Physics 880.05: Assignment #5

1. Effective action in one dimension.
   1. This requires you mainly to follow through the notes and understand the intermediate steps. You should be able to compare your answer to results for the one-D energy density derived in earlier problem sets. Do you expect the g factor in the leading-order term to be the same when you compare to past results (think about Hartree vs. Hartree-Fock)?
   2. Everything should go through like in the notes as long as you remember that the integration measures are different. When evaluating a quantity such as Pi₀, you will usually need to consider different ranges of q₀ and q, which define different integration regions (it helps to sketch them).
   3. If you have a non-interacting Fermi gas, it has non-zero pressure because non-zero momentum states are filled. What about a non-interacting Bose gas? How are the kinetic energy and momentum related?
2. Number fluctuations in the BCS ground state.
   • This is probably easiest to do in terms of the quasiparticle operators we introduced, but it is also possible directly in terms of the a's and a dagger's. You know how to write the number operator. You might want to separate out in the double sum in N-hat² the terms with the same k in both sums from those with different k's. Remember that 1 - v_k² = u _k².
   • Is |BCS> an eigenstate of N-hat? Is |F> an eigenstate of N-hat? What particle-number fluctuation do you expect in the normal ground state? Does it work out if you use the normal state values for u_k and v_k?
   • The true ground state ²⁰²Pb is an eigenstate of N-hat, with eigenvalue 202. We can also imagine other eigenstates of N-hat, such as ²⁰⁴Pb, ²⁰⁰Pb, and so on. If the state you calculate is not an eigenstate of N-hat, can you express it in terms of eigenstates?
3. Another BCS ground state problem.
   1. What does it mean that two states are the same? How do you prove orthogonality? This problem is quite straightforward if you rewrite the a's in terms of the alpha and beta operators introduced on pages 270 to 272 of the notes (note: there is a typo in the definition of beta; the second "a" should have a dagger). Both the alpha and beta acting on |BCS> give zero while alpha dagger and beta dagger acting to the left on < BCS| give zero. These properties are enough to show both parts of this problem.
   2. Use the definition of K-hat written in terms of the alpha's and beta's and remember that you are comparing energies, so you only need the difference between K-hat on one of these states and K-hat on |BCS> (remember that U is a c-number).
4. Feynman-Hellmann proof.
   1. The key is that the phi(x) wave functions are normalized (i.e., there integral is equal to one). So write out this normalization condition and take the functional derivative of it with respect to sigma_c(x).
   2. If you change sigma_c(x), does phi(x') change? If you got the first part correct, you'll have answered why these terms don't contribute!