PHYSICS 428: QUANTUM MECHANICS III

## PHYSICS 428: QUANTUM MECHANICS III Fall 2016

### Instructor: Prof. Michael S. Vogeley Department of Physics Office: Disque 811 Email: vogeley@drexel.edu Phone: (215)895-2710 Office hours: Thursday 2:30-3:30pm and by appointment Teaching Assistant: Justin Bird Department of Physics Office: Disque 808 Email: justin.bird@drexel.edu Phone: (215)895-2786 Office hours: Thursday 4:30-5:30pm

We are living in a quantum world. Get used to it.

### ANNOUNCEMENTS:

Welcome back to Drexel and welcome to the home page of QM III. This is your resource page for information about the course, including homework assignments, exams, and solutions. This web page is also the syllabus for the course. To save paper, I will not print and distribute copies of documents in class. You may read them on the web or your computer and print out if you need.

### Course Meetings

Lectures will be given on Mondays and Wednesdays 2:00-3:50 p.m. in Curtis 452 (aka "Cardiac Heights"). Our class meetings will include lectures on the course material and problem solving. You should do the reading assignments ahead of time so that you are prepared to ask and answer questions.

### Syllabus

This web page is the syllabus. Please print this out and save it and/or bookmark this website for the future. If you're reading a printed copy, and don't remember the URL, you can find the web page at http://www.physics.drexel.edu/~vogeley/Phys428. You should check the web page frequently for updates.

### Course Description and Philosophy

Quantum Mechanics (QM hereafter) is one of the foremost intellectual achievements of the 20th century and forms much of the foundation of modern Physics. Many of the giants of Physics (Einstein, Bohr, Pauli, Dirac, Feynmann, et al.) were responsible for its development. Study hard and you will be rewarded by sharing in their insight.

In the first two quarters of our three part sequence on QM, you studied the basic equations, discussed the similarities and differences between the classical and QM descriptions, and solved some simple, typically one-dimensional problems. In the second quarter you worked on QM in three dimensions, including description of the Hydrogen atom, from which you could first see how the QM formulation yields accurate predictions of the observed phenomena, and studied multi-particle systems.

Now you're ready to delve more deeply into QM. Read through the practice problems to tune up your brains after the Summer (and the coop cycle). Then we'll quickly step up to more interesting work. Most problems more complicated than those you solved in QM I and II require use of one approximation scheme or another. Clearly, learning to do Physics is more than just memorizing equations; you need to learn methods for applying them: perturbation theory and the variational principle , for example. We'll examine the interaction of radiation with matter to see how absorption and emission of photons arises from perturbation theory. Then we will study scattering theory which describes how particles interact with each other, as in collisions in a particle accelerator. We will also consider some deep questions and experiments that strike at the core of our understanding of quantum mechanics, including the EPR paradox and Bell's Theorem.

### Course Outline

Not necessarily in chronological order:
1. Review of Principles and Simple Problems
2. Time-Independent Perturbation Theory (Ch. 6)
• Nondegenerate Perturbation Theory
• Degenerate Perturbation Theory
• The Fine Structure of Hydrogen
• The Zeeman Effect
• Hyperfine Splitting
3. Time-Dependent Perturbation Theory (Ch. 9)
• Two-level systems
• Emission and absorption of radiation (Photons!)
• Spontaneous emission
4. Variational Principle (Ch. 7)
• Theory
• Ground state of Helium
• Hydrogen molecule ion
5. Scattering Theory (Ch. 11)
• Partial wave analysis
• Phase shifts
• Born approximation
6. Deep Questions in Quantum Mechanics (Ch. 12)
• Bell's theorem
• No-clone theorem
• Schroedinger's cat

Required Reading: Introduction to Quantum Mechanics, 2nd edition by David J. Griffiths, 2005, (Pearson Prentice Hall: Upper Saddle River, NJ) ISBN 0-13-111892-7 This is the same text that was used for Quantum Mechanics I and II, so most of you should already own it. Other books that I recommend include A Modern Approach to Quantum Mechanics by John S. Townsend, Principles of Quantum Mechanics by R. Shankar, and Quantum Physics by Stephen Gasiorowicz.

See the Course Schedule for the assigned readings, which you should do before class, so that you are prepared to ask and answer questions.

I will also hand out photocopies of selected passages from other QM texts, as necessary to supplement Griffiths. Here's one from David Mermin:

Grades will be based on the following weighting of different components of the course:
Problem Sets: 30%
Class Participation (in-class problems): 5%
Midterm Exam: 25%
Final Exam: 40%

### Course Rules of Conduct

Most of this is common sense, but some folks need a gentle reminder.

Electronic distractions: Silence your cell phone or leave it home. Only phone calls (to me) from the Nobel Prize committee will be tolerated. Laptop computers may be used only for taking notes. Web surfing, texting, reading/sending email is prohibited during class. I will ask you to leave the class if you violate this rule.

Plagiarism: Use your own very large brain (you're a physicist!) and don't even think about cheating. See homework rules below. The usual University rules apply. By stepping into the classroom, you agree to abide by Drexel's policy on Academic Integrity

### Problem Sets (check for hints down below!)

There will be seven problem sets. You will have a week to a week and a half to complete each. No late homework will be accepted. Please neatly and accurately write up your solutions to these problems; the notation of QM is quite compact in places and small differences in the equations can have large differences in meaning. I will hand out solutions to the problems on or shortly after their due dates, to give you feedback as quickly as possible.

You may discuss the homework with your classmates, but you and you alone are responsible for the work that you turn in. Please write up your own solutions to the problems. Breaches of this policy will result in homework scores being divided by the number of participants.'' Second offenses may result in failure (of the class).

Use of solutions to these problems from previous years constitutes plagiarism. You must attribute (by giving the correct reference) any significant help that you receive from outside sources.

Practice Problems 1 (PDF) Just for practice - I give you the solutions!

Problem Set 1 (PDF) Due in homework hand in folder or to Justin by 4:00 p.m. Friday, September 30.

Problem Set 2 (PDF) Due in homework hand in folder or to Justin by 4:00 p.m. Friday, October 7.

Problem Set 3 (PDF) Due in homework hand in folder or to Justin by 4:00 p.m. Friday, October 14.

Problem Set 4 (PDF) Due in homework hand in folder or to Justin by 4:00 p.m. Friday, October 21.

### Problem Set Solutions

Problem Set 1 solutions(PDF)

### Hints on Problems

Problem Set 1:
In Problem 2, the point is that the wavefunction is periodic, repeating over an interval of length L. This is what was meant by the particle being "defined on an interval of length L." This does not mean that the particle is "undefined" for x>L.
Problem 2c: Note that \sqrt{1+\epsilon^2} \approx 1 + (1/2)\epsilon^2 - (1/8)\epsilon^4 + ... [There's the hint, but you must show this,]
Problem 3c: When computing integrals over x to evaluate the matrix elements, use the approximation that the limits of the integral go from -\infty to \infty, which is justified if a is much smaller than L. Then the definite integral can be written in a fundamental form you can look up.
In Problem 4, I really do mean for you to find the eigenvalues (not the eigenvectors) and to determine their magnitude. The eigenvalues may be complex, so we want the norm of each.

Problem set 2:
Problem 4: Start by considering the hyperfine splitting \Delta E = (4 g_p \hbar^4)/(3 m_p m_e \mu c^2 a_0^4) where a_0=(4\pi \epsilon_0 \hbar^2)/(\mu e^2) and \mu is the reduced mass of electron-proton system (aka the hydrogen atom).

### Exams

Any topic covered in lecture, the assigned readings, or homeworks is fair game. I will distribute a non-exhaustive list of "questions to ponder" in advance of each exam to help guide your studying.

The midterm exam will be in class during week 6 and will cover material during the five weeks or so.

The final exam will be held during exam week, date/time TBA. It will be comprehensive and half closed and half open book. You may bring a calculator to perform numerical calculations only.

### Course Schedule

Please note the following schedule of readings and assignments. This schedule may be revised, so you should recheck this web page. Notation of "HW#" indicates that a homework is due by that Friday. Exact due dates for the homework will be announced in class. You should do the indicated reading before class.

 Week Class Dates Reading Homework Exams 1 September 19, 21 practice problems, Griffiths ch. 6 2 September 26, 28 Griffiths ch. 6 HW1 3 October 3, 5 Griffiths ch. 9 HW2 4 October 12 (Drexel closed on 10/10 for Columbus Day) Griffiths ch. 9 HW3 5 October 17, 19 Griffiths ch. 11 HW4 6 October 24, 26 Griffiths ch. 11 Midterm Exam in class 10/26 7 October 31, November 2 Griffiths ch. 11 8 November 9, 11 Griffiths ch. 11, 7 HW5 9 November 16, 18 Griffiths ch. 7, 12 HW6 10 November 21 (No Class 11/23 for Thanksgiving Break) Griffiths ch. 12 and handouts 11 November 28, 30 Griffiths ch. 12 and handouts HW7 12 No Class Final Exam, TBA

### Miscellaneous

Hear Schroedinger's cat meow (this one is still alive!)

Last update: October 13, 2016.