Physics 139B
Solutions to Homework Set 4
Fall 2009
1. Libo, problem 12.16 on page 594595.
Consider an atom whose electrons are LS coupled so that the good quantum
numbers are j s mj and eigenstates of the Hamiltonian H0 may be written as
|j s mj . In the
Physics 139B
Solutions to Homework Set 3
Fall 2009
1. Consider a particle of mass m attached to a rigid massless rod of xed length R
whose other end is xed at the origin. The rod is free to rotate about the origin.
Classical mechanics teaches us that the
Physics 139B
Fall 2009
The Variational Computation of the Ground State Energy of Helium
I. Introduction to the variational computation
The Hamiltonian for the two-electron system of the helium atom is:
2
H=
2m
( 2 + 2 )
1
2
2e2 2e2
e2
+
,
r1
r2
r12
(1)
w
Physics 139B/171. General Relativity/Quantum Mechanics.
Professor Dine
Fall, 2009. Handout: Vectors: Everything You Need to Know
This note is a concise summary of the things you need to know about vectors, matrices,
and rotations. If you master whats here
Physics 139B
Fall 2009
Quantum Mechanics of a Charged Particle in an Electromagnetic Field
These notes present the motivation for the Schrodinger equation for a charged
particle in an external electromagnetic eld. In order to obtain the relevant equation,
Physics 139B
Midterm Exam Solutions
Fall 2009
1. Consider a spin- 1 particle with magnetic moment = S . At time t = 0,
2
1
we measure Sy and nd a value of + 2 for its eigenvalue. Immediately after this
measurement, we apply a uniform time-dependent magnet
Physics 139B
Midterm Exam
Fall 2009
MIDTERM INSTRUCTIONS: You have one hour and forty ve minutes to complete
the midterm exam. This is an open book exam. You are permitted to consult the
textbook (Libo), your handwritten notes, and class handouts. No othe
Physics 195a
Course Notes
Path Integrals: An Example Solutions to Exercises
021024 F. Porter
1
Exercises
1. Show that the B eld is as given in Eqn. 6, and that the vector potential
is as given in Eqn. 7, up to gauge transformations.
Solution: We wish to s
Physics 139B
Final Exam
Fall 2009
FINAL EXAM INSTRUCTIONS: This is an open book exam. You are permitted to consult
the textbook by Libo, your handwritten notes, and class handouts. No other consultations
or collaborations are permitted during the exam. In
Physics 139B
Solutions to Homework Set 1
Fall 2009
1. Libo, problem 11.27 on page 498.
(a) Let A be an hermitian operator. We rst demonstrate that
(eiA ) = eiA .
To prove this, we use the series expansion that denes the exponential,
eiA =
n=0
(iA)n
.
n!
T
Physics 139B
Final Exam Solutions
Fall 2009
1. An electron is placed in a potential
e2
V (r ) =
+ (r 2 3 z 2 ) ,
r
where is a small parameter. Neglect the spin of the electron.
(a) Compute the shifts of the n = 2 energy levels (you may neglect ne-structur
Physics 139B
Solutions to Homework Set 5
Fall 2009
1. Libo, problem 13.51 on pages 749750.
A one-dimensional harmonic oscillator of charge-to-mass ratio e/m, and spring
constant K oscillates parallel to the x-axis and is in its second excited state at t <
Physics 139B
Solutions to Homework Set 2
Fall 2009
1. Libo, problem 9.32 on page 395.
(a) The key equation is given in Table 9.4 on p. 379 of Libo:
L | , m = [( m)( m + 1)]1/2 | , m 1 .
(1)
We will also need to use:
L2 | , m =
2
( + 1) | , m .
(2)
When co
D2-WWT02
Answer: The statement is incorrect, because the arrangement of the batteries determines the
brightness of the bulbs and not the number of bulbs.
The circuits in cases A, D, and E have the batteries connected in parallel with each other so
the pot