Problem set V, due Dec. 4
1. 30 pt Take problem 3 hydrogen atom in an electric eld of the
last problem set, use the matrix you worked out for the perturbation
V = ,eEz. Assume that E is small, determine the eigen energies
and eigen states within the subsp
Physics 221A
Fall 2007
Homework 2
Due Friday, September 9, 2011
Reading Assignment: Notes 1, pp. 1331, Notes 2 entire. I will post the relevant pages
in Sakurai later.
Please do Problems 1.5, 2.2, 2.3 and 2.4.
Physics 221A
Fall 2011
Homework 12
Due Friday, November 18, 2011
Reading Assignment:
Notes 22, entire; Notes 23, Secs. 16.
Please do Problems 19.2 (carried over from last week) and 22.2.
Physics 221A
Fall 2011
Homework 15
Due Monday, December 12, 2011
Reading Assignment:
Notes 27, Secs. 58; Notes 28, Sec. 10, handwritten notes on helium, covering the
perturbation analysis of the singly excited states and the variational treatment of the g
Physics 221A
Fall 2011
Homework 11
Due Monday, November 14, 2011
Reading Assignment:
Notes 19, Secs. 510, Notes 20 entire, Notes 21 entire.
This homework is due on Monday, November 14, since Friday, November 11 is ocially
a university holiday. There will
Physics 221A
Fall 2011
Homework 13
Due Monday, November 28, 2011
Reading Assignment:
Notes 23, Secs. 715; Notes 24, entire; Notes 26, entire; Notes 27, Secs. 15. I have
posted Notes 25, but we will skip them for now and come back to them later if there is
Physics 221B
Spring 2012
Homework 17
Due Friday, February 10, 2012
Reading Assignment:
A lot of the material from this week is covered only in hand written notes. See the web
pages for the lectures of the week. For Monday, January 30, please read the hand
Physics 221B
Spring 2012
Homework 20
Due Friday, March 9, 2012
Reading Assignment:
Notes 40, but you can skip the part on the orbital angular momentum of the eld;
handwritten notes from lecture of Feb. 27; Notes 41.
Please do Probs. 41.2 and 41.3.
Physics 221B
Spring 2012
Homework 18
Due Friday, February 17, 2012
Reading Assignment:
Notes 35, pp. 523, Notes 36, pp. 111.
Please do Probs. 36.1, 36.2 and 36.3.
Physics 221B
Spring 2012
Homework 19
Due Friday, February 24, 2012
Reading Assignment:
Notes 36, Secs. 8 and 10 (you can skip Sec. 9); Notes 38 entire. I am posting Notes 37
only for reference, we will skip that material this semester.
Please do Probs. 38
Physics 221A
Fall 2011
Homework 7
Due Friday, October 14, 2011
Reading Assignment:
Notes 13, Secs. 49; Notes 14, Secs. 15, 1014. You can skip Secs. 69 of Notes 14,
this set of notes needs some serious revision, but it wont happen this year.
Please do Prob
Physics 221A
Fall 2011
Homework 5
Due Friday, September 30, 2011
Reading Assignment: Notes 8, pp. 517; Notes 9, Secs. 13 only; Notes 10, Sec. 7 only
(on magnetic monopoles); Notes 11, Secs. 16. You will see that I am skipping a fair amount
of material. I
Physics 221A
Fall 2011
Homework 4
Due Friday, September 23, 2011
Reading Assignment: Notes 4, pp. 820, Notes 5, pp. 120, Notes 6, pp. 14, Notes 8,
pp. 15. You may skip Notes 7, which cover WKB theory.
I usually revise notes after I give my lectures, but s
Physics 221A
Fall 2011
Homework 3
Due Friday, September 16, 2011
Reading Assignment: Notes 3, pp. 118, Notes 4, pp. 18.
Please do Problems 2.4 (carried over from last week), 3.1, 3.2, 3.3.
