# Fermions in a well

I have two identical fermions in an infinite potential well. They are non-interacting. How should I show that the first excited state is four-fold degenerate? Is the wavefunction just the superposition of the wavefunction of each fermion?

• is it a one-dimensional potential well? – hft Feb 24 '15 at 22:01
• do they have spin? – hft Feb 24 '15 at 22:02
• Yes. Spin is 1/2. – Artemisia Feb 24 '15 at 22:10
• it is a one-dimensional or two-dimensional well? – hft Feb 24 '15 at 22:11
• I want to understand it for a 1D well first :) – Artemisia Feb 24 '15 at 22:14

For example, in the case of the ground state, the spatial part of the wavefunction is symmetric $$\sin(x_1\pi/L)\sin(x_2\pi/L)$$ and the spin part of the wavefunction is anti-symmetric $$|\uparrow\downarrow>-|\downarrow\uparrow>\;.$$