Problem Set 4
Phys 7280
Due: Apr 2
1 Plasma oscilations
Consider an electron gas with density nand chemical potential µwhich interacts via
Coulomb potential V(r) = e2/r where ris the distance between the electrons. Its Fourier
transform is given by V(q) = 4πe2/q2. An important parameter for this gas is the ratio of
its typical Coulomb and kinetic energies, denoted r0=e2n1
3/µ. In what follows, we assume
that r01 so that the interactions can be considered weak and perturbation theory in
powers of interaction can be used. As discussed, the interaction potential is modified by
the RPA diagrams as shown on Fig. 1. The basic building block of these diagrams is the
polarization operator which plays the role of the “self-energy” correction to the Coulomb
interactions.
The sum of these diagrams represents an approximation to the vertex function, as
is emphasized by the four external lines on each of the diagrams shown on Fig. 1. In
turn, that function can be thought of as describing the electron-hole excitations in the gas
(plasmons), and the poles in that function give their dispersion relation.
1. Use the explicit form of the polarization operator Π(E , p) found in Problem 3 of the
Problem Set 3 to sum all the diagrams shown on Fig. 1.
2. By analyzing the behavior of the polarization operator at ppF,Eµ, find the
spectrum of the plasmons (p) at low momentum. Hint: Assume that EppF/m.
3. There exists such pmax that the plasmon spectrum cannot be extended for larger p >
pmax, at least if Eis real. Find this pmax .
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the electron gas.
