Mathematical Fundamentals of Quantum Mechanics


Dr Judith A McGovern

Who should take this course?

This course is an alternative to PHYS30101 (Applications of Quantum Mechanics); all students must take one or the other.

There is considerable overlap in material, but the approach in this course will be much more mathematical, and as such it prepares students for future courses in QM and quantum field theory, as well as other course in which QM is used. MPhys students who wants to keep their options open should consider taking this course.

In spite of the title, this is a Physics course with relatively little new mathematics in it after the first three weeks, but an ability to use mathematics confidently is expected. Students who have taken some previous theory courses (advanced dynamics, Lagrangian, complex variables) are more likely to be comfortable with the style of the course. Students who failed to obtain good marks in PHYS20101, PHYS20141 and PHYS2017 (or MT10212) are likely to struggle.

The course starts with directed reading of Chapter 1 of Shankar. following the outline linked below under lectures 1-4. Anyone unsure whether to take this course should start working though these notes; indeed anyone intending to take the course can ease the pressure later by getting started on this.

Textbooks and websites

Here are details of useful textbooks and websites.

The course Blackboard site is only used as a repository for some material which cannot be made publically available, and for the visualiser notes . Do check it at the start.

Examples Classes

This course will be covered in the examples classes which cover third year core, series "A".

Examples classes will start in week 3. They are not like 1st year workshops, they are more like tutorials for which you do work in advance. It is very important that students work at examples sheets in their own time, and aim to complete the sheets as best they can before the classes. Working together is encouraged!

Weeks 4, 7, 9 and 11 are "Quantum" weeks and only MFQM sheets should be done in these, they alternate with "Particle and Nuclear" weeks after the first class. Solutions will be posted after the Wednesday classes.


Now available as single files: the lecture notes and examples, solutions and handouts.

A misprint in section 1.6, "Diagonalisation of Hermitian or unitary operators", has been pointed out; it should read S_ij=<x_i|w_j>


Guidance on the format of the exam (which is unchanged from previous years) and on the relevance of questions from the pre-2013/4 exams for PHYS20602 and PHYS30201 is provided here. This includes the formula sheet on the exam. These exams, along with bottom line answers and feedback are available in the Physics UG virtual common room on Blackboard, except for the 2009 bottom-line answers for PHYS20602 which are here. Do make use of the feedback, which indicates common errors made in the past.

Lecture Notes

Section 1 Lectures 1-4: An introduction to Vector Spaces (this material is actually set as directed reading, and the lectures will concentrate on examples)

Section 2 Lectures 5-6: Functions as vectors

Section 3.1-2 Lectures 7-8: Fundamentals of Quantum Mechanics

Section 3.3-4 Lectures 9-10: Operator techniques

Section 4.1-4 Lectures 11-13: Angular momentum and spin

Section 4.5-6 Lectures 14-15: Addition of angular momentum

Section 5.1-3 Lectures 16-17: Time-independent perturbation theory

Section 5.4-6 Lectures 18-20: The hydrogen atom

Section 6 Lecture 21: The EPR paradox and Bell’s inequalities


Useful notes on Hermite polynomials, hydrogen wave functions, delta functions, Gaussian integrals and units in EM

Clebsch-Gordan coefficients and instructions on usage.

A guide to matrix representations and basis changes for those who still find them puzzling..

Details of fine structure calculation..


Examples 1: covers the material of the first four lectures.

Solutions 1

Examples 2: covers the material of lectures 5&6.

Solutions 2

Examples 3: covers the material of lectures 7-10.

Solutions 3

Examples 4: covers the material of lectures 11-15.

Solutions 4

Examples 5: covers the material of lectures 16-20.

Solutions 5