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Quantum Mechanics (MATH 35500)

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Administrative Information

  1. Unit number and title: MATH 35500 Quantum Mechanics
  2. Level: H/6
  3. Credit point value: 20 credit points
  4. Year: 12/13
  5. First Given in this form: 2000/2001
  6. Unit Organiser: Stephen Wiggins
  7. Lecturer: Dr Anna Maltsev and Prof Stephen Wiggins
  8. Teaching block: 2
  9. Prerequisites: Either MATH20101 Ordinary Differential Equations 2 or MATH20402 Applied Partial Differential Equations 2

Unit aims

The aim of the unit is to provide mathematics students with a thorough introduction to nonrelativistic quantum mechanics, with emphasis on the mathematical structure of the theory.

General Description of the Unit

Quantum mechanics forms the foundation of 20th century and present-day physics, and most contemporary disciplines, including atomic and molecular physics, condensed matter physics, high-energy physics, quantum optics and quantum information theory, depend essentially upon it. Quantum mechanics is also the source and inspiration for various fields in mathematical physics and pure mathematics.

The aim of the unit is to provide mathematics students with a thorough introduction to nonrelativistic quantum mechanics, with emphasis on the mathematical structure of the theory. Additionally, in conjunction with other units, it should provide suitably able and inclined students with the necessary background for further study and research at the postgraduate level. Two relevant research fields, namely quantum chaos and quantum information theory are at present strongly represented in the Mathematics Department and in the Science Faculty as a whole.

Relation to Other Units

This unit cannot be taken by students who have taken or are taking relevant physics units at either level 2 or level 3. For mathematics students, it is a prerequisite for the Level M unit Quantum Chaos and a useful preparation for the Level M unit Quantum Information Theory.

Teaching Methods

Lectures, problem sheets

Learning Objectives

At the end of the unit the student should:

  • be familiar with the time-independent and time-dependent Schroedinger equations, and be able to solve them in simple examples
  • be familiar with the notions of Hilbert space, self-adjoint operators, unitary operators, commutation relations, understand their relevance to the mathematical formulation of quantum mechanics and be able to use the notions to formulate and solve problems
  • understand the probabilistic interpretation of quantum states, and basic aspects of the relation between classical and quantum mechanics
  • understand the quantum mechanical description of angular momentum and spin

Assessment Methods

The assessment mark for Quantum Mechanics is calculated from a 2½-hour written examination in May/June consisting of FIVE questions. A candidate's best FOUR answers will be used for assessment. Calculators are NOT permitted for this examination.

Award of Credit Points

Credit points are gained by:

  • either passing the unit,
  • or achieving an examination mark of 30 or over, and also handing in satisfactory attempts at selected questions from at least six of the weekly homework sheets.

Transferable Skills

Expressing physical axioms mathematically and analysing their consequences.

Texts

K. Hannabuss, An Introduction to Quantum Theory, Oxford 1997

Students may also find the following books interesting for further reading:

C. J. Isham, Lectures on Quantum Theory, Imperial College Press, 1995

A. Peres, Quantum Theory: Concepts and Methods, Kluwer, 1995

Syllabus

  • motivation
  • time-dependent and time-independent Schroedinger equations with characteristic examples including square wells and barriers (illustrating bound and scatttering systems and tunneling) and the harmonic oscillator
  • the mathematical structure of quantum mechanics, including Hilbert space as state space, observables as self-adjoint operators and the probabilistic interpretation of quantum states
  • commutation relations and the uncertainty principle
  • unitary transformations, including time evolution
  • measurement
  • classical/quantum correspondence
  • angular momentum and spin
  • the Einstein-Podolsky-Rosen paradox and Bells inequalities