Pset 1 - iSites

) Use the result of part a) to prove the f-sum rule for the dynamic form factor S(q, w):
Problem 5
∫ ∞
o
ωS(⃗q, ω) = Nq2
2m
a) In order to deal with quasiparticles with spins quantized along general directions, it
is necessary to use a 2 x 2 density matrix of “occupation numbers” δn ⃗pαβ for each ⃗p. What
are the δn ⃗pαβ in terms of expectations of quasiparticle creation and annihilation operators
C (†)
pσ ? The free energy can be written up to second order in δn ⃗pαβ in terms of an interaction
function f ⃗pαβ⃗p′α′β′ Use spin rotational invariance to express this in terms of f S,A
⃗p⃗p′
b) The group of Jun Ye at JILA recently obtained a quantum degenerate gas of 87 Sr atoms
with nuclear spin I = 9 . Other groups are working on getting other quantum degenerate
2
alkaline-earth fermionic atoms with different values of I. Such systems provide examples of
fermionic many-body states with higher spins. How many Landau parameters do we have
in a Fermi liquid with spin I? Explain.
c ∗ ) Discuss properties of spin- 9 2
Fermi liquid.
d ∗ ) Explain why this system may have higher symmetry than SU(2). How many collective
modes do you expect?