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Physical Review A

Permutation Symmetries of Wave Functions with Some Applications to Density-Matrix Theory

F. David Peat

Published Dec 1, 1971• 4 citations
Publisher

Abstract

The formation of wave functions of a particular symmetry out of particle-subgroup functions is discussed as an eigenvector problem to an interchange matrix. Some properties of this interchange matrix are derived and found to be insufficient to fully characterize the matrix and a connection with the $N$-representability problem is made. A number of special cases are discussed in detail and an indication of a systematic numerical approximation is made.

DOI
10.1103/physreva.4.2206

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Physical Review•Jul 26, 1965•208 citations
The formation of wave functions of a particular symmetry out of particle-subgroup functions is discussed as an eigenvector problem to an interchange matrix. Some properties of this interchange matrix are derived and found to be insufficient to fully characterize the matrix and a connection with the NNN-representability problem is made. A number of special cases are discussed in detail and an indication of a systematic numerical approximation is made.
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 usual restriction on the permutational symmetry of density matrices is removed. An Nth-order density matrix of general permutational symmetry is reduced and the symmetry of the resulting pth-order density matrix is found. This gives rise to necessary conditions for the N-representability of spin-free density matrices. We indicate the importance of these results for symmetry under the groups GL(n), U(n), and SU(n). We generalize the question of N-representability to the tensor algebra, where contraction of tensors plays the role of reduction of density matrices. Symmetry properties of the tensors are studied and density matrices are considered as special cases.
The NNN-representability problem for the first- and second-order density matrices is considered when NNN is an odd integer. Attention is restricted to those density matrices derivable from functions whose natural spin orbitals have certain paired properties. Neither pairing of spatial orbitals nor special spin properties are required in general, but they are discussed as special cases. New sufficient conditions for NNN-representability of the first-order density matrix are given. The second-order density matrix is studied in detail. Special cases in which the NNN-representability problem for the second-order density matrix can be solved are considered.
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.