Solution to Assignment 1 - Physics 251a
1
Problem 1
(i) Start with Schrodingers equation
i
h
(x, t)
h2
= (
t
2m
2
+ V (x)(x, t) .
(1)
Take the complex conjugate
h2
(x, t)
= (
i
h
t
2m
2
+ V (x) (x, t) .
(2)
Now multiplying (1) by and (2) by and subtracti
Physics 251a
PROBLEM SET 4
Fall, 2012
Due: Friday October 26, 2012
1. Derive the WKB formula for transmission through a smooth potential barrier given in
the notes, using the WKB matching formulas.
2. Calculate the transmission coecient at energy E throug
Physics 251a
PROBLEM SET 3
Fall, 2012
Due: October 12, 2012
1. A particle of mass m is incident on a planar barrier which we represent by a potential
V (z ) = (z ). If the energy of the particle is E , and the incident velocity makes an angle
with the z
Physics 251a
PROBLEM SET 2
Fall, 2012
Due Friday, September 28
1. Let U and V be unitary operators in Hilbert space. Show that U V is also unitary.
2. Let (r, t) obey the time-dependent Schroedinger Equation for a particle of mass m, in
the potential V (r
Physics 251a
PROBLEM SET 1
Fall 2012
Due: Monday, September 17
1. Show that in classical mechanics, for circular motion, in an arbitrary central potential
V (r), one has = dE/dL where E is the energy, L is the angular momentum, and is the
angular frequenc
Phys 251a Lecture 33
28 November 2012
B. I. Halperin / Gordon Ritter
printed to pdf November 27, 2012
Diamagnetic and Paramagnetic Susceptibility via Perturbation Theory1
Consider an atom or a molecule in a DC magnetic eld, in the non-relativistic approxi
Phys 251a Lecture 32
26 November 2012
B. I. Halperin / Gordon Ritter
printed to pdf November 25, 2012
Bound State Perturbation Theory (continued)
Brillouin-Wigner Perturbation Theory for the Degenerate Case
Again, we consider a Hamiltonian of the form H =
Phys 251a Lecture 31
19 November 2012
B. I. Halperin / Gordon Ritter
printed to pdf November 18, 2012
Bound State Perturbation Theory
Consider a Hamiltonian of the form H = H0 + gV , where the Schrdinger equation for H0 (but
o
not for H ) can be solved ex
Phys 251a Lecture 30
16 November 2012
B. I. Halperin
printed to pdf November 15, 2012
PARTIAL NOTES FOR LECTURE
Measurement in Quantum Mechanics
When we rst introduced the postulates of quantum mechanics, we stated that any physical
observable should be r
Phys 251a Lecture 29
14 November 2012
B. I. Halperin
printed to pdf November 11, 2012
PARTIAL NOTES FOR LECTURE
Density Matrix for a 2-Level System
We consider the density matix w for a two-level system, such as a single spin with s=1/2. Since
w is Hermit
Physics 251a
PROBLEM SET 5
Fall 2012
Due: November 9, 2012
1. A particle of mass m is scattered by a spherical square-well potential of form
V (r) = V0 < 0 for r < a.
V (r) = 0 for r > a.
Show that there exist choices of V0 , other than V0 = 0, such that
Physics 251a
PROBLEM SET 6
Fall, 2012
Due: Friday, Nov. 30
1. In the region of space 0 < x < a there is a magnetic eld B0 which points in the z
direction. Elsewhere the magnetic eld vanishes. A spinless quantum mechanical particle of
mass m and charge q i
PHYSICS 251A (FALL 2012)
PROBLEM SET 5
SOLUTIONS
Problem 1. We rst consider s-wave scattering ( = 0). The radial wave function Rk, =0 (r)
is given by
Rk, =0 (r) =
where k =
1
j0 (k r),
r
a
.
C [j (kr) cos n (kr) sin ] r a
0
0
0
0
2m(E V0 ), k = 1 2mE . Th
Solution to Problem Set 06- Physics 251a
1
Problem 1
(i) Lets check directly whether it satises the time-independent Schrodinger equation.
(E H0 )|
V |
= (E H0 )(
1
V | + | )
E H0
= V | .
(1)
(ii) Insert a complete momentum eigenbasis and dene p = hq, we
Solution to Problem Set 02 - Physics 251a
1
Problem 1
(i) As a trace over the eigenstates of the Hamiltonian, the partition function is given by
n|eH |n =
Z =
en =
n=0
n=0
1
.
1 e
(1)
(ii-a) Coherent States Method: We want to compute the kernel B(x, y) =
Harvard Physics 251a, Fall 2008
Assignment 6
Due Tuesday, December 16.
Reminder: The take-home Exam can be picked up on
Tuesday, January 6, between 2:30 and 4:00, from Barbara in J348.
Problem 1. In class we discussed the time-dependent perturbation theor
Harvard Physics 251a, Fall 2008
Assignment 3
Due Thursday, October 30, before the start of class.
Reading: Merzbacher Chapters 11 and 12.
1
Problem 1. Consider a one-particle Hamiltonian of the form H = 2m
the potential as a function of r = |x| leads to a
Harvard Physics 251a, Fall 2008
Assignment 4
Due Thursday, November 13, before class.
(Note there will be class on Tuesday, November 11.)
Reading: Merzbacher, Chapter 12.
Problem 1. (Bound States in 1 and 3 Dimensions.) The point of this problem is
to stu
Physics 251a, Fall 2008
Problem Set 2
Due before class, October 16, 2008
Reading: We will cover selected topics in Merzbacher, Chapters 5-10, leading to Symmetries
and Angular Momentum starting in Chapter 11. Some of the corresponding material can be
foun
Harvard Physics 251a, Fall 2008
Assignment 5
Due Tuesday before Thanksgiving, November 25.
Hand in to Barbara Drauschke in Jeerson 348,
BETWEEN 2:304:00pm.
(No class that day.)
Reading: Merzbacher, Chapters 17, 18.
Problem 1. In class we calculated the cl
Harvard Physics 251a, Fall 2008
Assignment 1
Due Thursday, October 2, before the start of class.
Spread the work out over the two weeks!
Reading: Week 1: Halperin 251a lectures 1-6; Weeks 12: Merzbacher chapters 1-5.
Problem 1. Consider the one-particle S