# 2022/23 Taught Postgraduate Module Catalogue

### 20 creditsClass Size: 35

Module manager: Vincent Caudrelier
Email: v.caudrelier@leeds.ac.uk

Taught: Semester 2 (Jan to Jun) View Timetable

Year running 2022/23

### Pre-requisite qualifications

MATH2375, or equivalent.

### This module is mutually exclusive with

 MATH3385 Quantum Mechanics

This module is approved as an Elective

### Module summary

In the early years of the 20th century, it became clear that certain experimental results of atomic physics could not be adequately explained within the framework of the classical mechanics of Newton. This led to the development of quantum mechanics, which attained in the mid-1920s a coherent mathematical form which provided a basis for a proper understanding of atomic physics. This module provides the basic theory, explaining how in quantum mechanics the states and observables of a single-particle system is described and how predictions are made about the probable outcomes of experiments. Furthermore, at the advanced level a different formulation of quantum mechanics, due to R. Feynman, will be described, leading to the concept of path integrals, and simple examples of this approach to compute quantum propagators will be presented.

### Objectives

On completion of this module, students should be able to:
a) understand the basic principles of quantum mechanics and be able to apply them in simple physical situations;
b) prove and use basic results about inner products and Hilbert space and linear operators on them;
c) calculate the quantum mechanical wave function and probability distribution in a given state;
d) calculate quantum mechanical expectation values of observables;
e) solve simple eigenfunction problems for Hermitian operators;
f) calculate reflection and transmission coefficients in simple cases;
g) predict how the state of an undisturbed quantum system evolves with time;
h) compute quantum mechanical propagators using Feynman's approach.

### Syllabus

1. Summary of the classical theory: Hamiltonian formalism, conservation laws.
2. The need for quantum mechanics, uncertainty principle, wave-particle duality, wave packets.
3. The quantum mechanics of structureless particles: the Schrödinger equation, single-particle wave function, position and momentum representation, Heisenberg uncertainty relation and the minimal wave packet.
4. Fourier integrals and the Dirac delta-function.
5. Hilbert spaces and linear operators: unitary and Hermitian operators, differential operators, results from spectral theory.
6. The mathematical formulation of quantum mechanics; eigenvalues and eigenstates, bra- and ket notation, quantum mechanical propagator, Heisenberg versus Schrödinger picture.
7. The stationary Schrödinger equation and eigenstates of the Hamiltonian: simple one-dimensional potentials and their energy levels.
8. The quantum mechanical harmonic oscillator, orthogonal polynomials.
9. Angular momentum operator and central forces: the hydrogen atom.
10. Feynman's path integral approach to quantum propagators.

### Teaching methods

 Delivery type Number Length hours Student hours Lecture 44 1.00 44.00 Private study hours 156.00 Total Contact hours 44.00 Total hours (100hr per 10 credits) 200.00

### Private study

Studying and revising of course material.
Completing of assignments and assessments.

### Opportunities for Formative Feedback

Regular example sheets.

### Methods of assessment

Exams
 Exam type Exam duration % of formal assessment Standard exam (closed essays, MCQs etc) 3 hr 00 mins 100.00 Total percentage (Assessment Exams) 100.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated