1
INTRODUCTION TO MANY BODY PHYSICS: 620. Fall 2012
Questions 4. (Due Fri, Nov 2nd)
Figure 1: Honeycomb structure of graphene. See Problem 1
1. Electrons on graphene move on a Honeycomb lattice as shown in Fig. (1). The vertices of each
unit cellform a tr
INTRODUCTION TO MANY BODY PHYSICS: 620. Fall 2012
Questions I. (Due Weds, Sept 19th.)
These questions can all be done with a minimum of algebra. They will familiarize you with the
method of second-quantization, as applied to free bosonic elds. Please choo
INTRODUCTION TO MANY BODY PHYSICS: 620. Fall 2012
Answers to Questions 4. Friday, Nov 9.
1. (a) We begin by noting that the matrix elements for the tight-binding Hamiltonian are
(i = j )
t (i, j, nearest neighbors)
i|H |j =
0 (otherwise)
(1)
where is the
1
INTRODUCTION TO MANY BODY PHYSICS: 620. Fall 2012
Answers to Questions II. Oct 2nd
Here is an outline of the solutions.
1. (i) Expanding | = |1111100 . . . = c 5 c 4 c 3 c 2 c 1 |0 we obtain (in gory detail)
c 3 c6 c4 c6 c3 |
c 3 c6 c4 c6 c 5 c 43 c 2
INTRODUCTION TO MANY BODY PHYSICS: 620. Fall 2012
Questions 2. (Due Oct 3)
1. In this question ci and ci are fermion creation and annihilation operators and the states are
fermion states. Use the convention
|11111000 . . . = c5 c4 c3 c2 c1 |vacuum .
(i) E
INTRODUCTION TO MANY BODY PHYSICS: 620. Fall 2012
Questions 3. (Due Mon , Oct 15th)
1. (a) Show that for a general system of conserved particles at chemical potential, the total particle
number in thermal equilibrium can be written as N = F/, where F = kB
INTRODUCTION TO MANY BODY PHYSICS: 620. Fall 2012
Questions 5: Due Mon 19th Nov
1. Consider a gas of particles with interaction
V = 1/2
Vq c k q c k + q c k c k
kk q
(a) Let |0 = |k|<kF , c k |0 represent the lled Fermi sea, i.e. the ground state of the
INTRODUCTION TO MANY BODY PHYSICS: 620. Fall 2012
Questions 6. Finite Temperature. (Due Weds, Dec 5th.)
1. Use the method of complex contour integration to carry out the Matsubara sums in the following:
(i) Derive the density of a spinless Bose Gas at nit
INTRODUCTION TO MANY BODY PHYSICS: 620. Fall 2012
Exploratory Quiz
Welcome to Physics 620, introduction to Many Body physics. It would be very helpful to me if
I could have some idea of your interests and backgrounds. The quiz here is not going to be used
INTRODUCTION TO MANY BODY PHYSICS: 620. Fall 2012
Answers to Questions I. Sept 24th
1. For the Harmonic oscillator H = h [a a + 1 ], we know that
2
aa
aa
= n( ) =
1
h
e
= n( ) + 1,
1
,
(1)
where = 1/(kB T ) . Relating the displacement to the creation and
INTRODUCTION TO MANY BODY PHYSICS: 620. Fall 2012
Answers to Questions III. Oct. 22nd.
Here is an outline of the solutions.
1. (a) We begin by writing down the partition function
Z = Tr e (H N )
(1)
Z
= Tr e (H N ) N
(2)
Dierentiating w.r.t. , we have
So
INTRODUCTION TO MANY BODY PHYSICS: 620. Fall 2011
Solution to Problems 5.
1. (a) Suppose we write the interaction in the form
V = 1/2
Vijkl c i c j cl ck ,
ijkl
then using Wicks theorem
| V |
Vijkl |c i c j cl ck |
= 1/2
ijkl
Vijkl
= 1/2
| c i c j c l c