Ground State Wave Function for Identical Spin 1/2 Particles in a Potential Well

[lion8172]

Homework Statement

Find the wave function of the ground energy state for a system of two non-interacting, identical spin 1/2 particles in a potential well extending from x=0 to x=a. Don't forget to consider spin.

Homework Equations

The Attempt at a Solution

Since the particles are fermions, they must occupy a state that is antisymmetric with respect to particle exchange. Their wave function will be of the form \psi (r) \chi(s), where \chi is a spinor. Either the spatial part or the spin part of the wave function must be antisymmetric (and the other part must be symmetric). If the spin state is symmetric, then the particles will occupy a state in the spin triplet. Otherwise, they will occupy the singlet. At this point I am confused, however. How do we know which component of the wave function must be anti-symmetric (spatial or spin)? Moreover, if the particles are in the spin triplet, how do we know which of the three possible spin states corresponds to the lowest energy?

 

[Avodyne]

Since we want the ground state, we want to put each particle into the single-particle state of lowest energy. So their single-particle spatial wave functions are the same. What does that tell you about spatial symmetry?