1
8.511 Theory of Solids I
1.
Problem Set 11
Due December 9, 2004
(a) Start with the reduced Hamiltonian
H = H0 V 0
c c k ck ck
k
k,k
where H0 =
k, k ck, ck,
W=
. Show that W < |H N | > is equal to
2(k )|vk |2 V0
uk vk uk vk ,
(1)
k,k
k
where | > is the B

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8.511 Theory of Solids I
Problem Set 10
Due November 30, 2004
1. Electron Lifetime: For simplicity, we treat the problem at T = 0 and consider an added
electron at energy E above the Fermi energy. The more physical problem of electron lifetime
at nite t

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8.511 Theory of Solids I
Problem Set 9
Due November 23, 2004
Consider the twosite Hubbard model
H=t
ni ni
c c2 + h.c. + U
1
i=1,2
where c creates an electron with spin on site i and ni = c ci is the number operator of
i
i
electrons with spin on site

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8.511 Theory of Solids I
Problem Set 8
Due November 16, 2004
1. (a) Solve the HartreeFock equations for an electron gas in a uniform positive background
charge with density n. Calculate the HartreeFock eigenvalues and the ground state
energy.
(b) Show

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8.511 Theory of Solids I
Problem Set 7
Due November 2, 2004
1. Marder, Chapter 17, Problem 6
To do this problem, you may start with the following equation relating the heat current
density jQ and the charge current density j .
T
jQ = j
x
(1)
where is

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8.511 Theory of Solids I
1.
Problem Set 6
Due October 26, 2004
1
(a) Calculate the period B of the ShubnikovdeHaas oscillation of potassium assuming
the free electron model.
(b) What is the area in real space of the extremal orbit for B = 1 tesla?
2. C

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8.511 Theory of Solids I
Problem Set 5
Due October 19, 2004
1. Breakdown of Semiclassical Theory: Zener Tunneling
Consider a semiconductor with a bandgap (Fig. a) sub ject to a strong electric eld E . We
can consider that the energy level is tilted in s

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8.511 Theory of Solids I
Problem Set 4
Due October 12, 2004
1. The electronic structure of La2 CuO4 is believed to be well described by a two dimensional
square planar model of copper with oxygen mid-way between the nearest neighbor copper
atoms. (See t

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8.511 Theory of Solids I
Problem Set 2
Due September 28, 2004
1. Problem 2.5(b) from page 38 of Marder.
(a) The hcp also has a glide plane, which is parallel to a plane containing both the a and c
axes. Describe where this plane may be located, so that

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8.511 Theory of Solids I
Problem Set 1
Due September 21, 2004
1. Prove that the product of the volume of the rst Brillouin zone and the volume of the unit
cell of the Bravais lattice equals (2 )3 .
2. Show that rotations about any axis that takes a Brav