Physics 216: Topics in many-body theory, spring 2002
Problem set 1: assigned 2/6/03, due 2/20/03, based on lectures I-VI
1. A warmup on variational methods: (similar to Ashcroft and Mermin problems 17.1 and
17.2) Show that Hartree-Fock equations like those given in lecture I are obtained by variational
minimization over N-electron Slater determinants of the Hamiltonian
He=
N
X
i=1 −¯
h2
2m∇2
i+Vb(ri)!+1
2X
i6=j
e2
|ri−rj|.(1)
It may help to show first that the Hartree equations come from minimization over product wave-
functions (not antisymmetrized).
2. Show that, in BCS theory, it does not modify the fluctuations in the number operator while
preserving |uk|2and |vk|2by introducing phases into ukand vk. Can you find a different way to
minimize the fluctuations?
Hint: define
|ΨBCS (θ)i=Y
k
(uk+vkeiθb†
k)|0i(2)
and integrate
Z2π
0
e−iNθ |ΨBCS (θ)idθ (3)
where here Nis the desired number of Cooper pairs. How does the energy of this state compare
to the original state with ukand vkreal? How does the number compare?
3. How much is the kinetic energy increased in the BCS state compared to the Fermi sea, for
the simple solvable model
Vkk0=−V < 0 if |k−µ|and |k0−µ|< ωc
0 otherwise .(4)
How much is the interaction energy reduced?
4. Obtain the compressibility formula in Fermi liquid theory
∂P
∂ρ =pF2
3mm∗(1 + F0) = pF2
3m2
1 + F0
1 + F1
.(5)
You will probably want to start from the relation
∂µ
∂N =−V2
N2
∂P
∂V (6)
so that the compressibility is just (N/m)∂µ
∂N . The next step is to write (you should justify this)
δµ =Zf(pF,p0)δn0δτ0+∂ F
∂pF
δpF.(7)
Feel free to consult chapter 2 of Landau and Lifshitz volume 9 if you get stuck.
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