Phys 500, Quantum Mechanics
Homework 4
O
* The Clebsch-Gordan Issue *
Posted: Friday, November 7; due: Monday, November 17
Problem 1: The Heisenberg spin chain and SU(2)-symmetry. (5 points)
Remark 1: In this problem, we go back to the Heisenberg spin c

Phys 500, Quantum Mechanics
Reference Solution for Homework 3
November 9, 2014
Problem 1. (3 points) Demonstrate the following three properties of density operators, starting
from their definition (where all states in the ensemble are normalized to unity)

Phys 500, Quantum Mechanics
Reference solution for Homework 1
Posted: Friday, September 26, 2014
Problem 1 (5 points): In class we introduced the Hermitian adjoint X of an operator X through the dual
correspondence
X|i h|X , |i H.
(1)
Denote by [X] the ma

Phys 500, Quantum Mechanics
Homework 5 Reference Solution
Solution to Problem 1. Because H must be Hermitian, V12 is real. Energies up to second order
perturbation are given by
En =
En(0)
+ hn
(0)
|V |n
(0)
2
i+
X hk (0) |V |n(0) i2
k6=n
so
(0)
E1 = E1 +

T-symmetry
In theoretical physics, T-symmetry is the theoretical symmetry of physical laws under a time reversal
transformation:
T : t 7 t.
Although in restricted contexts one may nd this symmetry, the observable universe itself does not show
symmetry und

Phys 500, Quantum Mechanics
Homework 3
Posted: Tue, Oct 16, 2012 Due: Mon, Oct 22, 2012, 1PM.
Problem 1: Quantum states and tensor product Hilbert spaces. (5 points) How
many physically relevant real parameters are required to specify the state vector of

Mixed, pure, and entangled quantum states.
Density matrix
Quantum Information and Quantum Optics course, 2013
Goran Johansson, Thilo Bauch, Jonas Bylander
Chalmers, MC2
September 24, 2013
The density operator or density matrix is a more general way of des

Phys 500, Quantum Mechanics
Homework 1
Posted: Wed, September 17, 2014 Due: Fri, September 26, 2014, 1PM.
Problem 1 (5 points): In class we introduced the Hermitian adjoint X of an operator X
through the dual correspondence
X|i h|X , |i H.
(1)
Denote by [

Phys 500, Quantum Mechanics
Homework 2
Posted: Wed, October 1, 2014 Due: Fri, October 10, 2014, 1PM.
Problem 1 (5 points). Show that the Gaussian wave packet |i with
x2
2 1/4
(x) := hx|i = (2d )
exp 2
4d
p
p
saturates the Heisenberg uncertainty relation,

Lecture notes for Phys 500 QM I
*
Fall term 2013
From: Robert Raussendorf, November 30, 2013.
Abstract. These lecture notes cover material which is not in Sakurais book. We discuss probabilistic mixtures of quantum states, density operators and their prop

Phys 500, Quantum Mechanics
Reference solution for Homework 2
Posted: October 12, 2014
Problem 1 (5 points). Show that the Gaussian wave packet |i with
x2
(1)
(x) := hx|i = (2d2 )1/4 exp 2
4d
p
p
saturates the Heisenberg uncertainty relation, i.e., h(x)2

Phys 500, Quantum Mechanics
Homework 5
Posted: Fri, Nov 21, 2014 Due: Fri, Nov 28, 2014, 1PM.
Problem 1: Time-independent perturbation theory. (5 points) Consider the Hamiltonian H = H0 + V of a two-state system
!
(0)
E1
0
0 V12
Ho =
, V =
,
(0)
V12 0
0 E

Phys 500, Quantum Mechanics
Reference solution for Homework 1
Posted: Wednesday, September 25, 2013
Problem 1 (5 points): In class we introduced the Hermitian adjoint X of an operator X through the dual
correspondence
X|i ! h|X , 8 |i 2 H.
(1)
Denote by [

8. WKB Approximation
The WKB approximation, named after
Wentzel, Kramers, and Brillouin, is a method
for obtaining an approximate solution to a
time-independent one-dimensional differential
equation, in this case the Schr
odinger
equation. Its principal a

