Physics 711 - Fall 2001

Contact Information
Goals and Requirements
Method of Evaluation
Course Content
Course Schedule

Contact Information

Lectures: Tuesday and Thursday 9:30 AM - 10:45 AM
Lecture Room: PSC 203
Instructor: Dr. Jeeva S. Anandan
Office: PSC 703
Phone: 777-4985, 788-4302
Home Page: "Jeeva S. Anandan's Home Page"

Assisted By: Dr. Alonso Botero
Office: PSC 711
Phone: 777-2634

Goals and Requirements

The goal of this course is to present non-relativistic quantum mechanics to a student who has already taken an introductory undergraduate course in the subject.

Students are required to know how to solve simple problems, such as motion of the free particle, a particle in a finite and infinite well, the one-dimensional simple harmonic oscillator and the hydrogen atom using the Schrodinger equation. They are expected to already know how to find the energy eigenvalues and eigenfunctions for these problems. They should be familiar with the time-dependent Schrodinger equation and know how to apply it to simple two-state systems such as a spin-1/2 particle. Students are also expected to have been exposed to the WKB approximation. It is desireable for them to have some exposure to angular momentum in quantum mechanics, the elements of perturbation theory, scattering theory and identical particles. It is expected that all these topics were covered in an earlier course using wavefunctions and the Schrodinger equation.

This course will differ in a fundamental way from an introductory course. The emphasis here is not on wave-mechanical methods but on employing Dirac notation and using state vectors to describe quanta. The course deals with many of the same topics as an elementary course, but uses kets instead of wavefunctions, emphasizes operator algebra and formal techniques. Unitary operators are employed to describe transformations such as spatial translation, time evolution and rotation. Operator algebra is used to illustrate a different approach to the harmonic oscillator and to angular momentum. The theory of angular momentum is developed extensively so the student may gain familiarity with formal techniques and be comfortable with them.

A secondary but nevertheless critically important goal of this course arises from the fact that it is a core graduate course. Therefore one of the goals of this course is to prepare students for the Quantum Mechanics Section of the Admission to Candidacy Examination. This is why there are so many homework problems. Students who do well in this course can reasonably expect to do well in the Quantum Mechanics Section of the Admission to Candidacy Examination.

Methods of Evaluation

Students are evaluated through the semester using both homeworks and examinations. Roughly one-third of the grade is based on performance on homework assignments. These assignments are typically, but not always, problems from the text. Students are encouraged to seek help in solving these problems should the need arise - they may see me during office hours posted above, or at any other time that I am available in my office. Details of the grading scheme are listed below.

Homework = 35%, In Class Exams = 30%, Final Exam = 35%
you will need at least 90% for an A, 85% for a B+, 75% for a B, 70% for a C+ and 60% for a C.

You must read roughly 10 pages or 1 section of Sakurai as preparation for each lecture. Ideally, students should read material prior to a lecture, pay close attention during the lecture and ask questions if they are still unclear about anything. Do not hesitate to ask questions - even when many students in class are puzzled only one may be brave enough to ask! Don't think you are the only one who is confused and / or that asking a question reflects poorly on you in any way. Remember, grades are earned via homeworks and exams, questions are merely to understand the material better.

Two or more homework problems will be assigned every day and they will be due in one week.
you lose 5% for every lecture that the homework is late.

Attendance: Mandatory!

Course Content:

Modern Quantum Mechanics by J. J. Sakurai. (Addison-Wesley, 1994), chapters 1-3, and part of chapter 4. ISBN 0-8053-3163-8

Quantum Mechanics by Leonard I. Schiff
Quantum Mechanics by L.D.Landau and E.M.Lifshitz
Lectures on Quantum Mechanics by Gordon Bayn
Quantum Mechanics by Albert Messiah

As mentioned above, the emphasis in this course is on formal techniques and different approaches to quantum mechanics are explained.

The course begins with a description of the two-slit and Stern Gerlach experiments and their implications. We proceed to build a formal structure using Dirac notation and state vectors. Expansion coefficients, matrix representations and operators are explained, with special emphasis on hermitian and unitary operators.

Next, we focus on the time development of quantum systems. Time development is illustrated using the Schrodinger and Heisenberg pictures, using the time evolution operator, using the Schrodinger equation and using the Feynman propagator and path integral approaches. Issues of deep importance such as the Aharonov-Bohm effect and gauge invariance are introduced.

