Final PHY405, Fall 2012
1. A one-dimensional harmonic oscillator of frequency is formed by
a particle of mass m moving in a quadratic potential. At time t = t0 ,
the oscillator is in the ground state |0 . It is acted upon by the force
2
2
F (t) = F et / ,
PRACTICE FINAL FOR MAT 341
In problems (1)-(4) below we consider a cylindrical rod centered along the
x-axis in 3-dimensional space, from x = 0 to x = a; for each 0 x a the
intersection of this rod with the plane containing (x, 0, 0) and perpendicular
to
Spectral correlations of the QCD Dirac operator at nite
temperature
Talk given by
Bertram Klein
in the RMT course of Jac Verbaarschot, spring 99
Stony Brook, May 6 1999
1 Introduction
We want to calculate spectral correlation functions for a simpli ed ran
The Color Flavor Transformation
talk given by Achim Schwenk
in Jac Verbaarschot's Topics on Random Matrix Theory course
April 30, 1999
1 The Color Flavor Transformation
The Color Flavor Transformation was discovered by Martin Zirnbauer and relates an inte
Midterm PHY405, Fall 2012
1. Consider quantum one-dimensional harmonic oscillator of frequency
formed by a particle of mass m attached to a spring with spring constant k:
H=
p2
kx2
+
.
2m
2
In terms of the raising and lowering operators a , a, the oscill
1 Hirota Equations for 2D Quantum Gravity Models
Lecture and lecture notes by Olindo Corradini
The idea of using Random Matrices to perform calculations in Quantum Gravity was
originally due to the need of having non perturbative solutions for the string
20
Partial Quenching and the Valence Quark Mass Dependence
of the Chiral Condensate
In this chapter we study the spectrum of the QCD Dirac operator by means of the valence quark
mass dependence of the chiral condensate. For simplicity we will only conside
19 Infrared limit of the QCD Partition Function and LeutwylerSmilga Sum-Rules
19.1 The chiral Lagrangian
For light quarks the low energy limit of QCD is well understood. It is given by the chiral
Lagrangian describing the interactions of the pseudoscalar
18 Symmetries QCD partition function for two colors
The QCD partition function for two colors is invariant under a larger group than the three-color
QCD partition function. This well-known fact for QCD with fundamental fermions relies on the
pseudo-real n
14 Hilbert Space Quantization of a Random Hopping Model
There are two di erent ways of doing eld theory. First, by means of path integrals where the
elds are represented by complex functions or Grassmann valued functions. Second, in a second
quantized for
16 Symmetries of the QCD Partition Function
It is well-known that the QCD action is greatly constrained by gauge symmetry, Euclidean
Poincare invariance and renormalizability. These symmetries determine the structure of the
Dirac operator. In this section
13
The Calogero-Sutherland-Moser Model
In this chapter we explore the connection between exactly integrable systems knows as the
Calogero-Sutherland-Moser models 1, 2, 3] and Random Matrix Theory. We consider the partition function
Z=
Z
d ( )e S ( ( ) ;
(
Homework 11, PHY405, Fall 2012
Name and ID#
1. Consider a one-dimensional quantum particle of mass m moving on a
ring of length L, x [L/2, L/2], in the repulsive delta-function potential
V (x) = u(x) , u 0 .
Aharonov-Bohm ux = ( /e) threads the ring, so t
Homework 5, PHY405, Fall 2012
Name and ID#
1. Problem 8.2 of the textbook.
2. Consider a 1D particle of mass m and energy E scattered by the
potential formed by two identical -functions:
V (x) = u[(x + d/2) + (x d/2)] ,
u > 0.
(a) Calculate the transmissi
Homework 9, PHY405, Fall 2012
Name and ID#
1. Problem 9.15 of the textbook.
2. Calculate the entropy S = Tr[ ln ] of a one-dimensional harmonic
oscillator of frequency in the state of thermodynamic equilibrium at temperature T .
Homework 2, PHY405, Fall 2012
Name and ID#
1. Problem 6.4 (a) of the textbook.
2. A particle of mass m moves in a three-dimensional spherically-symmetric
potential
V (r) = u(r a) , u > 0.
(a) Reduce the radial part of the Schrdinger equation to the eectiv
PRACTICE FINAL EXAM
MAT 341, Fall 13
1. (a) Find the Fourier series of the function f (x), periodic of period 2,
equal to 0 on [0, /2], 1 on [/2, 3/2] and 0 on [3/2, 2].
(b) Use this Fourier series to deduce the following series representation
for :
1 1 1
Homework 1, PHY405, Fall 2012
Name and ID#
1. Problem 6.1 of the textbook.
2. A particle of mass m is moving in a one-dimensional potential
cfw_
U (x) =
u(x),
+,
for b < x < b,
for
|x| b,
with u > 0.
(a) Find the eigenenergies En and the eigenfunctions n
PRACTICE MIDTERM FOR MAT 341
(1) Consider the function f (x) = 2x 1, 0 < x < 2.
(a) Compute the Fourier sine series for f (x). At each x [4, 6] compute the value of this series.
n
Hint: The Fourier sine series looks like
n=1 bn sin( 2 x), where
formulae f