The solution of the exercise problem in Chapter 1.3.5:
For free particles, we have of course n ( q ) = e
ik n q
, n ( q ) = e ikn q , where kn is related with the
momentum p by the relation kn = pn / h . In fact, for free particles, k and p are continuum,
PHY4221 Quantum Mechanics (Fall Term Of 2004)
Problem Set 5 (Answers)
Question 1:
As following:
J
h
+
= ( x, t ) ( x, t ) +
2mi x ( x, t ) x ( x, t ) ( x, t ) x ( x, t )
t x t
= ( x, t ) ( x, t ) + ( x, t ) ( x , t )
t
t
h
2
2
+
( x, t ) x 2 ( x, t
PHY4221 Homework Assignment 1 Solution
(Written by: Leung Shing Chi)
3
1. (In this problem, we use Einsteins summation convention, i.e. Ai Ai Ai Ai , also we use
i 1
the following notation: Ai , j
Ai
)
x j
1
q
mv v q (r , t ) v A(r , t ) , we apply the
PHY4221
Quantum Mechanics I Fall term of 2004
Problem Set No.5
(Due on November 24, 2004)
1. If we dene the probability particle density and probability current density J
as follows:
(x, t) = | (x, t) |2
h
J (x, t) =
(x, t) (x, t) (x, t) (x, t)
2mi
x
x
PHYS4221
Quantum Mechanics I
Fall term of 2010
Problem Set No.1
(Due on September 20, 2010)
1. Consider a charge q of mass m moving in an electromagnetic eld. The Lagrangian
of such a system is given by
q
1
L = mv v q (r, t) + v A (r, t)
2
c
where c is th
PHY4221 Quantum Mechanics (Fall Term Of 2004)
Problem Set 6 (Answers)
Question 1:
The Spherical Harmonic Functions (SHFs) are the common eigen functions of L2 and LZ:
L2Ylm = l ( l + 1) h2Ylm
LZ Ylm = mhYlm
(1)
Or Equation (1) is sometimes written as:
L2
PHYS4221
Quantum Mechanics I
Fall term of 2010
Problem Set No.2
(Due on September 29, 2010)
1. Show that two linear operators A and B in the Hilbert space H are equal if u|A|u =
|u , |u H.
u|B
2. Given the commutator a, a = 1, evaluate the commuator an ,
PHY4221
Quantum Mechanics I Fall term of 2004
Problem Set No.6
(Due on December 1, 2004)
1. Given
Y21 (, ) =
15
sin cos ei
8
,
apply the raising operator L+ to nd Y22 (, ).
2. Determine graphically the allowed energies for the innite spherical well when
PHYS4221
Quantum Mechanics I
Fall term of 2010
Problem Set No.3
(Due on October 15, 2010)
1. If we dene the probability particle density and probability current density J as
follows:
(x, t) = | (x, t) |2
h
J (x, t) =
(x, t) (x, t) (x, t) (x, t)
2mi
x
x
PHY4221 Quantum Mechanics (Fall Term Of 2004)
Problem Set 7 (Answers)
Question 1:
The unperturbed eigen state and eigen energy are respectively:
3/ 2
l x
m y
n z
2
l(,0m) , n = sin
sin
sin
a
a
a
a
2
2
2
22
(l +m +n ) h
El(,0m) , n =
2ma 2
(1)
The ground
PHYS4221
Quantum Mechanics I Fall term of 2010
Problem Set No.4
(Due on November 3, 2010)
1. Consider a quantum particle trapped inside the one-dimensional potential well
V ( x) =
1
m 2 x2
2
,
,
for x > 0
for x < 0 .
(a) Find an upper bound of the ground
PHY4221
Quantum Mechanics I Fall term of 2004
Problem Set No.7
(Due on December 10, 2004)
Reading assignment:
Read the chapters on Time-independent Perturbation Theory, of the following
reference books:
D.J. Griths, Introduction to Quantum Mechanics
B.H.
