Need help finding fermion energies and probabilities

[L52892]

<Moved from a technical forum, therefore no template>

For two non-interacting fermions confined to a 1d box of length L. Construct the antisymmetric wave functions (Slater determinant) and compare ground state energies of two systems, one in the singlet state and the other in the triplet state. For both states, evaluate the probability of find the two particles at the same position. Here's what I have so far...

ψ_n(x) = √2/L*sin(πnx/L)
E_n = (h²π²))/2m * (n/L)²

E_grnd = E₁ + E₁ = 2E₁ = (h²/m)(π/L)²
ψ_grnd = ψ₁(x₁)ψ₁(x₂)*1/√2(σ_1↑σ_2↓-σ_1↓σ_2↑)

E_1st = E₁ + E₂ = (5h²/m)(π/L)²
ψ_1st^singlet = 1/√2 * ψ₁(x₁)ψ₂(x₂)+ψ₂(x₁)ψ₁(x₂) * 1/√2(σ_1↑σ_2↓-σ_1↓σ_2↑)
ψ_1st^triplet = 1/√2 * ψ₁(x₁)ψ₂(x₂)-ψ₂(x₁)ψ₁(x₂)

• σ_1↑σ_2↑
• 1/√2(σ_1↑σ_2↓+σ_1↓σ_2↑)
• σ_1↓σ_2↓

Any help would be appreciated.

 

[blue_leaf77]

For the triplet state, the spatial wavefunction should be anit-symmetric.

 

[L52892]

That was supposed to be a minus. Thanks.

I need help

• determining how to find the ground state energies of each of the systems (I think you integrate each of the wavefunctions...possibly?)
• and also calculating the probability of finding the two particles at the same position

 

[blue_leaf77]

Each system means each of the singlet and triplet series independently. You have found the answer for the singlet series, now you are left with the triplet. What is the lowest energy level for triplet state? No need for integrating the wavefunctions.
The probability of finding the two particles at the same place is $|\psi(x_1,x_1)|^2$ but you also need to take the spin into account.