Winter 2015-2016

Instructor: Prof. Michael S. Vogeley
Department of Physics
Office: Disque 811
Phone: (215)895-2710
Office hours: Thursday 2:30-3:30 p.m. and by appointment

Teaching Assistant: Justin Bird
Department of Physics
Office: Disque 808
Phone: (215)895-2786
Office hours: Thursday 4:30-5:30pm

Animation of an excited state of Hydrogen, by Drexel student Glenn Winship.

Course Meetings
Course Description and Philosophy
Course Outline
Textbook and Reading Assignments
Course rules of conduct
Problem Sets
Problem Set Solutions
Problem Hints
Course Schedule


This web site is the home page of QM II. 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 Wednesdays and Fridays 12:00-1:50 p.m. in Lebow 135. Our class meetings will be a mix of lecture and problem solving.


This web page is the syllabus. Please print this out and save it and/or bookmark this website for the future (no printed copies will be distributed). If you're reading a printed copy, and don't remember the URL, you can find the web page at 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 this second quarter of our three part sequence on QM, we'll move on to three dimensional problems, and the QM description of the Hydrogen atom, from which you could first see how the QM formulation yields accurate predictions of the observed phenomena, and begin study of multi-particle systems and (if there's time) perturbation theory.

Course Outline

Here are the topics we'll cover in QM II

  1. Quantum Mechanics in Three Dimensions (Ch. 4)
    • The Schroedinger Equation in Three Dimensions
    • The Hydrogen Atom
    • Angular Momentum
    • Spin
  2. Identical Particles (Ch. 5)
    • Two-Particle Systems
    • Atoms
    • Solids
    • Quantum Statistical Mechanics
  3. Time-Independent Perturbation Theory (Ch. 6)
    • Nondegenerate Perturbation Theory

Textbook and Reading Assignments

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 text will also be used for Quantum Mechanics III, so buy it if you do not already own a copy from QMI.

See the course outline above for the chapters that correspond to the material covered in this course.

I will also hand out photocopies of selected passages from other QM texts, as necessary to supplement Griffiths.


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

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.

Food: Our class meetings are at lunchtime and everyone has to eat sooner or later. So, if you must bring your lunch, you may do so, provided that you can still takes notes while eating it and the smell is not unbearable (or so tasty that I'm tempted to steal it - triathletes are always hungry).

Plagiarism: Use your own very large brain (you're a physicist!) and don't even think about cheating. See homework rules below.

Problem Sets (check for hints down below!)

There will be eight 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 or any other source constitutes plagiarism. You must attribute (by giving the correct reference) any significant help that you receive from outside sources.

Problem Set 1 (Due Friday, January 15):
Griffiths 4.3, 4.8, 4.9, 4.38 and the following (problem 5):
Find the l=0 energy and total wave function (radial and angular parts together and properly normalized) of a particle of mass m that is subject to the central potential V(r)=0 in the region from r=a to r=b and V=infinity elsewhere (below r=a and above r=b), thus the particle is trapped in a spherical annulus.

Problem Set 2 (Due Friday, January 22):
Griffiths 4.11, 4.13, 4.17, and the following two problems:
Problem 4: (a) For the following cases, calculate the value of r at which the radial probability density of the hydrogen atom reaches its maximum: (i) n=1, l=0, m=0 (ii) n=2, l=1, m=0 (iii) l=n-1, m=0
(b) Compare the values obtained with the Bohr radius for circular orbits.
Problem 5: Assume that the eigenstates of a hydrogen atom isolated in space are all known and designated as usual by psi_{nlm}(r, \theta, phi)=R_{nl}(r) Y_l^m(theta, phi), as we derived in class. Now suppose that the nucleus is located at a distance d from an infinite potential wall, which tends to distort the hydrogen atom. Find the explicit form of the ground state wave function of this hydrogen atom as d approaches zero. Your answer should be written in terms of the of eigenstates of unperturbed hydrogen. (Hint: choose a coordinate system in which the z axis is perpendicular to the wall.)

Problem Set 3 (Due Friday, January 29):
Griffiths 4.18, 4.22, 4.27, 4.28, 4.57

Problem Set 4 (Due Friday, February 5):
Griffiths 4.34, 4.49, 4.52, 4.55 (parts a-d only) and the following:
For a particle with spin s=1/2, find the eigenvectors and eigenvalues of the operator S_x + S_y.

Problem Set Solutions

Problem Set 1 solutions(PDF)

Problem Set 2 solutions(PDF)

Problem Set 3 solutions(PDF)

Problem Set 4 solutions(PDF)

Hints on Problems

Problem set 1: In problem 3, note that you've solved this before! See 2.29 from Quantum 1, problem set 6.In problem 5, the spherical annulus cavity, note that, in general, the sinusoidal solutions to the Schroedinger equation have a degree of freedom that we have not frequently used: a phase shift. In other words, if you have a solution like sin or cos, there is no rule that says it has to be sin(kr). It could be sin(kr + delta), which is the same as sin(k(r-c)) where c is some constant. What scale in the problem would be an obvious offset (phase shift) to help match the boundary conditions?

Problem set 2: In problem 4a(iii), you should find that the Laguerre functions are all constants in this case: L^{2n-1}_{2n-1}=-(2n-1)! In problem 5, remember that the eigenstates of Hydrogen are a complete basis for whatever the new wave functions are. If you follow the directions and make the infinite wall in the x-y plane, so that V=infinity at/below z=0, which of the \psi_{nlm} satisfy the boundary condition? Which of those has the lowest energy? Note that once you find which l,m satisfy the boundary condition, you can work backwards to find the lowest possible n.

Problem set 3: In problems 3 and 4, be careful when you multiply (vector)(matrix)(vector). The first of those is the adjoint of the column vector (make it a row vector and take the complex conjugate). In Problem 3, note that S_x can be written as a linear combination of S_+ and S_-, so first construct the matrices for those operators then construct the matrix for S_x.


The midterm will be in class on Friday, February 12.

The final exam (time and date TBA) will be roughly 1/3 from the first half of the course, 2/3 from the second half.

Both exams will be half closed and half open book (textbook, your notes, my handouts, your problem sets, my solutions).

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 that Friday at the start of class. 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 January 6, 8 4.1
2 January 13, 15 4.2, 4.3 HW1
3 January 20, 22 4.4 HW2
4 January 27, 29 4.4 HW3
5 February 3, 5 5.1 HW4
6 February 12 (no class 2/10 for Kaczamarczik Lecture) 5.2 Midterm in class 2/12
7 February 17, 19 5.3 HW5
8 February 24, 26 5.4 HW6
9 March 2, 4 5.4 HW7
10 March 9, 11 6.1 HW8
11 Exams begin on Tuesday Final Exam, TBA

Last update: February 8, 2016