Problem 1 [15 points]
In lecture we computed the two particle density matrix in real space, ⟨r1,r2|ρ2|r1,r2⟩, for spinless particles obeying either Bose-Einstein or Fermi-Dirac statistics. Here you will calculate it for spin 1/2 fermions, where each particle i has an intrinsic spin that can take on one of two possible values, si = ↑ or ↓.
For two free fermions in a box, we can write the spin part of the energy eigenstates as |s1⟩|s2⟩, which specify the spin of each particle. However we can also write the spin part in terms of eigenstates of the total angular momentum, |ℓ, mz⟩. These are then:
| spin singlet: | |0, 0⟩ = ( |↑⟩|↓⟩ − |↓⟩|↑⟩ )/sqrt(2) | |
spin triplet: | |1, 1⟩ = |↑⟩|↑⟩
|1, -1⟩ = |↓⟩|↓⟩
|1, 0⟩ = ( |↑⟩|↓⟩ + |↓⟩|↑⟩ )/sqrt(2) |
For this problem you want to compute the two particle density matrix using the above basis for the spin states,
⟨r1, r2, ℓ, mz|ρ2|r1, r2, ℓ, mz⟩
To compute the above matrix element you will need the matrix elements,
⟨r1, r2, ℓ, mz|k1, k2 , ℓ′, m′z⟩
where k1 and k2 give the wavevectors of the real space part of the energy eigenstates.
a) Compute the above matrix elements ⟨r1, r2, ℓ, mz|k1, k2 , ℓ′, m′z⟩ for all four of the spin basis states |ℓ, mz⟩. You will have to take into account that the total wavefunction must be antisymmetric under the exchange of the two particles.
b) Using your results from (a), compute the density matrix ⟨r1, r2, ℓ, mz|ρ2|r1, r2, ℓ, mz⟩ for all four of the spin basis states |ℓ, mz⟩.
You should find that for any of the spin triplet states, the density matrix has the same dependence on |r1 − r2| as we found previously for two spinless fermions, i.e. there is an effective repulsion at small separations.
However for the spin singlet state, the density matrix has the same dependence on |r1 − r2| as we found previously for two spinless bosons, i.e. there is an effective attraction at small separations. Give a simple physical argument as to why this is a reasonable result.
This is the origin of Hund's rule of atomic physics. As one fills up a partially full energy subshell of an atom, the electrons first go in with parallel spins since this gives an effective repulsion between them that lowers the electrostatic repulsive energy.
Hint: Once you understand what the problem is asking for, there really is not much of a calculation that needs to be done. All the parts you need can be taken from the calculation done in lecture.
Bonus: (no points, just for fun) Consider the two particle density matrix where we use the spin states |s1, s2⟩ for the basis, instead of the total spin states |ℓ, mz⟩. Compute ⟨r1, r2, ↑, ↑ |ρ2|r1, r2, ↑, ↑⟩ and ⟨r1, r2, ↑, ↓ |ρ2|r1, r2, ↑, ↓⟩. Can you explain your results?