The permutation symmetry of Newtonian stationary-state functions – such as the total momentum and the Hamiltonian – is crucial for correct state counting in classical statistical mechanics, but plays virtually no role for classical many-body systems. In contrast, it yields novel effects for quantum many-body systems of equal/identical particles. In order to avoid the misunderstandings Jaynes (1992) has rightly stressed, basic notions, such as equal and identical particles, (in)distinguishability and exchangeability (the really relevant notion) are discussed first within classical mechanics. Then, they are applied to the weight (limiting) functions. The permutation symmetry of the wave functions follow largely from that of the weight functions. The different effects for bosons and fermions are mentioned and illustrated. For the generalization towards anyons and the fractional quantum Hall effect, some ideas for further exploration are proposed. Entanglement is treated as a consequence of the conservation laws, in particular, for the angular momentum.The permutation symmetry of Newtonian stationary-state functions – such as the total momentum and the Hamiltonian – is crucial for correct state counting in classical statistical mechanics, but plays virtually no role for classical many-body systems. In contrast, it yields novel effects for quantum many-body systems of equal/identical particles. In order to avoid the misunderstandings Jaynes (1992) has rightly stressed, basic notions, such as equal and identical particles, (in)distinguishability and exchangeability (the really relevant notion) are discussed first within classical mechanics. Then, they are applied to the weight (limiting) functions. The permutation symmetry of the wave functions follow largely from that of the weight functions. The different effects for bosons and fermions are mentioned and illustrated. For the generalization towards anyons and the fractional quantum Hall effect, some ideas for further exploration are proposed. Entanglement is treated as a consequence of the conservation laws, in particular, for the angular momentum.

BENTHAM SCIENCE PUBLISHERS eBooksFeb 12, 20190 citations

Permutation Symmetry of Many-Particle Wave Functions

Marvin D. Girardeau

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 $3n$ -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 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 $3n$-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.

Physical ReviewJul 26, 1965208 citations

3n

-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.