Physics 880.05: Assignment #5
Here are some hints, suggestions, and comments on the assignment.
- Effective action in one dimension.
- 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)?
- 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).
- 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
- 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
the terms with the same k in both sums from those with different
k's. Remember that 1 - vk2 =
Is |BCS> an eigenstate of N-hat? Is |F> an eigenstate
What particle-number fluctuation do you expect in the normal
ground state? Does it work out if you use the normal state values
for uk and vk?
- 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?
- Another BCS ground state problem.
- 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
- 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).
- Feynman-Hellmann proof.
- 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
- 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
Physics 880.05: Assignment #5 hints.
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