Identical particles, spin, fermions, etc.

[Void123]

Homework Statement

I got two particles, spin-(1/2), in a box of finite length and I must compute the energy and wavefunctions for the three lowest states. The particles are in a singlet spin state.

Homework Equations

$$E = \epsilon_{1} + \epsilon_{2} +...$$

The Attempt at a Solution

I got the wavefunctions down.

Just want to clarify some uncertainty here, if the ground state is just going to be for n = 1, 2, then would the first excited state be n = 1, 3, and second excited state n = 2, 3?

I am also assuming they're fermions.

 

[Void123]

Also, I have assumed that since the total spin is in the singlet state, therefore the wavefunction will be the product of the symmetric spatial part and the antisymmetric spin part. Is this correct?