Ph 129a
Problem Set 4
Due: Nov. 7, 2013
Please put your solutions in Hee-Joong Chungs mailbox on the fourth oor of Lauritsen.
1) Show explicitly that the dening matrices of SO(1,1) give a reducible representation
of the group and that SO(1,1) is isomorphi
Ph 129a
Problem Set 6
Due: Nov. 21, 2013
Please put your solutions in Hee-Joong Chungs mailbox on the fourth oor of Lauritsen.
1) By generalizing the construction in the last problem of the previous problem set
to the complex case, construct the generator
Ph 129a
Problem Set 7
Due: Dec. 3, 2013
Please put your solutions in Hee-Joong Chungs mailbox on the fourth oor of Lauritsen.
1) In class we showed that the rotational SO(3) symmetry of the Bohr atom extends
to an SO(4) symmetry, when considering the boun
Physics 129A, Fall 2013
Problem set 1 Solution
Shu-Ping Lee, Chan Y. Park
Problem 1
Which of the following dene groups? For those that do not, indicate the reason why:
(a) The set of all complex n n matrices when the group operation is matrix multiplicati
Physics 129A, Fall 2013
Problem Set 3 Solutions
Shu-Ping Lee
Problem 1. Irreducible representations of cyclic group
By Schurs rst lemma, every complex irreducible representation of Abelian groups is one-dimensional.
so we could construct irreducible repre
Physics 129A, Fall 2013
Problem Set 2 Solution
H.-J. Chung
Problem 1
Given that H is a subgroup of G and N is a normal subgroup of G, prove that N H is a normal
subgroup of H.
Solution (revised by Chan Y. Park):
It is easy to show that the intersection of
Physics 129A, Fall 2013
Problem Set 4 Solutions
H.-J. Chung
Problem 1
Show explicitly that the dening matrices of SO(1, 1) give a reducible representation of the group
and that SO(1, 1) is isomorphic to T1 .
Solution:
The dening matrices of SO(1, 1) is
na
Physics 129A, Fall 2013
Problem Set 5 Solutions
Shu-Ping Lee, Chan Y. Park
Problem 1.
Solution (Shu-Ping Lee):
z1
z2
ej (
m
j+m jm
z 1 z2
)=
(1)
(j + m)!(j m)!
If g is an element of SU(2), then g = exp(iJ), where J is the generator of SU(2), so Jx = 1 x ,
Physics 129A, Fall 2013
Problem Set 6 Solution
H.-J. Chung, Chan Y. Park
Problem 1
By generalizing the construction in the last problem of the previous problem set to the complex
case, construct the generators of U (n) in the dening representation. Use th
Ph 129a
Problem Set 5
Due: Nov. 14, 2013
Please put your solutions in Hee-Joong Chungs mailbox on the fourth oor of Lauritsen.
1) Recall that the spin j UIR of SU (2), D(j) (g), was constructed in class by considering
the transformation of 2j + 1 basis fu
Ph 129a
Problem Set 3
Due: Oct. 31, 2013
Please put your solutions in Hee-Joong Chungs mailbox on the fourth oor of LauritsenDowns.
1) Construct all the irreducible representations of the cyclic groups Cn and verify that
the orthogonality relations satise
Ph129 HW1 solution:
Problem 1:
for t > 0:
There is one pole ( = E i) located inside contour CR1
Figure 1. contour CR1
Let = Rei , R
(1)
eit
d
<
E + i 2
CR1
0
eRtsin()
Rd = 2
RE
/2
0
eRtsin()
Rd < 2
RE
/2
0
eRt2/
Rd
RE
as R we could get:
(2)
CR1
eit
d
Physics 129A, Fall 2010
Problem Set 2 Solution
H. -J. Chung
October 17, 2010
1. Convolution theorem
Let Fcfw_f (k ), Fcfw_g (k ) be the Fourier transform of functions f (x), g (x), respectively. Then the
convolution of two functions f (x) and g (x) is den
Physics 129A, Fall 2010
Problem Set 3 Solution
Shu-Ping Lee
October 31, 2010
Problem 1:
xt
= (x, t)ln(x)
t
x
(1)
(x0 , t) = (x(x0 ,t) , t)
(2)
dx(x0 ,t)
d
=
+
=
(xt) +
dt
x dt
t
x
t
(3)
dx(x0 ,t)
= (xt)
dt
(4)
dx
= (tdt)
x
(5)
(a)
(b)
by initial condit
Physics 129A, Fall 2010
Problem Set 4 Solution
H. -J. Chung
November 4, 2010
1. The eld of a tip
(a)
We put a metallic wedge of angle and take conformal mapping z = w 2 as in gure (1). The
positive real axis in w plane is mapped to the positive real axis
1
Ph 129 Mid-term
1. Dissipative string. Consider the modied wave equation for the displacement of a string:
2
2
+ 2
+ 2
= 0.
t2
t
x2
(1)
with > 0.
(a) What is the Green function, G(x, t) of this equation?
(b) The string is made to oscillate by applying
Ph 129a
Problem Set 2
Due: Oct. 24, 2013
Please put your solutions in Hee-Joong Chungs mailbox on the fourth oor of Lauritsen.
1) Given that H is a subgroup of G and N is a normal subgroup of G, prove that
N H is a normal subgroup of H.
2) Given a group G
Ph 129a
Problem Set 1
Due: Oct. 17, 2013
Please put your solutions in Hee-Joong Chungs mailbox on the fourth oor of Lauritsen.
1) Which of the following dene groups? For those that do not, indicate the reason
why:
(a) The set of all complex n n matrices w
Physics 129A, Fall 2013
Problem Set 7 Solution
H.-J. Chung, Chan Y. Park
Problem 1
Solution (Chan Y. Park):
When we consider an n n matrix O in the dening representation of SO(n), we act this on pi
and xi as
pi = Oij pj , xi = Oij xj ,
(1)
where summation