Coupled Harmonic Oscillators: Indistinguishable Particles

Abstract

The probability density ${\rho _D}({x_1},{x_2},t) = |\psi ({x_1},{x_2},t){|^2}$ used in the last chapter described the joint probability of observing particle 1 at position x ₁ and particle 2 at x ₂. There is no difficulty with this notion as long as particle 1 can be unambiguously attributed to position x ₁ and particle 2 to x ₂. To so attribute them, however, presupposes that particles 1 and 2 have different identities, that they can be distinguished by properties other than being at different locations or having different momenta. They must have different intrinsic properties, for instance, different masses or different electric charges. A system consisting of an electron and a proton is one in which the two particles have different intrinsic properties. A system consisting of two electrons is not. For such a system it is impossible in principle to distinguish the two particles if they are close to each other.