Why two-particle wavefunctions are separable and their corresponding particles are indistiguishable at the same time?

Question

If the wavefunction $\psi(r_1,r_2)$ doesn't represents an entangled state, it is separable: $$\psi(r_1,r_2)=\psi_a(r_1)\psi_b(r_2)$$ In this treatment, we ignore the interaction between two particles so that the initial wavefunction can be written as a product. However, since they are indistinguishable particles, their wavefunction must overlap more or less, otherwise we need infinite potential wells.

My question is: what is the relationship between the interaction of particles and the overlap of their wavefunctions? I know they are not the same thing but I feel confused about their relation. And how both of them affect the indistinguishablility of particles?

Answer 1

The way you wrote it, they are distinguishable (unless $a=b$ of course). For the particles to be indistinguishable their wavefunction must be of the form $$ \psi(r_1,r_2) = \frac{1}{\sqrt{2}} [ \psi_a(r_1)\psi_b(r_2) \pm \psi_a(r_2)\psi_b(r_1) ]$$ where the sign depends on the fermionic/bosonic nature of the particles.

If the particles are described by a separable wave-function, they must be distinguishable: you can measure one without affecting the other.

A separable wavefunction like the one you wrote can describe a couple of non-interacting, distinguishable particles. If indistinguishable they are always in some sense "interacting" (or more correctly, correlated), meaning that you can't affect one without affecting the other. A separable wavefunction is sometimes used as a first approximation (for example in the Hartree method), but strictly speaking, again, one should use a wavefunction that takes into account the indistinguishability (like a Slater determinant for electrons in the Hartree-Fock method).

Two particles are said to be interacting if the Hamiltonian contains coupling terms, for example a potential that depends on the relative positions of the particles. When this is the case, the stationary states are not separable and a correlation must always occur.

Also related is this discussion.