Mathematical Fundamentals of Quantum Mechanics

2017-2018

Dr Judith A McGovern



About the course

This course is an alternative to PHYS30101 "Applications of Quantum Mechanics"; all students must take one or the other (but not both). The syllabus for 2017/18 is in the bluebook entry.

There is considerable overlap in material between PHYS30101 and PHYS30201, 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. In spite of the title, this is a Physics course with rather little new mathematics in it, but an ability to use mathematics confidently is expected.

The second year course PHYS20672 Complex Variables and Vector Spaces is a prerequisite for the course for non-maths-physics students (Math/Phys cover most of the necessary material in Linear Algebra). If, exceptionally, a student without the prerequisite wants to be considered for this course, they should study the material in the "prerequisites" section below over the summer and submit the two revision problem sheets for marking before the start of the semester. Please contact the lecturer in good time if you plan to do 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 from the lectures.

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.

News

Exam

The format of the exam is unchanged from previous years. However the syllabus has changed a little, and more detailed guidance will be provided in due course. As in past years there will be a formula sheet on the exam, which is unlikely to change significantly and which will be made available well in advance of this year's exam.

Lecture Notes

Table of contents

Fundamentals of Quantum Mechanics

Angular momentum and spin

The variational method and WKB approximation

Time-independent perturbation theory and the hydrogen atom

The EPR paradox and Bell's inequalities

Appendices

Useful (mostly revision) background notes covering

Index notation,

Vector spaces and functions as vectors,

Recap of 2nd year QM,

Hermite polynomials,

Basic angular momentum in QM,

Hydrogen wave functions,

Delta functions,

Gaussian integrals,

Airy functions,

Units in EM.

Examples

Examples 1

Solutions 1

Examples 2

Solutions 2

Examples 3

Solutions 3

Examples 4: As of 17th November, questions 1-9 have been covered [note for part 2 of qu 9, the WKB condition for semi-infinite wells is (n+3/4)pi].

Prerequisite material on vector spaces

The following two sections from last year's notes are now lectured in PHYS20672, and should be covered through self-study by any student who has not taken that course or covered linear algebra elsewhere. A shortened version of the material is in the background notes.

An introduction to Vector Spaces

Functions as vectors

Revision examples 1

Revision solutions 1

Revision examples 2