Physics 582, Fall Semester 2008 Professor Eduardo Fradkin Problem Set No. 7/ Final Exam Due Date: Friday, December 19, 2008; 5:00 pm
This is a take home nal exam. As noted in the course website you will have to send me your solutions by email. This can be
UIUC Physics 582: Problem Set 4
Solutions
Problem 4.1: Supersymmetry (the Wess-Zumino model):
Consider a complex scalar , a left-handed Weyl spinor and a complex scalar F with Lagrangian
L = + i
+ F F
(1.1)
F is called an auxiliary field (it does not ha
Quantum Field Theory
PHYS582/583 Lecture Notes (V4.2)
R.G. Leigh
University of Illinois
2015-16
These notes are for the sole use of registered students of UIUC Physics 582-3, 2015-16. Before using the
enclosed material for any other purpose, contact the a
Quantum Field Theory
PHYS582/583 Lecture Notes (V4.2)
R.G. Leigh
University of Illinois
2015-16
These notes are for the sole use of registered students of UIUC Physics 582-3, 2015-16. Before using the
enclosed material for any other purpose, contact the a
Quantum Field Theory
PHYS582/583 Lecture Notes (V4.2)
R.G. Leigh
University of Illinois
2015-16
These notes are for the sole use of registered students of UIUC Physics 582-3, 2015-16. Before using the
enclosed material for any other purpose, contact the a
Solutions to Homework Set 1
Differential calculus: The point of the exercise was to make sure that you know how to
differentiate integrals with respect to their limits:
Z b
Z
d
d b
f (x) dx = f (a),
f (x) dx = f (b),
da a
db a
and in general
d
dt
Z
b(t)
a
Solutions to Homework Set 3
1) Bent bars: First recall some elementary calculus results: the curvature , and the radius
R of the osculating circle, at a point (x, y) on a curve y(x) are given by
=
1
d
1
y
d tan1 y
=
=p
=
.
R
ds
dx
(1 + y 2 )3/2
1 + y 2
Solutions to Homework Set 8
1) Critical mass: We expand
n(x, t) =
X
am (t) sin
m=1
and also
mx
X 4
=
,
sin
m
L
m,odd
mx
L
,
0 < x < L.
Substituting in the given equation, and using the linear independence of the sine functions,
then gives
Dm2 2
4
a m
Physics 508
Handout 1
Fall 2016
Mathematical Methods in Physics I
Course Material
Homework 1
Prof. M. Stone
2117 ESB
University of Illinois
Revision and Warm-up exercises
These problems are designed to reactivate your mathematical skills after a relaxing
Solutions to Homework Set 4
Test functions and distributions: For part a) we take any test function (x) and look
at
Z
0
0
(x) cfw_f (x) 0 (x) + f 0 (x)(x) dx
(, f + f )
Z
cfw_[0 (x)f (x) (x)f 0 (x)](x) + (x)f 0 (x)(x) dx
=
0
= (0)f (0),
and compare it w
Physics 508
Handout 3
Fall 2016
Mathematical Methods in Physics I
Course Material
Homework 3
Prof. M. Stone
2117 ESB
University of Illinois
1) Elastic Rods. The elastic energy per unit length of a bent steel rod is given by 21 Y I/R2 .
Here R is the radiu
Quantum Field Theory
PHYS582/583 Lecture Notes (V4.2)
R.G. Leigh
University of Illinois
2015-16
These notes are for the sole use of registered students of UIUC Physics 582-3, 2015-16. Before using the
enclosed material for any other purpose, contact the a
Quantum Field Theory
PHYS582/583 Lecture Notes (V4.2)
R.G. Leigh
University of Illinois
2015-16
These notes are for the sole use of registered students of UIUC Physics 582-3, 2015-16. Before using the
enclosed material for any other purpose, contact the a
UIUC Physics 582: Problem Set 2
Solutions
Problem 2.1: Identifying Symmetry Groups
Describe the symmetry groups (discrete or continuous) of the following Lagrangian densities, and identify
any conserved currents. In each case, a1 , a2 , . are real scalar
UIUC Physics 582: Problem Set 1
Solutions
Problem 1.1: Axion Models
The action
Z
d 4x
S[a] =
M
2
1
a(x)
2
has a continuous global invariance a(x) a(x) + , where is constant.
(a) Find the associated conserved current.
