Strongly Correlated Systems in Condensed Matter Physics
PHYS 8.514

Fall 2004
8.514: Manybody phenomena in condensed matter and atomic physics Problem Set # 5 Due: 10/14/03
Interacting Fermions
1. Shortrange interaction.
a) Consider Schrodinger equation in D = 1 for two spinless fermions moving in an external
1
potential U (x) =
Strongly Correlated Systems in Condensed Matter Physics
PHYS 8.514

Fall 2004
8.514: Manybody phenomena in condensed matter and atomic physics Problem Set # 10 Due: 12/02/03
Quantum tunneling and escape
Reading: S. Coleman, Aspects of Symmetry
1. Decay of a metastable state.
Consider a quantum particle tunneling through a barrier,
Strongly Correlated Systems in Condensed Matter Physics
PHYS 8.514

Fall 2004
8.514: Manybody phenomena in condensed matter and atomic physics Problem Set # 9 Due: 11/20/03
(released late due to instructor il lness)
Path Integral
Reading: R. P. Feynman and A. P. Hibbs, Quantum Mechanics and Path Integrals
M. Stone, The Physics of
Strongly Correlated Systems in Condensed Matter Physics
PHYS 8.514

Fall 2004
8.514: Manybody phenomena in condensed matter and atomic physics Last modi ed: September 24, 2003
1 Lecture 2. Squeezed States
In this lecture we shall continue the discussion of coherent states, focusing on their properties as a basis in Hilbert space.
Strongly Correlated Systems in Condensed Matter Physics
PHYS 8.514

Fall 2004
Manybody phenomena in condensed matter and atomic physics Last modi ed: September 24, 2003
1 Lectures 4, 5. Bose condensation. Symmetrybreaking and quasiparticles.
In an ideal Bose gas, at su ciently low temperature, the lowest energy state becomes
occ
Strongly Correlated Systems in Condensed Matter Physics
PHYS 8.514

Fall 2004
8.514: Manybody phenomena in condensed matter and atomic physics Last modi ed: September 24, 2003
1 Lecture 1. Coherent States.
We start the course with the discussion of coherent states. These states are of interest
because they provide
a method to desc
Strongly Correlated Systems in Condensed Matter Physics
PHYS 8.514

Fall 2004
8.514:
Manybody phenomena in condensed matter and atomic physics Problem Set # 6 Due: 10/21/03
BardeenCooperSchrie er theory
1. Quasiparticles.
Consider quasiparticles of a BCS superconductor,
H=
X
p
+
p ap
ap +
X
p
ap " a;p # + h:c: =
X
p
Ep b+ bp
p
(
Strongly Correlated Systems in Condensed Matter Physics
PHYS 8.514

Fall 2004
8.514: Manybody phenomena in condensed matter and atomic physics Problem Sets # 7,8 Due: 11/04/03
(released late due to instructor illness)
Quasiparticle transport in a superconductor
1. Electron tunneling.
Consider two metals that can be in a normal or
Strongly Correlated Systems in Condensed Matter Physics
PHYS 8.514

Fall 2004
8.514: Manybody phenomena in condensed matter and atomic physics Problem Set # 3 Due: 9/30/03
Bose condensation
1. Quasiparticles.
Consider a Bose gas at T = 0 with one quasiparticle with momentum p = 0 added on the top.
6
Quasiparticle state can be obta
Strongly Correlated Systems in Condensed Matter Physics
PHYS 8.514

Fall 2004
8.514: Manybody phenomena in condensed matter and atomic physics Problem Set # 2 Due: 9/23/03
Squeezed states
1. Squeeze operators.
bb b b
Consider a unitary operator U ( ) = exp ( (aa ; a+a+) =2).
a) Prove that
b
b
b
b
b
U + ( )aU ( ) = cosh a ; sinh a+
Strongly Correlated Systems in Condensed Matter Physics
PHYS 8.514

Fall 2004
8.514: Manybody phenomena in condensed matter and atomic physics Problem Set # 1 Due: 9/16/03
Coherent states
1. Operator identities.
Here we prove two useful theorems from operator algebra that will be used in the problems of
this homework and later in
Strongly Correlated Systems in Condensed Matter Physics
PHYS 8.514

Fall 2004
8.514: Manybody phenomena in condensed matter and atomic physics Problem Set # 4 Due: 10/07/03
Bose condensation
1. Vortices.
H
h
h
a) Starting from the super ow equations away from singularities, v = m , v dr = 2 m l with
integer l, show that the veloci
Strongly Correlated Systems in Condensed Matter Physics
PHYS 8.514

Fall 2004
Manybody phenomena in condensed matter and atomic physics Last modi ed: September 29, 2003
1 Lecture 6. Vortices, super uidity. Trapped gases.
BEC at nite temperature.
To treat hydrodynamics and BEC in spatially varying background, need a more general