Distinguishable equivalent particles with symmetrized wave functions

A. Bijl

Yu Shi

We consider entanglement in a system with a fixed number of identical particles. Since any operation should be symmetrized over all the identical particles and there is the precondition that the spatial wave functions overlap, the meaning of identical-particle entanglement is fundamentally different from that of distinguishable particles. The identical-particle counterpart of the Schmidt basis is shown to be the single-particle basis in which the one-particle reduced density matrix is diagonal. But it does not play a special role in the issue of entanglement, which depends on the single-particle basis chosen. The nonfactorization due to (anti)symmetrization is naturally excluded by using the (anti)symmetrized basis or, equivalently, the particle number representation. The natural degrees of freedom in quantifying the identical-particle entanglement in a chosen single-particle basis are occupation numbers of different single-particle basis states. The entanglement between effectively distinguishable spins is shown to be a special case of the occupation-number entanglement.

F. J. Bloore, S. J. Swarbrick

Yoshio Ohnuki, S. Kamefuchi

Don S. Lemons

Symmetries and wave functions An important difference between the classical and quantum perspectives is their different criteria of distinguishability. Identical particles are classically distinguishable when separated in phase space. On the other hand, identical particles are always quantum mechanically indistinguishable for the purpose of counting distinct microstates. But these concepts and these distinctions do not tell the whole story of how we count the microstates and determine the multiplicity of a quantized system. There are actually two different ways of counting the accessible microstates of a quantized system of identical, and so indistinguishable, particles. While these two ways were discovered in the years 1924–1926 independently of Erwin Schrödinger’s (1887–1961) invention of wave mechanics in 1926, their most convincing explanation is in terms of particle wave functions. The following two paragraphs may be helpful to those familiar with the basic features of wave mechanics. A system of identical particles has, as one might expect, a probability density that is symmetric under particle exchange, that is, the probability density is invariant under the exchange of two identical particles. But here wave mechanics surprises the classical physicist. A system wave function may either keep the same sign or change signs under particle exchange. In particular, a system wave function may be either symmetric or antisymmetric under particle exchange.