Winter 2017-2018

Instructor: Prof. Michael S. Vogeley
Department of Physics
Office: Disque 811
Phone: (215)895-2710
Office hours: TBA

Teaching Assistant: Joseph Fabritius
Department of Physics
Office: Disque 808
Phone: (215)895-2786
Office hours: TBA

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

Course Meetings
Course Description and Philosophy
Course Outline
Course Learning Outcomes
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 2:00-3:50 p.m. in Pearlstein Business Center 101. 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
    • (maybe) Degenerate Perturbation Theory

Course Learning Outcomes

  1. Solve for the eigenstates and energies of a single particle in three dimensions for the case of central potential.
  2. Understand the detailed structure of the wave function for Hydrogen and hydrogen-like atoms.
  3. Compute properties of systems including orbital angular momentum and spin, including magnetic moments.
  4. Solve for the eigenstates and energies of multiple particle systems such as occur in simple atoms and solids.
  5. Find most probable configurations and energies of many-particle, many-level systems using quantum statistical mechanics.
  6. Use knowledge of the eigenstates and energies of simple systems and approximation methods to solve for energies of perturbed systems.

Textbook and Reading Assignments

Required Reading: Introduction to Quantum Mechanics, 2nd edition by David J. Griffiths, 2016, (Cambridge University Press) ISBN 1107179866. 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.

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

Students with disabilities requesting accomodation and services at Drexel University need to present a current accomodation letter (AVL) to faculty before accomodations can be made. This cannot be done retroactively. AVL's are issued by the Office of Disability Services (ODS). For additional information, contact ODS at 3201 Arch St., Suite 210, 215-895-1401 (V), or 215-895-2299 (TTY).

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 19):
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 26):
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, February 2):
Griffiths 4.18, 4.22, 4.27, 4.28, 4.57

Problem Set 4 (Due Friday, February 9):
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 5 (Due Friday, February 23):
Griffiths 5.4, 5.5, 5.6 and two problems in the following PDF document: Problem Set 5 assignment (PDF)

Problem Set 6 (Due Friday, March 2):
Griffiths 5.7, 5.33, and two more problems in the following PDF document: Problem Set 6 assignment (PDF)

Problem Set 7 (Due Friday, March 9):
Griffiths 5.16, 5.17, 5.34, 5.35 (see Homework Hints below re problem 5.35).

Problem Set 8 (Due Friday, March 16):
Griffiths 5.22, 5.23, 5.24, 5.30

Problem Set Solutions

Problem Set 1 solutions(PDF)

Problem Set 2 solutions(PDF)

Problem Set 3 solutions(PDF)

Problem Set 4 solutions(PDF)

Problem Set 5 solutions(PDF)

Problem Set 6 solutions(PDF)

Problem Set 7 solutions(PDF)

Problem Set 8 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 4, part (b), it may help to know that the sum of a series of j consecutive integers is \sum_{i=1}^{j} i = j(j+1)/2.
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}_{0}=-(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).

Problem set 4: In Problem 3 (4.52), 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.
In problem 4 (4.55) note again that you only have to do parts a-d!

Problem set 7, 5.35:
In (a) use Eq. 5.45 to write the total energy in terms of N, q, and m. In part (b) you can compute the total (negative!) gravitational energy of a uniform sphere by considering how much work must be done to add an amount of mass m to the star (this work is negative) and then integrating. You should get E=-(16 \pi^2 \rho^2 R^5 G/15) where \rho is the density of the star. In part (c), add up the two energy terms and find the minimum by taking the derivative (duh). Then plug in all the numbers and check that you get the given answer. In part (d), the mass of the Sun is 1.989 X 10^{30} kg. In part (e), there are 1.602 X 10^{-19} Joules/eV. The rest energy of an electron is 0.511 MeV (memorize this!).


The midterm will be in class on Wednesday, February 14.

The final exam (Friday, March 23, 8:00-10:00am, Disque 108) 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).

Midterm exam solutions(PDF)

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 12 (no class 1/10) 4.1
2 January 17, 19 4.2, 4.3 HW1
3 January 24, 26 4.4 HW2
4 January 31, February 2 4.4 HW3
5 February 7, 9 5.1 HW4
6 February 14 (no class 2/16) 5.2 Midterm in class 2/14
7 February 21, 23 5.3 HW5
8 February 28, March 2 5.4 HW6
9 March 7, 9 5.4 HW7
10 March 14, 16 6.1 HW8
11 Exam Week Final Exam Friday, March 23 8:00-10:00am

Last update: March 16, 2018