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
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.
Due Monday, December 12, 2011
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
Due Monday, November 14, 2011
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
Due Monday, November 28, 2011
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
Due Friday, February 10, 2012
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
Due Friday, March 9, 2012
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.
Due Friday, February 24, 2012
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
Due Friday, October 14, 2011
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
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
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
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
@yA transform under rotation around the x; y; z axes as
0x1 01 0
R @y A = @0 cos , sin A @y A
0xz1 0 0 sin 0 cos 1 0xz1
Physics 221A Midterm Solution
1. A set of basis states for two spin- 1 particles is fj + +i; j + ,i; j , +i; j , ,ig.
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
Let jvi be an eigenvector of the skew-Hermitian operator A, with eigenvalue . Then
hvjAjvi = hvj,Ay jvi
since A = ,Ay
using hujvi = hvjui ; with jui = ,Ajvi
hvjvi = , hvjvi
Problem Set 4 Solution
The rotation operator Rx is de ned by
Rx = @0
cos , sin A = e, ~ L
0 sin cos
Expand both sides of this to rst order in :
1 ,A = I , ~ Lx;
i~ @ 0
so i~ @0
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
2s 0 2
2p; m = 0 a b
Problem Set 2 Solution
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
Representations of the Angular Momentum Operators
In Notes 12 we introduced the concept of rotation operators acting on the Hilbert space of
some quantum mechanical system, and set down postula
The Propagator and the Path Integral
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
Problem Set 3 Solution
H = X x + Y y + Z z = X + iY X ,ZiY = R sin ei sin cos
has eigenvalues R, with corresponding orthonormal eigenvectors
cos 2 ;
ju+ i = ei sin
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
Due Friday, October 28, 2011
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.