## Pre-requisite modules

- None

## Co-requisite modules

- None

## Prohibited combinations

Occurrence | Teaching cycle |
---|---|

A | Autumn Term 2021-22 |

This module aims to:

introduce quantum mechanical commutators and their significance for the compatibility of measurements.

introduce the quantum mechanical treatment of angular momentum

demonstrate co-ordinate transformation from Cartesian to spherical polar co-ordinates and apply this to the angular momentum operators and time-independent Schrodinger equation find solutions of the time-independent Schrodinger equation for a spherically symmetrical potential

solve the time-independent Schroedinger equation for the Hydrogen atom (full analytical solution) and extend quantum mechanics to incorporate spin

introduce matrix mechanics, with particular application to spin-spin operators, Pauli spin matrices

discuss the theory of measurement with the Stern-Gerlach measurement of spin as an example

develop approximate methods for solving the Schrodinger equation when no analytic solutions exist, such as time-independent perturbation theory.

Understand the physical significance of commutators in terms of compatibility of measurements

Perform simple commutator algebra, in order to obtain commutators for operators expressible in terms of the position and momentum operators.

Derive operators for the angular momentum components L_x, L_y, L_z, and for L^2, in terms of position and momentum operators in Cartesian coordinates

Understand how the angular momentum operators are transformed from Cartesian into spherical polar coordinates

Derive the operators for L_z and L^2 in spherical polar co-ordinates

Derive and interpret the eigenvalues and eigenvectors of the operators for angular momentum, L_z, and L^2 in terms of possible measurement results.

Explain the use of the central force theorem for a spherically symmetric potential within the context of the time-independent Schrodinger equation written in spherical polar co-ordinates and applied to hydrogen-like atoms

Discuss the relationship between the operators L_z, L^2 and the above Hamiltonian for a hydrogen-like atom system

Apply the above to solving the full analytical eigensolution for the case of the Hydrogen atom

Reproduce and interpret a labelled diagram showing the energy levels and angular momentum states of the hydrogen atom

Provide a physical interpretation of the quantum numbers *n, l *and m_l and be able to sketch the wavefunction solutions of the hydrogen atom for a given *n, l *and m_l )

Understand the matrix formalism of quantum mechanics and apply this to the case of spin

Apply the Pauli spin matrices to find the eigenvalues and eigenvectors of spin operators

Interpret generalised Stern-Gerlach experiments in terms of eigenvector superposition, illustrating the theory of measurement.

Derive the first and second order energy corrections in non-degenerate perturbation theory, and first order eigenvector correction and apply these formulae to simple problems, e.g. anharmonic oscillators

**Syllabus**

- An introduction to quantum mechanical commutators and their significance for the compatibility of measurements.
- An introduction to the quantum mechanical treatment of angular momentum.
- Time-independent Schrödinger equation for a spherically symmetrical potential, and application of the results to the Hydrogen atom.
- Extension of quantum mechanics to incorporate spin.
- Introduction to matrix mechanics, with particular application to spin.
- Discussion of the theory of measurement as illustrated by the Stern-Gerlach measurement of spin.
- Approximate methods for solving the Schrödinger equation when no analytic solutions exist (time-independent).

Task | Length | % of module mark |
---|---|---|

Essay/coursework(NS) Quantum Mechanics II Assignment 1 |
N/A | 40 |

Essay/coursework(NS) Quantum Mechanics II Assignment 2 |
N/A | 60 |

Non-reassessable

Task | Length | % of module mark |
---|---|---|

Essay/coursework(NS) Quantum Mechanics II Assignment 1 |
N/A | 40 |

Essay/coursework(NS) Quantum Mechanics II Assignment 2 |
N/A | 60 |

Our policy on how you receive feedback for formative and summative purposes is contained in our Department Handbook.

A I M Rae: *Quantum mechanics* (McGraw-Hill) ***

R C Greenhow: *Introductory quantum mechanics* (Taylor & Francis/IoP Publishing) **

B H Bransden and C J Joachain: *Introduction to quantum mechanics *(Prentice Hall)*