BSc Physics with Theoretical Physics / Course details

Mathematical Fundamentals of Quantum Mechanics (M)

For recommneded theory units following this module please see PHYS40481 and 40682.

To develop an understanding of quantum mechanics and in particular the mathematical structures underpinning it.

On completion of the course, successful students should be able to:

1. Use Dirac notation to represent quantum-mechanical states and manipulate operators in terms of their matrix elements.
2. Solve a variety of problems with model and more realistic Hamiltonians, demonstrating an ability to use the mathematical underpinning of quantum mechanics.
3. Work with angular momentum operators and their eigenvalues both qualitatively and quantitatively.
4. Use perturbation theory and other methods to find approximate solutions to problems in quantum mechanics, including the fine-structure of energy levels of hydrogen.

1. The Fundamentals of Quantum Mechanics (6 lectures)

Postulates of quantum mechanics

Time evolution: the Schrödinger equation and the time evolution operator

Ehrenfest’s theorem and the classical limit

The simple harmonic oscillator: creation and annihilation operators

Composite systems and entanglement

2. Angular Momentum (7 lectures)

General properties of angular momentum

Electron spin and the Stern-Gerlach experiment

Higher spins

Addition of angular momentum

Vector Operators

3. Approximate methods I: variational method and WKB (3 lectures)

Variational methods

WKB approximation for bound states and tunneling

4. Approximate methods II: Time-independent perturbation theory (5 lectures)

Non-degenerate and degenerate perturbation theory

The fine structure of hydrogen

External fields: Zeeman and Stark effect in hydrogen

5. The Einstein-Poldosky-Rosen “paradox” and Bell’s inequalities (1 lecture)

Feedback will be available on students’ solutions to examples sheets through examples classes, and model answers will be issued.

Shankar, R. Principles of Quantum Mechanics 2nd ed. (Plenum 1994)
Gasiorowicz, S. Quantum Physics, 3rd ed. (Wiley, 2003)
Mandl, F. Quantum Mechanics (Wiley, 1992)
Griffths, D. J. Introduction to Quantum Mechanics, 2nd ed (CUP, 2017)