1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Need help finding fermion energies and probabilities

  1. Mar 12, 2017 #1
    <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. jcsd
  3. Mar 12, 2017 #2

    blue_leaf77

    User Avatar
    Science Advisor
    Homework Helper

    For the triplet state, the spatial wavefunction should be anit-symmetric.
     
  4. Mar 12, 2017 #3
    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
     
  5. Mar 12, 2017 #4

    blue_leaf77

    User Avatar
    Science Advisor
    Homework Helper

    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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted