Goals and Requirements
Method of Evaluation
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.
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.
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:
|Average of 2 in class tests||45%|
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
|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|