(b) Now add a term
S1 [a, A ] =
1
f
Z
Quantum Field Theory
PHYS582/583 Lecture Notes (V4.2)
R.G. Leigh
University of Illinois
2015-16
These notes are for the sole use of registered students of UIUC Physics 582-3, 2015-16. Before using the
enclosed material for any other purpose, contact the a
UIUC Physics 582: Problem Set 4
Solutions
Problem 4.1: Supersymmetry (the Wess-Zumino model):
Consider a complex scalar , a left-handed Weyl spinor and a complex scalar F with Lagrangian
L = + i
+ F F
(1.1)
F is called an auxiliary field (it does not ha
UIUC Physics 582: Problem Set 5
Solutions
Problem 5.1: Tunneling
Consider a particle of mass m moving in a one-dimensional double well potential
V (q) = (q 2 q02 )2
(1.1)
where q0 is constant. You will use imaginary time path integral methods to compute t
UIUC Physics 582: Problem Set 3
Solutions
Problem 3.1: Vector and Axial Vector Currents
Suppose we have a Lagrangian in 3+1 dimensions with a global U(1) symmetry:
L = i
+ i
m
m
(1.1)
where is a left-handed 2-component spinor and
is right-handed
UIUC Physics 582: Problem Set 0
Solutions
Problem 0.1: Lorentz Transformations
The components of the position vector, x , where (x 0 , x 1 , .) = (ct, x, .), transform under a Lorentz
transformation according to
x 0 = x
(1.1)
is the Lorentz transformati
UIUC Physics 582: Problem Set 5
Solutions
Problem 5.1: Tunneling
Consider a particle of mass m moving in a one-dimensional double well potential
V (q) = (q 2 q02 )2
(1.1)
where q0 is constant. You will use imaginary time path integral methods to compute t
Quantum Field Theory
PHYS582/583 Lecture Notes (V4.2)
R.G. Leigh
University of Illinois
2015-16
These notes are for the sole use of registered students of UIUC Physics 582-3, 2015-16. Before using the
enclosed material for any other purpose, contact the a
Quantum Field Theory
PHYS582/583 Lecture Notes (V4.2)
R.G. Leigh
University of Illinois
2015-16
These notes are for the sole use of registered students of UIUC Physics 582-3, 2015-16. Before using the
enclosed material for any other purpose, contact the a
Solutions to Homework Set 6
1) Missing state:
The continuum eigenfunctions are are
k = (ik + tanh x)eikx ,
so
k (x) =
(k i)eikx /i
x 0,
(k + i)eikx /i x 0.
It now helps to draw a phasor diagram
Im
i
k
Re
from which we see that (k) = tan1 (1/k) and A = i
Physics 508
Handout 7
Fall 2016
Mathematical Methods in Physics I
Course Material
Homework 7
Prof. M. Stone
2117 ESB
University of Illinois
1) Flexible rod again: A flexible rod is supported near its ends by means of knife edges
that constrain its positio
Solutions to Homework Set 11
Dielectric Sphere: The solution outside will be of the form
C1
out (r, ) = E0 rP1 (cos ) + 2 P1 (cos )
r
C
=
E0 r + 2 cos .
r
while, inside
in (r, ) = A1 rP1 (cos ).
These terms have been selected to match the asymptotically
Physics 508
Handout 8
Fall 2016
Mathematical Methods in Physics I
Course Material
Homework 8
Prof. M. Stone
2117 ESB
University of Illinois
1) Critical Mass: An infinite slab of fissile material has thickness L. The neutron density
n(r) in the material ob
Physics 508
Handout 12
Oct 2016
Mathematical Methods in Physics I
Course Material
Homework 12
Prof. M. Stone
2117 ESB
University of Illinois
Here are some optional problems on integral equations. They are taken verbatim from
Paul Goldbarts homework sets.
11
Perturbation Theory and Feynam Diagrams
We now turn our attention to the dynamics of a quantum eld theory. All of the results that we will derive in this section apply equally to both relativistic and non-relativistic theories with only minor changes.
8
8.1
8.1.1
Coherent State Path Integral Quantization of Quantum Field Theory
Coherent states and path integral quantization.
Coherent States
Let us consider a Hilbert space spanned by a complete set of harmonic oscillator states cfw_|n , with n = 0, . .
Physics 508
Handout 2
Fall 2016
Mathematical Methods in Physics I
Course Material
Homework 2
Prof. M. Stone
2117 ESB
University of Illinois
1) Fermats principle: According to Fermats principle, the path taken by a ray of light
between any two points makes
Solutions to Homework Set 7
1) Flexible rod again: This is a somewhat tedious, but ultimately rewarding exercise.
a) Look back at your solution for homework set 4, where you showed that
Z 1
Z 1
(4)
(3)
(3) 1
u v dx = [u v (u ) v + (u ) v (u ) v]0 +
Physics 508
Handout 4
Fall 2016
Mathematical Methods in Physics I
Course Material
Homework 4
Prof. M. Stone
2117 ESB
University of Illinois
1) Test functions and distributions:
a) Let f (x) be a smooth function.
i) Show that f (x)(x) = f (0)(x). Deduce th
Physics 508
Handout 9
Fall 2016
Mathematical Methods in Physics I
Course Material
Homework 9
Prof. M. Stone
2117 ESB
University of Illinois
1) Pantograph Drag: A high-speed train picks up its electrical power via a pantograph
from an overhead line. The lo
Solutions to Homework Set 10
1) Conducting strip:
A(k) =
Z
+
ikx
V (x)e
dx = V0
Z
a
eikx dx =
a
From this we have
V (x, 0, y) = 2V0
Z
2V0 sin ka
k
dk sin(ka) ikx |k|y|
e e
.
2 k
Taking the y gradients to get Ey , and then getting = 0 (Ey |y=+ Ey |y= ) giv