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 /
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 )
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
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
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
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 =
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. Giv
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 t
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
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
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
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 book
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 pe
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 met
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 e
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 (Ehrenfes
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 ge
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 re
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 a
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 ] +
x
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, th
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 con
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 sudden
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 wit
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 it