In my lecture on indistinguishable particles, my lecturer is trying to illustrate to me the notion of particles being indistinguishable when considering that when we swap two particles in a box the superposition state looks absolutely the same. She then goes on to note:
If we consider the phase of the wavefunction, then it turns out that there are two possible outcomes for the overall two-particle wavefunction, which either maintains or reverses its sign every time a swap between particles is made
I don't see how this happens. A global phase shouldn't change the wavefunction appreciably. Can someone explain why what she is saying is the case?