Phys 500, Quantum Mechanics
Homework 4 Reference Solution
November 25, 2012
Solution to Problem 1: (a). First, Sz
=
[H, Sz ] =
n 1
g~3 X
8
i=1
=
=
"
n 1
g~3 X h
8
i=1
n 1
g~3 X
8
= 0.
(i) (i+1)
x x
~
2
+
Pn
k=1
(k)
z .
(i) (i+1)
y
y
Thus,
+
(i) (i+1)
,
z

Kramers theorem
1 See also
In quantum mechanics, the Kramers degeneracy theorem states that for every energy eigenstate of a timereversal symmetric system with half-integer total spin,
there is at least one more eigenstate with the same energy. In other w

Phys 500, Quantum Mechanics
Reference Solution for Homework 3
November 6, 2013
Problem 1 (5 points): Consider a particle subject to a one-dimensional simple harmonic oscillator
potential. Suppose at t = 0 the state vector is given by
!
aP
exp
i
|0i,
~
fo

Phys 500, Quantum Mechanics
Reference Solution for Homework 3
October 24, 2012
Problem 1: Quantum states and tensor product Hilbert spaces. (5 points) How
many physically relevant real parameters are required to specify the state vector of n spin
1/2 part

Lecture notes for Phys 500 QM I
*
2014
From: Robert Raussendorf, October 8, 2014.
Abstract. These lecture notes cover material which is not in Sakurais book. We discuss probabilistic mixtures of quantum states, density operators and their properties, tens

Phys 500, Quantum Mechanics
Homework 5 Reference Solution
November 26, 2012
The problems are appended for reference.
Solution to Problem 1: If the barrier is of infinite height, every energy level is two-fold degenerate. For finite barrier height, all deg

Phys 500, Quantum Mechanics
Homework 5 Reference Solution
The problems are appended for reference.
Solution to Problem 1: If the barrier is of infinite height, every energy level is two-fold degenerate. For finite barrier height, all degeneracies are lift

Phys 500, Quantum Mechanics
Reference solution for Homework 2
Posted: Tuesday, October 16, 2012
Problem 1 (5 points). By considering position eigenkets |xi, with X|xi
= x|xi, show that the
momentum operator P is the generator of translations, i.e.,
i
exp

Phys 500, Quantum Mechanics
Reference solution for Homework 2
Posted: October 19, 2013
Problem 1. (3 points) Demonstrate the following three properties of the trace Tr A :=
with B an ONB.
P
|ii2B hi|A|ii,
1. Tr A is independent of the choice of ONB B.
2.

Phys 500, Quantum Mechanics
Homework 3
Posted: Sun, October 12, 2014 Due: Wed, October 22, 2014, 1PM.
Problem 1. (3 points) Consider the P
density operator E , specified by an ensemble E =
n
cfw_(pi , |i i), i = 1, ., n, namely E =
i=1 pi |i ihi |. Show t

Phys 500, Quantum Mechanics
Reference solution for Homework 1
Posted: Wednesday, October 3rd, 2012
Problem 1 (5 points): In class we introduced the Hermitian adjoint X of an operator X through the dual
correspondence
X|i ! h|X , 8 |i 2 H.
(1)
Denote by [X

Phys 500, Quantum Mechanics
Intro Test Reference Solution
1
Things you should know
Question 1. (a) List the eigenvalues and eigenvectors
0
0
1
of the matrix
0 1
0 0 .
0 0
(b) How is a measurement of an observable O described in quantum mechanics? What ar

Phys 500, Quantum Mechanics
Homework 1
Posted: Mon, September 17, 2012 Due: Fri, September 28, 2012, 1PM.
Problem 1 (5 points): In class we introduced the Hermitian adjoint X of an operator X
through the dual correspondence
X|i h|X , |i H.
(1)
Denote by [

Phys 500, Quantum Mechanics
Homework 5
Posted: Sat, Nov 10, 2012 Due: Fri, Nov 23, 2012, 1PM.
Problem 1. (5 points) Consider a symmetric rectangular double-well potential,
, for |x| > a + b,
0, for a < |x| < a + b,
V =
V0 > 0, for |x| < a.
Assuming that t

Phys 500, Quantum Mechanics
Homework 4
* The Clebsch-Gordan Issue *
Posted: Wed, Oct 31, 2012. Due: Fri, Nov 9, 2012, 1PM.
Problem 1: The Heisenberg spin chain and SU (2)-symmetry. (5 points)
Remark 1: In this problem, we go back to the Heisenberg spin ch