Quantum Mechanics
Dung-Hai Lee
Summer 2000
Contents
1 A brief reminder of linear Algebra
1.1 Linear vector space . . . . . . . . . . . . . . . . . . . .
1.2 Linear operators and their corresponding matrices . . .
1.3 Function of an operator . . . . . . .
Physics 221A Problem Set 4, due Nov. 17
1. 30 pts
a 0 1 the fact that the three components of the position vector
Using
x
@yA transform under rotation around the x; y; z axes as
z
0x1 01 0
1 0x1
0
R @y A = @0 cos , sin A @y A
0xz1 0 0 sin 0 cos 1 0xz1
cos
Physics 221A Midterm Solution
1. A set of basis states for two spin- 1 particles is fj + +i; j + ,i; j , +i; j , ,ig.
2
+ ,
, +
The e ect of the Hamiltonian H = J S1 S2 + S1 S2 on these states is
H j + +i = 0; H j + ,i = J ~2 j , +i; H j , +i = J ~2 j + ,
Physics 221A, Fall 2000
Problem Set 1 Solution
Problem 1
Let jvi be an eigenvector of the skew-Hermitian operator A, with eigenvalue . Then
hvjAjvi = hvj,Ay jvi
= hvj,Ajvi
since A = ,Ay
using hujvi = hvjui ; with jui = ,Ajvi
and therefore
hvjvi = , hvjvi
Physics 221A
Problem Set 4 Solution
Problem 1a
The rotation operator Rx is de ned by
01
Rx = @0
1
0
0
cos , sin A = e, ~ L
0 sin cos
i
Expand both sides of this to rst order in :
01
@0
1
0 0
i
1 ,A = I , ~ Lx;
0 1
and similarly
00
i~ @ 0
x
00
so i~ @0
1
Physics 221A
Problem Set 5 Solution
Problem 1: Stark e ect in hydrogen atom
From Problem Set 4, the matrix of the total Hamiltonian H0 + V in the space
spanned by n = 1; 2 is
1s 2s 2p; m = 0 2p; m = +1 2p; m = ,1
1s 1 0
a
0
0
2s 0 2
b
0
0
2p; m = 0 a b
0
Physics 221A
Problem Set 2 Solution
Problem 1
Given a two-photon state jS i = 12 j + 1; ,1i + j , 1; +1i, we want to nd the
probability that the rst photon carries angular momentum in the +z direction.
Experimentally, this would be done by putting a lter
Physics 221A
Fall 2011
Notes 13
Representations of the Angular Momentum Operators
and Rotations
1. Introduction
In Notes 12 we introduced the concept of rotation operators acting on the Hilbert space of
some quantum mechanical system, and set down postula
Physics 221A
Fall 2011
Notes 9
The Propagator and the Path Integral
1. Introduction
The propagator is basically the x-space matrix element of the time evolution operator U (t, t0 ),
which can be used to advance wave functions in time. It is closely relate
Physics 221A
Problem Set 3 Solution
Problem 1
The Hamiltonian
,
cos
e,i
H = X x + Y y + Z z = X + iY X ,ZiY = R sin ei sin cos
,
has eigenvalues R, with corresponding orthonormal eigenvectors
,i
cos 2 ;
,e sin
2
ju+ i = ei sin
ju,i =
cos
2
2
Z
The
Physics 221A
Fall 2011
Homework 1
Due Friday, September 2, 2011
Reading Assignment: Notes 1, pp. 113. Although I am posting all of Notes 1, the
lecture on Friday only covered the rst 13 pages. I may revise pages 14 to the end after
lectures next week. Rel
Physics 221A
Fall 2011
Homework 9
Due Friday, October 28, 2011
Reading Assignment:
Notes 16, Secs. 9-11, Handwritten notes on hydrogen, handwritten notes on coupling
of spatial and spin degrees of freedom, Notes 17, Secs. 1-9.
Please do Problems 16.1, 16.