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# Quantum & Continuum Dynamics - MAT00049I

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• Department: Mathematics
• Module co-ordinator: Prof. Kasia Rejzner
• Credit value: 20 credits
• Credit level: I
• Academic year of delivery: 2023-24
• See module specification for other years: 2024-25

## Module summary

Quantum Dynamics begins the development of quantum mechanics, and various relevant techniques of differential equations which have numerous other applications in pure and applied mathematics.

Continuum Dynamics explores the dynamics of continuous media, focusing on elementary fluid dynamics and the motion of waves.

This lays the foundations for the full development of fluid dynamics in stages 3 and 4, as well as for modules on electromagnetism and quantum mechanics. The mathematical techniques of vector calculus are employed and further developed, as are Fourier methods.

## Related modules

• None

• None

### Additional information

Pre-requisite modules:

• Classical Dynamics
• Vector and Complex Calculus

This module is the second part of the Applied Mathematics stream, and as such must be taken with the first part (Classical Dynamics).

## Module will run

Occurrence Teaching period
A Semester 2 2023-24

## Module aims

This lays the foundations for the full development of fluid dynamics in stages 3 and 4, as well as for modules on electromagnetism and quantum mechanics. The mathematical techniques of vector calculus are employed and further developed, as are Fourier methods

## Module learning outcomes

By the end of the module, students will be able to:

1. solve the wave equation in 1 and 2 dimensions with various boundary and interface conditions and understand harmonic travelling waves and wave packets,

2. describe a variety of ideal fluid flows,

3. solve Schroedinger’s equation for simple quantum-mechanical systems,

4. apply power series methods to solve ordinary differential equations,

5. compute the spectra of the harmonic oscillator and the hydrogen atom

## Module content

Quantum Dynamics

• Introduction to quantum mechanics. Schroedinger's equation (motivated by brief discussion of Planck-Einstein and de Broglie relations). Time-independent Schroedinger equation. The one-dimensional box. Probability interpretation of the wavefunction and the orthogonality of distinct energy eigenfunctions.

• Eigenvalue problems of Sturm-Liouville type Reality of eigenvalues, orthogonality of eigenfunctions for distinct eigenvalues, eigenfunction expansions. Applications, including the heat equation and the quantum mechanics of square wells and boxes in one and three dimensions.

• Series solution methods (e.g., motivated by the one-dimensional harmonic oscillator) Power series solutions of first and second order equations. Legendre's equation, Hermite's equation etc. Regular singular points: the method of Frobenius. Application to the quantum harmonic oscillator.

• Spherical harmonics and the hydrogen atom

Introduction to Waves

• The 1D wave equation, particularly vibrations of a string. Waves on an infinitely long string: characteristics, D’Alembert’s formula. Concepts: standing waves, the wave number, the wave energy, wave packets. Waves on a semi-infinite string: boundary conditions and reflection of waves. Vibrations of a finite string: Fourier methods.

• The 2D wave equation. Vibrations of rectangular and circular membranes. Vector calculus: operators in curvilinear coordinates, particularly polar coordinates. (Limited statements on Bessel functions.)

Elementary fluid dynamics

• Continuum fields: density, velocity. Particle paths, streamlines and streaklines. Material derivative. Conservation of mass, incompressibility. 2D flows and the stream function.

• The Euler momentum equation, surface and body forces, pressure. Bernoulli’s theorem and applications.

• Vorticity equation, irrotational flow, circulation. Complex potentials. Examples.

• Kinematic and dynamic conditions at a free surface. Surface gravity waves. (Limited statements on potential flow)

## Assessment

Task Length % of module mark
Closed/in-person Exam (Centrally scheduled)
Quantum & Continuum Dynamics
3 hours 100

None

### Additional assessment information

There will be five formative assignments with marked work returned in the seminars. At least one of them will contain a longer written part, done in LaTeX.

### Reassessment

Task Length % of module mark
Closed/in-person Exam (Centrally scheduled)
Quantum & Continuum Dynamics
3 hours 100

## Module feedback

Current Department policy on feedback is available in the student handbook. Coursework and examinations will be marked and returned in accordance with this policy

## Indicative reading

Coulson, C. A. & Jeffrey, A. Waves: A Mathematical Approach to the Common Types of Wave Motion. (Longman Higher Education, 1977).

Acheson, D. J. Elementary Fluid Dynamics. (Clarendon Press, 1990).

“Quantum mechanics”, Davies, P. C. W., Betts, David S., (Chapman & Hall, 1994)

The information on this page is indicative of the module that is currently on offer. The University is constantly exploring ways to enhance and improve its degree programmes and therefore reserves the right to make variations to the content and method of delivery of modules, and to discontinue modules, if such action is reasonably considered to be necessary by the University. Where appropriate, the University will notify and consult with affected students in advance about any changes that are required in line with the University's policy on the Approval of Modifications to Existing Taught Programmes of Study.