Quantum Mechanics II (PHYS*4040)

Course Information

Instruction

Office Hours: Tuesdays, 11 am – 12 pm and Wednesdays, 1 pm – 3 pm

Lectures

Course Materials

Required texts

• David J. Griffiths, Introduction to Quantum Mechanics, 2nd ed. (Pearson Prentice Hall, 2005)
• Richard L. Liboff, Introductory Quantum Mechanics, 4th ed. (Addison Wesley, 2003)

Other texts

• R. Shankar, Principles of Quantum Mechanics
• J. J. Sakurai, Modern Quantum Mechanics
• E. Merzbacher, Quantum Mechanics
• L. I. Schiff, Quantum Mechanics
• C. Cohen-Tannoudji, B. Diu, and F. Lalo¨e, Quantum Mechanics

Evaluation

Assignments are due weekly. There will be no extensions and all deadlines are hard. The use of Maple to do assignment problems or make graphs is forbidden, unless otherwise noted.

Students may discuss problems amongst themselves but their written solutions must not be shared with anyone (this would be an example of plagiarism).

Plagiarism is the act of appropriating the “...composition of another, or parts or passages of his [or her] writings, or the ideas or language of the same, and passing them off as the product of one’s own mind...” (Black’s Law Dictionary). A student found to have plagiarized will receive zero for the work concerned. Collaborators shown to be culpable will be subject to the same penalties.

Midterm test date: February 26th, 7 – 9 pm
Essay due date: March 19th, in class
Exam date: Friday, April 9th, 8:30 – 10:30 am
Medical Certificate: Required if the exam is missed.

Course Outline

In this course, students will build on the foundations of Quantum Mechanics I by developing their skills with the mathematics and formalism of quantum mechanics, by solving problems in three dimensions, and by learning how angular momentum is treated in quantum mechanics. Students will also be introduced to perturbative approaches.

Part I: Mathematical Structure of the Theory, and Measurement (4 weeks)

Griffiths Ch. 3, Appendix; Liboff 3.1-3.3, 4.3-4.6, Ch. 5, 11.1-11.3

• Measurement and postulates, fundamental problem
• Linear algebra: the language of quantum mechanics
• Structure of state space: basis, subspaces, representation of operators
• Approach to predict the results of measurements
• Degenerate subspaces and complete sets of commuting operators
• Continuous basis: position and momentum representations
• Quantum mechanics in three dimensions

Part II: Angular Momentum (5 weeks)

1) Orbital angular momentum
Griffiths 4.1 - 4.3; Liboff 9.1-9.3, 10.1-10.3, 10.6, 11.5

• Commutation relations, interpretation, consequences
• Eigenproblem for L2 and Lz, interpretation, consequences
• Eigenproblem in the position basis, measurement outcomes
• Eigenproblem in the |nlmi basis
• Application to the hydrogen atom

2) Spin angular momentum
Griffiths 4.4; Liboff 11.6, 11.8, 11.9

• Experimental observations, algebraic theory for spin, spin state space
• Spin 1/2, spinor representation
• Spin precession in a magnetic field

3) Total angular momentum of a system
Griffiths 4.4; Liboff 9.4-9.5, 11.10, 12.3, 12.5

• Multi-particle systems, direct product spaces, identical particles, symmetrization
• Addition of two angular momenta
• Two spin-1/2 particles; singlets and doublets, interpretation
• Complete description of a particle with spin and orbital angular momentum
• General case

Part III: Perturbation Theory (3 weeks)

1) Time-independent perturbation theory
Griffiths Ch. 6; Liboff 13.1-13.3

• The idea of a perturbation, formulation of the perturbation series
• How to compute approximate eigenvalues and eigenvectors
• Degenerate perturbation theory
• Example: the Stark effect in hydrogen

2) The variational method
Griffiths Ch. 7

• Context, principle of the method
• Application: the ground state of the Helium atom