# Need help finding fermion energies and probabilities

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1. Mar 12, 2017

### 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...

ψn(x) = √2/L*sin(πnx/L)
En = (h2π2))/2m * (n/L)2

Egrnd = E1 + E1 = 2E1 = (h2/m)(π/L)2
ψgrnd = ψ1(x11(x2)*1/√2(σ1↑σ2↓1↓σ2↑)

E1st = E1 + E2 = (5h2/m)(π/L)2
ψ1stsinglet = 1/√2 * ψ1(x12(x2)+ψ2(x11(x2) * 1/√2(σ1↑σ2↓1↓σ2↑)
ψ1sttriplet = 1/√2 * ψ1(x12(x2)-ψ2(x11(x2)
• σ1↑σ2↑
• 1/√2(σ1↑σ2↓1↓σ2↑)
• σ1↓σ2↓
Any help would be appreciated.

Last edited: Mar 12, 2017
2. Mar 12, 2017

### blue_leaf77

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

3. Mar 12, 2017

### 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

4. Mar 12, 2017

### 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.