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\markboth{{\bf 561 F 2005 Homework 2}}{{\bf 561 F 2005 Homework
2}}
\begin{document}

{\bf 561 Fall 2005 Homework 2}

Passed out September 8, 2005;  Due September 22, 2005


\medskip
{\bf 1.  Problem  5.2 of Phillips.}  Note that the specific heat
depends on on the density of states in a range $\approx T$ around
the Fermi energy. The Hartree-Fock approximation leads to a
non-analytic form for the specific heat that is correction to
well-known form for non-interacting particles.  It is tricky to
work out a detailed answer to this problem; you are asked to give
the correct reasoning and an estimate for the temperature range
below which the correction is $> \approx 10\%$.

\medskip
{\bf 2.  Problem  8.3 of Phillips.}  In this problem assume that
the Coulomb interaction is the usual 3d $1/r$ interaction, but the
particles are confine to move in 2 dimensions. This is the actual
problem for plasmons due to charged particles in a layer.  [NOT
REQUIRED: Also work out the last part of Problem 8.1 to show that
the Thomas-Fermi screening length in 2 dimensions is independent
of density.]


\medskip
{\bf 3.  Phonons in metals }

a.  Derive the formula for the plasma frequency for the long-wavelength
longitudinal
oscillation frequency of charged particles of charge $Ze$, mass $M$,
and density $n$,
%
\begin{equation}
\omega_p^2 = \frac{4\pi n e^2 Z^2}{M}.
\end{equation}
%

b.  Show that if the interaction is screened according to the
Thomas-Fermi approximation, then the frequency vanishes at long
wavelength as it should for an acoustic mode in a metal, with
a longitudinal velocity of sound $v_L$ given by a simple formula in terms of the
Thomas-Fermi screening constant $k_{TF}$.

c.  Finally, using the
relation of $k_{TF}$ and the electron velocity at the Fermi
surface $v_F$, derive the ratio of  $v_L$ to $v_F$.  Does this appear
to be a reasonable approximation?

\medskip
{\bf 4.  Green's Functions for the Harmonic Lattice }

There are many types Green's functions defined with different
boundary conditions (as discussed in Mahan, Fetter, and Donaich).
For a harmonic lattice with the hamiltonian,
\begin{equation}
H = \frac{1}{2M} \sum_I p_I^2 + \frac{1}{2} \sum_{I,J} D_{I,J} u_I u_J,
\end{equation}
%
the  retarded Green's functions for the displacements is
%
\begin{equation}
G^R_{I,J}(t) = -i \Theta(t) \langle [u^H_I(t),u^H_J(0)] \rangle,
\end{equation}
%
and the time ordered Green's function is
%
\begin{equation}
G^T_{I,J}(t) = -i \langle T [u^H_I(t) u^H_J(0)] \rangle,
\end{equation}
%
\noindent where $[a,b]$ is a commutator, $T[ab]$ is a time ordered
product, and $u^H$ means a Heisenberg operator.  (In this case it
is not important to distinguish between Heisenberg and interaction
representations.)

a.  Show that each of these Green's functions satisfies the
equation
%
\begin{equation}
- M \frac{d^2}{dt^2} G_{I,J}(t) - \sum_K D_{I,K} G_{K,J}(t)
= \delta_{I,J} \delta(t) .
\label{eq-G}
\end{equation}
%
Hint:  use the usual relation for Heisenberg operators
$i \frac{d O^H}{dt} = [O^H,H]$.  For commutators such as
$[u,p^2]$, it may be convenient to use the identity
%
\begin{equation}
[A,BC] = ABC - BCA = [A,B]C + B[A,C]
\end{equation}
%

b.  Transform to normal modes labeled by wavevector $k$
and give the equation obeyed by the Green's function
$G_k(t)$  analogous to Eq. \ref{eq-G}.

c.  Give an explicit expression  for the retarded function
$G^R_k(t)$ valid for temperature $T \neq 0$.

d.  Find an expression valid for temperature $T \neq 0$
for the dynamic correlation function
%
\begin{equation}
J_{1,q}(\omega) = \int^{\infty}_{-\infty} dt e^{i \omega t}
\langle u^H_q(t) u^H_{-q}(0)] \rangle
\label{eq-J}
\end{equation}
%
which governs the intensity of scattering in an experimental
measurement.
(Leave the matrix elements which multiply the scattering
intensity as unspecified multiplying constants.  These depend upon
the particular scattering problem.  You need to give only the
frequency dependent part given in equation \ref{eq-J}.)


\medskip
{\bf 5.  Relation of Structure factor, Interaction Energy, and
Dielectric function }

a. From the definition of $\epsilon^{-1}$  in terms of the
density-density response function, derive the result that the
Fourier transform $S(k)$ of the pair correlation function $g(r)$
is given by the following formula (and correct my errors if I have
missed some factors).  The purpose for this problem is for everyone
to go through the derivation of this important result.
%
\begin{equation}
S(k) = - \frac{\hbar}{\pi}\frac{k^2}{4 \pi e^2} \int_{0}^{\infty}
d \omega Im \epsilon^{-1}(k,\omega)
\end{equation}
%

b.  Give the total potential energy of interaction between the
electrons $E_{int}$ in terms of $S(k)$, and derive the final
expression for the total energy of an electron system in terms
of integrals over the dielectric function
%
\begin{equation}
E = E^0 - \frac{\hbar}{\pi^3} \int d^3 k \int_0^{e^2}
\frac{d \alpha}{\alpha} \left[ \int_0^{\infty} d \omega
Im \epsilon_{\alpha}^{-1}(k,\omega) + \frac{2 \pi n \alpha}{k^2} \right],
\end{equation}
%
This formula is analogous to Pines A-39, but it is general and
not restricted to the RPA approximation.  Note that the dielectric
function is a function of the coupling constant $\alpha$.  Also
there may be errors in my formula of the factors of $2 \pi$, etc.


\medskip
{\bf EXTRA - NOT REQUIRED TO TURN IN\\
  Density Functional Theory (DFT) }

These are small practice problems to make you think about the
logic of DFT.  (If you have trouble with these questions, Prof.
Martin will be glad to discuss them with you. We will not treat
DFT  further in this course.)

a.  The general proof of Hohenberg and Kohn is that the ground
state density of any interacting electron system uniquely
determines (except for a constant) the "external potential", i.e.,
the potential acting on the electrons from other sources.  Give
the proof explicitly for the simplest case, a non-interacting
system of electrons.

b.  For a non-interacting system of electrons, the "internal
energy" of the electrons is just the kinetic energy. This is
easily written in terms of the orbitals (as is done in the notes
in the section of Kohn-Sham theory).  Show that the arguments of
Hohenberg and Kohn applied to this case show that the kinetic
energy is also a universal functional of the ground state density.

c.  Not all densities are possible ground state densities. For
example, show explicitly that the 2s state of hydrogen cannot be
the ground state of any potential that does not have delta
function singularities.


\end{document}