric or antisymmetric. is can be seen by introducing the permutation operator P12, which permutes the coordinates of the particles.
A general connection between nodelessness and symmetry of a function is pointed out. It is proved that a real nodeless energy eigenfunction with energy E has a nonzero part which also is an eigenfunction with energy E and which under coordinate transformations has the full symmetry of the Hamiltonian. This result can be applied to many systems of physical interest for which the ground-state energy eigenfunction is known from the nature of the Hamiltonian to be nodeless. Simple counterexamples are given however, to show that not all Hamiltonians have a nodeless ground-state energy eigenfunction.
On the basis of reasonable physical assumptions, it is proved that for systems of identical spinless particles, the permissible nonrelativistic Schr\"odinger wave functions are either all symmetric or all antisymmetric. The proof is a generalization of that given by Girardeau. Although connectivity properties of the configuration space play an important role in the proof, it is not true, as was previously believed, that a connected space is always necessary. In particular, for one-dimensional systems, connectivity of the configuration space is not necessary for the proof to hold.