Quantum Mechanics I - 2023 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

An original assessment that is based on both examination and coursework, tests, etc., is considered as a single element for the purpose of referral; i.e., the referred mark is based on the referred examination only, discounting all previous marks. In the event that the mark for a referred assessment is lower than that of the original assessment, the original higher mark will be retained.

 

Physics Modules with PHY Codes

Referred examinations will only be available in PHY3064, PHYM004 and those other modules for which the original assessment includes an examination component - this information is given in individual module descriptors.

ELE:

 

http://newton.ex.ac.uk/redirects/ELE/phy2022