Download Phys 518 Homework Set I, Jan

Phys 518 Homework Set I, Jan., 18, 2006 (Due Feb., 1, 2006)
Prob. 1
1) The BCS wavefunction usually involves states with different numbers of particles.
Given the coherence factor (u,v) for each k, calculate explicitly the average
number of particles and fluctuations for a given chemical potential.
2) For Fermi gases, there is a definite relation between the number density and the
chemical potential (at zero temperature that is called the Fermi energy). When
interactions are present, the chemical potential is renormalized. Compare the
chemical potential of a superconducting state (in the weakly interacting) for a
given number density to that of a Fermi gas. Estimate the difference between
these two.
Prob. 2
1) Calculate the condensation energy of a BCS state explicitly.
2) Calculate explicitly the energy cost to remove a particle from the BCS ground state.
Compare it with the energy cost to remove a particle from a Fermi gas.
Prob. 3
1) Show explicitly that quasi-particles we studied are fermions satisfying the usually
commutation relations.
2) Show that annihilation operators do annihilate BCS states.
3) Use (u,v) obtained, find out the spatial structure of the two-body wave functions
BCS used to construct the many-body wavefunction.