PHYS4221
Quantum Mechanics I Fall term of 2010
Problem Set No.5
(Due on November 16, 2010)
1. Consider a two-dimensional isotropic simple harmonic oscillator of mass m0 and
frequency 0 subject to a perturbation xy :
H=
2
p2 + p2 m0 0 (x2 + y 2 )
x
y
+
+ x
PHY4221 Quantum Mechanics I (Fall 2004)
Lecture Exercise
Question 1:
Discuss how to obtain the 1 st few excited states and ground state of the following Hamiltonian
using numerical diagonalization method (or linear variational method):
H=
p2 x2
+ + x4
2
2
PHY4221
Quantum Mechanics I Fall term of 2004
Problem Set No.4
(Due on November 13, 2004)
1. Use the Heisenbergs equation of motion to show that
d
dV
xp = 2 T x
dt
dx
,
where T and V are the kinetic energy operator and potential energy operator
associated
10/PH04/QM.-
Markus Bobrowski
Abstracts on Quantum Mechanics
Ehrenfests Theorem and Quantum Virial Theorem
We rst state a formula representing a general form of Ehrenfests theorem:
Theorem 1 (Ehrenfest) For any quantum mechanical operator, the equation
i
Solution of the exercise in the Simple Harmonic Oscillator:
Most of you may have been familiar with the following relation:
1
1+
x | n =
x | a + n 1 =
a x | n 1 ,
n
n
(1)
So we could go further and get:
1+
x | n =
a x | n 1
n
1+
=
a n 1 ( x )
n
d
m x h
1
PHYS4221
Quantum Mechanics I
Spring 2010
1.3 Representations
In the preceding section we have developed the basic mathematical framework as
used in Diracs formalism of quantum mechanics. We are now ready to apply the
abstract theory to tackle real physica
PHYS4221
Quantum Mechanics I
Fall 2010
2. Examples in One Dimension:
1. Free Particle
The Hamiltonian operator H of a free particle consists of the kinetic energy term
only:
p2
H=
2m
.
(1)
Then what are the eigenvalues and eigenvectors of H ? Since any ei
PHY4221 Quantum Mechanics (Fall Term Of 2004)
Problem Set 4 (Answers)
Question 1:
The calculation process is:
d
d
xp =
xp
dt
dt
=
xp +
xp + xp
t
t
t
H
H
xp +
xp + xp
ih
t
ih
1
= [ xp, H ] +
xp
ih
t
1
= [ xp, H ]
ih
=
(1)
In above the last = is
PHY4221
Quantum Mechanics I
Fall term of 2004
Problem Set No.1
(Due on October 2, 2004)
Prove the following theorems:
1. Theorem 1: If A and B are two xed noncommuting operators and is a
parameter, then
exp (A) B exp (A)
2
3
= B + [B, A] + [B, A] , A] + [
PHY4221
Quantum Mechanics I
Fall term of 2004
Problem Set No.2
(Due on October 16, 2004)
1. A free particle has the initial wave function
(x, 0) = A exp cfw_a|x|
,
where A and a are positive real constants.
(a) Normalize (x, 0). Then nd the corresponding
PHY4221 Quantum Mechanic I (Fall term of 2004)
Problem Set No. 3
(Answers)
Question 1:
(a): You could see from the time-depended Schrodinger equation (1) that, although the potential
is changed suddenly, the wave function should be continuous as a functio
PHYS4221
Quantum Mechanics I
Fall 2010
Supplementary notes on Perturbation Theory
More often than not, the vast majority of problems in quantum mechanics cannot
be solved rigorously in closed form with the present resources of mathematics. For
such proble
PHY4221
Quantum Mechanics I
Fall term of 2004
Problem Set No.3
(Due on November 1, 2004)
1. A particle of mass m is in the ground state of the innite square well. Suddenly
the well expands to twice its original size the right wall moving from L to
2L leav