Identical spin 1/2 particles in a box.

[phyky]

Homework Statement

two identical particles of spin 1/2 that confined in a cubical box of side L. find the energy and wave function (non-interacting between particles)

Homework Equations

for a cubic boxby and reducing the Schrödinger equation:
ψ(x,y,z)=√(8/L³ ) sin((n_x πx)/L)sin((n_y πy)/L)sin((n_z πz)/L)
E= (ħ² π²)/(2mL² ) (n_x²+n_y²+n_z² )

The Attempt at a Solution

E=ε₁+ε₂=(ħ² π²)/(2mL² ) [(n_x1²+n_y1²+n_z1² )+(n_x2²+n_y2²+n_z2² ) ]
n how about the wave function? should i find it anti-symmetry wave funtion？
ψ_a(r1,r2;S1,S2)=ψ_s(r1,r2)χ_a(S1,S2)
=ψ_a(r1,r2)χ_s(S1,S2)
then the wave function is combination of this?

 

[vela]

ψ_a(r1,r2;S1,S2)=ψ_s(r1,r2)χ_a(S1,S2)
=ψ_a(r1,r2)χ_s(S1,S2)
then the wave function is combination of this?

I think you have the basic idea. You need to find the combinations that result in an antisymmetric wave function. If you meant to say that ψ_s(r1,r2)χ_a(S1,S2)=ψ_a(r1,r2)χ_s(S1,S2), however, that's obviously wrong.