Physics 5153 Homework #6
Fall 2012
Professor Bruno Uchoa
Due date: November 9, 2012
November 1, 2012
1
Harmonic oscillator
The Hamiltonian of an Harmonic oscillator is
H=
p2
1
+ m 2 x2 .
2m 2
where p is the generalized momentum. Introducing the complex ge
Statistical Mechanics: Homework 3
due Monday February 25 in class
1. Using the quantum mechanical canonical distribution nd the heat capacity C = dE/dT , in
terms of the energy moments, i.e., C = function(< E >, < E2 >, .) and how it relates to
energy dis
1
Solutions to the problems in Chapter 6 and 7
6.3 Pressure of a Fermi gas at zero temperature
The number of electrons N and the internal energy U , in the volume V , are
N =V
0
D()f ()d,
U =V
D()f ()d,
0
(1)
The Fermi distribution function f () and the d
Statistical Mechanics: Homework 2
due Friday February 18 at the beginning of class
1. How likely is it to deviate from the maximum entropy state? This simple calculation should
give you an idea.
If a point is chosen at random in an M-dimensional unit cube
Statistical Mechanics: Homework 1
due Wednesday January 30 at the BEGINNING of class
1. Random walking.Consider a one dimensional lattice, with lattice constant a. The probability of going to the right is p. The probability of going to the left is q = 1 p
Dear Students,
Send me a reply to acknowledge that you have received this message.
A pdf file is in the attachment that has Review for Weeks 9 & 10.
In addition, here is a short list of important topics:
1 The Uncertainty Principle [Chapter 9]
2 Ehrenfest
PHYS 5393: Quantum Mechanics I, Autumn 2012
The Ninth and Tenth Weeks
Announcements and Assignments
Reading: Chapters 7 and 12 in Shankars book and My notes on Harmonic Oscillator
Homework: Problem Set 8 due October 25 (Thursday)
Handouts: (a) Graded H
PHYS 5393: Quantum Mechanics I, Autumn 2012
The Sixth Week
Announcements and Assignments
Reading: Chapters 4, 5, 6, and 7 in Shankars book
Exam I: October 5 (F), 4:00 PM6:00 PM.
Lecture 12: September 25 (T)
Topics for today:
3.7 Equation of continuity
3
PHYS 5393: Quantum Mechanics I, Autumn 2012
The Fifth Week
Lecture 10: September 18 (T)
Announcements and Assignments
Reading: Chapters 4, 5, and 6 in the textbook by Shankar
Homework: Problem Set 5 due Thursday
Exams: October 5 and November 2 (F), 4:0
PHYS 5393: Quantum Mechanics I, Autumn 2012
The Fourth Week
Chapter 3 Postulates of Quantum Mechanics and Schrodinger
Equation
3.1 The Postulates
3.2 Implication of the Postulates
3.3 Expectation Value
3.4 Ehrenfests Theorem
3.5 Uncertainty Principle
3.6
PHYS 5393: Quantum Mechanics I, Autumn 2012
The Third Week
Chapter 2: Mathematical Tools
(A) Linear Vector Space
(B) Inner Product and Inner Product Space, Norm,
(C) Dirac Notation and Hilbert Space
(D) Linear and Unitary Operators
(E) Eigenvalue Equation
PHYS 5393: Quantum Mechanics I, Autumn 2012
Instructor:
Professor Chung Kao [Jung Gau]
Oce: Nielsen Hall 341
E-mail: Kao@physics.ou.edu
Web page: http:/www.nhn.ou.edu/kao/phys5393.html
Oce Hours: Monday and Wednesday, 03:30 PM05:00 PM
Class Times:
August
PHYS 5393: Quantum Mechanics I
Problem Set 10Due November 29, 2012
Problem (1): Legendre Polynomials
Starting from the Rodrigues formula, derive the orthonormal condition for Legendre
polynomials:
1
1
P (z )P (z ) dz =
2
.
2 + 1
Problem (2): The Runge-Le
PHYS 5393: Quantum Mechanics I
Problem Set 9Due November 15, 2012
Problem (1)
In the Hilbert space, the eigenvectors of L2 and Lz are denoted by |, m . They obey
the following eigenvalue equations
L2 |, m = ( + 1) 2 |, m
h
and L3 |, m = mh|, m
and have ad
PHYS 5393: Quantum Mechanics I
Problem Set 8Due October 25, 2012
Problem (1)
In a system with a harmonic oscillator, a particle starts out in
1
| (0) = ( |0 + |1 ) .
2
(1)
at t = 0 with En = h (n + 1/2).
(a) Find | (t) in terms of .
(b) Find X (0) , P (0)
PHYS 5393: Quantum Mechanics I
Problem Set 7Due October 18, 2012
Problem (1)
In the energy basis of a harmonic oscillator, we have
h
(a + a ) and
2m
X=
P = i
hm
(a a )
2
where a and a are the lowering and raising operators, respectively.
Calculate the fol
PHYS 5393: Quantum Mechanics I
Problem Set 6Due October 12, 2012
Problem (1)
Let us consider the following operators on a Hilbert space V 3 :
10
0
010
0 i 0
1
1
0 .
0 i , Lz = 0 0
Lx = 1 0 1 , Ly = i
2 010
20
0 0 1
i0
(i) Consider the state
1/2
| = 1/2
1
PHYS 5393: Quantum Mechanics I
Problem Set 5Due September 27, 2012
Problem (1): Normalization of Wave Functions
(a) Find the normalization constant A for
(x) = A exp[
i
1
(x X )2 + P x]
2ch
h
such that
| (x)|2 dx = 1
where c is a constant.
(b) Calculate
PHYS 5393: Quantum Mechanics I
Problem Set 4 Due September 20, 2012
Problem (1)
If A and B are two operators satisfying [A, B ], A] = 0, apply Mathematical Induction
and show that the relation
[Am , B ] = mAm1 [A, B ]
holds for all positive integers m.
Pr
Physics 5393
Problem Set 3 Due September 13, 2011
(1). Find the eigenvalues and the normalized eigenvectors for a matrix operator ,
101
= 0 2 0 .
101
There is a degeneracy. Apply orthonormal relations to nd the third eigenvector |c
then normalize it.
(2).