The second half of the course concentrates on developing concepts and techniques of angular momentum. We begin with rotation operators and density matrices. Later, we introduce the addition of angular momenta, a topic important both for its physical applications and for its use of operator techniques. Further, it will serve as a basis or prototype for group theoretical techniques in future exposure to quantum mechanics. The treatment concludes with a discussion of tensor operators and the Wigner Eckart theorem.

Essentially all of the material is contained in chapters 1-3 of the text by Sakurai, except for the first lecture on the two-slit experiment and its implications for Quantum Mechanics.


A problem set will be issued approximately each week and you will have a week to submit the solution.

The most important step in learning any field of physics is to solve problems. Reading the text is essential, but it would give you only a superficial understanding of the subject. Apart from the pleasure you will get in solving the problems, the understanding you will thereby gain will be with you forever.


Two tests during regular class hours and a two-hour final test only, scheduled in advance, will be given. The tests will be on what is covered in class and the homework.


Your grades will be derived from the factors below with the weighting shown:

Criteria Percentage
Average of 2 in class tests 45%
Final Exam 35%
Home Work 20%
Total 100%


You are expected to prepare for class by reading the chapter ahead of time and by working problems. You will be responsible for all material covered and/or assigned in class. Attendance of classes is mandatory. But if you should miss class, it is your responsibility to determine assignments, etc., and get them completed.

Please do not hesitate to ask questions. Any questions

Course Schedule

Week Date Topics Section comments
1 Thu. Aug. 23, 2001 Historical introduction Last day to drop/add without "W"
is August 29
2 Tue. Aug. 28, 2001 Double-Slit, Stern-Gerlach experiments 1.1
2 Thu. Aug. 30, 2001 Quantum state vectors 1.2
3 Tue. Sep. 4, 2001 Matrix representation 1.3
3 Thu. Sep. 6, 2001 Measurements, Observables 1.4
4 Tue. Sep. 11, 2001 Uncertainty relations 1.5
4 Thu. Sep. 13, 2001 Position, Momentum, and Translations 1.6
5 Tue. Sep. 18, 2001 Wave function in position and momentum space 1.7
5 Thu. Sep. 20, 2001 Schrodinger's wave equation 2.1
6 Tue. Sep. 25, 2001 Test 1, in class
6 Thu. Sep. 27, 2001 Schrodinger and Heisenberg pictures 2.2
7 Tue. Oct. 2, 2001 Simple Harmonic Oscillator 2.3
7 Thu. Oct. 4, 2001 Classical and semi-classical limits 2.4 Last day to drop without "WF"
8 Tue. Oct. 9, 2001 Feynman path integral formulation 2.4, 2.5
8 Thu. Oct. 11, 2001 Potentials and gauge transformations 2.5, 2.6 Midpoint of Semester
9 Tue. Oct. 16, 2001 No class Oct. 15,16: Fall Break
9 Thu. Oct. 18, 2001 Aharonov-Bohm effect and gauge fields 2.6
10 Tue. Oct. 23, 2001 Rotations and angular momentum 3.1
10 Thu. Oct. 25, 2001 Spin 3.2
11 Tue. Oct. 30, 2001 Test 2, in class
11 Thu. Nov. 1, 2001 The rotation group 3.3
12 Tue. Nov. 6, 2001 Density operators, pure and mixed states 3.4
12 Thu. Nov. 8, 2001 Angular momentum eigenstates and eigenvalues 3.5
13 Tue. Nov. 13, 2001 Orbital angluar momentum 3.6
13 Thu. Nov. 15, 2001 Addition of angular momenta 3.7
14 Tue. Nov. 20, 2001 EPR correlation and Bell's inequality 3.9
14 Thu. Nov. 22, 2001 No class Nov. 21-25: Thanksgiving Holidays
15 Tue. Nov. 27, 2001 Tensor operators 3.10
15 Thu. Nov. 29, 2001 Symmetries and conservation laws 4.1
16 Tue. Dec. 4, 2001 Discrete symmetries 4.2 Graduate Student Day
16 Thu. Dec. 6, 2001 Review Dec. &: Last day of classes
17 Tue. Dec. 13, 2001 Final Examination 9-11 AM, PSC 203 Final Exam

This page is maintained by Jeeva S. Anandan