Quantum Mechanics I - 2024 entry

Quantum Mechanics is one of the fundamental building-blocks of Physics. It affects profoundly the way we think about the universe and is the basis for much of condensed-matter, nuclear and statistical physics. It also has a strong influence on technological developments, for instance in optical and electronic devices. This module aims to give students a firm grounding in the subject and to prepare them for future modules such as PHY3052 Nuclear and High-Energy Particle Physics.

A student who has passed this module should be able to:

 

Module Specific Skills and Knowledge:

1. describe the definition and interpretation of the wavefunction and of operators in quantum mechanics;

2. discuss the origin of energy quantisation and quantum tunnelling effects;

3. describe the general properties of the stationary states of quantum particles confined to simple symmetric potentials;

4. perform calculations on wavefunctions, and solve the Schrödinger equation for a range of problems;

5. use time-independent perturbation theory to solve problems and interpret results;

6. explain the origin of the un-coupled set of quantum numbers for the hydrogen atom and the form of the associated eigenfunctions;

 

Discipline Specific Skills and Knowledge:

7. use the principles of quantum mechanics to solve problems;

8. explain quantum mechanics to a lay-person in an informed manner;

 

Personal and Key Transferable / Employment Skills and Knowledge:

9. construct arguments that explain observations;

10. solve problems by using mathematics;

11. use a range of resources to develop an understanding of topics through independent study.

12. meet deadlines for completion of work for problems classes and develop appropriate time-management strategies.

 

I. Introduction

Brief historical survey; recap of PHY1022; what is required of the theory; the wave equation; time-dependent Schrödinger equation

II. Wave Functions and their Interpretation

The Born probability interpretation; normalization of the wave function; first postulate; wave function of a free particle; wave function of a confined particle; Gaussian wave packets (Self-study pack): the uncertainty principle; time evolution of wave packets

III. Dynamical Variables

Observables as operators; the second postulate; the third postulate; physical significance of eigenfunctions; Schrödinger equation revisited

IV. Stationary States and the Time-Independent Schrödinger Equation

Time-independent probability distributions; the time-independent Schrödinger equation; stationary states: eigenfunctions of the Hamiltonian; example: region of constant potential; method of solution ; boundary conditions

V. Particle in a Box - the Infinite Square Well

Internal solution; boundary conditions; energy quantization; normalized wave functions

VI. The Finite Square Potential Well (Self-study pack)

Interior and exterior solutions; boundary conditions; symmetric solutions - energies and wave functions; antisymmetric solutions - energies and wave functions

VII. Flow of Particles

Probability flux; continuity equation; persistence of normalization; derivation of probability flux

VIII. Barrier Problems

Boundary conditions at a potential discontinuity; a potential step; tunnelling: reflection and transmission by a barrier; practical examples of tunnelling

IX. Quantum Measurement and the Structure of Quantum Mechanics

Properties of Hermitian operators; the superposition principle: fourth postulate; measurements of a general quantum state; commutation relations and simultaneous observables; the uncertainty principle; commutation with the Hamiltonian; summary: the postulates of quantum mechanics

X. The Quantum Harmonic Oscillator

Hamiltonian in operator form; ladder operators; eigenvalues and eigenfunctions

XI. The 3D Time-Independent Schrödinger Equation

Momentum eigenfunctions in 3D; Schrödinger equation in 3D Cartesian coordinates (Self-study pack); example: particle in a 3D box; Schrödinger equation in spherical polar coordinates

XII. Angular Momentum

Cartesian representation of angular momentum operators; commutation relations; polar representation of angular momentum operators; eigenfunctions and eigenvalues; example: Rotational energy levels of a diatomic molecule

XIII. The Hydrogen Atom

Solutions of the angular equation; solutions of the radial equation; energy eigenvalues - the hydrogen spectrum; electron density distributions

XIV. First-Order Time-Independent Perturbation Theory

Perturbation theory for non-degenerate levels; perturbation theory for degenerate levels

The following list is offered as an indication of the type & level of information that students are expected to consult. Further guidance will be provided by the Module Instructor(s).

Core text:

Rae A.I.M. (2007), Quantum Mechanics (5th edition), Chapman and Hal, ISBN 1-584-88970-5 (UL: 530.12 RAE)
 

Supplementary texts:

McMurry S.M. (1994), Quantum Mechanics, Addison Wesley, ISBN 0-201-54439-3 (UL: 530.12 MCM)

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