Symmetric, antisymmetric and parity

Sacroiliac

Let me see if I can make it clearer.

Problem 5.5 In David Griffiths “Introduction to Quantum Mechanics” says:

Imagine two non interacting particles, each of mass m, in the infinite square well. If one is in the state psi_n and the other in state psi_m orthogonal to psi_n, calculate < (x₁ - x₂) ² >, assuming that (a) they are distinguishable particles, (b) they are identical bosons, (c) they are identical fermions.

(a) a² [1/6 – (1/2pi²)(1/n² + 1/m²)]

(b) The answer to (a) - (128*a²*m²n²) / (pi⁴(m² - n²)⁴)

But this last term is present only when m,n have opposite parity.

(c) The answer to (a) plus the term added in (b) with the same stipulation as in (b)

What does this mean? It seams to be saying that all three particles would have the same separation unless their states have opposite parity. Is this correct? Bosons and Fermions would have the same separation unless their states have odd parities? I never heard of this before, how does this work?