# TOPICS IN THEORETICAL PHYSICS - 2024/5

Module code: PHYM039

## Module Overview

This 15-credit M-Level module introduces important topics and techniques in theoretical physics that have a wide range of applications in many areas physics and engineering and which the students will not have met before. Both the mathematical techniques and their applications are covered at a level appropriate for Masters level students coming to the end of their degree and who should be able to pull many different ideas in theoretical physics together.

### Module provider

Mathematics & Physics

ROCCO Andrea (Biosciences)

### Module cap (Maximum number of students): N/A

Independent Learning Hours: 78

Lecture Hours: 26

Tutorial Hours: 10

Guided Learning: 10

Captured Content: 26

Semester 2

None

## Module content

I. Functions of complex variables

• Continuity and differentiability

• The Cauchy-Riemann conditions

• Analyticity, singularities, poles.

• Complex integration

• Cauchy's theorem

• Residues and the residue theorem

• Taylor and Laurent series

• Laplace's equation in 2D and conformal mapping

• Laplace and Fourier transforms

• Dispersion relations

II. Calculus of Variations

• Integral principles in physics

• Principle of least action and other minimisation problems

• Lagrangian mechanics

• Euler-Lagrange equations

• Applications in configuration space

• Variation subject to constraints

• Extension to functions of more than one variable

• Isoperimetric problems and Lagrange multipliers

III. Integral Transforms

• Fourier transforms

• The Dirac delta function

• Laplace transforms

• Solving differential equations with Laplace transforms

• Convolutions

## Assessment pattern

Assessment type Unit of assessment Weighting
Coursework Coursework 30
Examination End of Semester Examination (Final Unit of Assessment) - 2 hours 70

N/A

## Assessment Strategy

The assessment strategy is designed to provide students with the opportunity to demonstrate their understanding of mathematical techniques, their derivation (bookwork), their applications in physical examples, both of a type they have encountered in lectures and in the process of solving the examples in problem sheets and more original problems not encountered.

Thus, the summative assessment for this module consists of:

• A Coursework submission to be completed during the semester.

• A written end of semester Examination (Final Unit of Assessment)), two hours long, in which the student must answer 3 from 4 questions covering all areas of the course.

Formative assessment and feedback

Regular feedback on previously taught material at the beginning of a lecture and discussion of problems and issues encountered in working through the problem sheets. These sheets will be discussed during the tutorial sessions. Model solutions to all problem sheet questions are made available after the students have had sufficient time to tackle them themselves. A revision class is set at the end of the module to go through past examination papers.

## Module aims

• To provide a sound grounding two important topics mathematical physics: Complex Variable Theory and Calculus of Variations. In particular, the basic theorems, methods and applications of functions of a complex variable, a range of advanced integration techniques and theorems and their applications in a range of physical examples and variational principles in classical mechanics leading to both Lagrangian and Hamiltonian formulations.

## Learning outcomes

 Attributes Developed 001 On successful completion of this module, students will have a solid understanding of complex variable theory. KC 002 Students will have a solid grounding in both Lagrangian and Hamiltonian mechanics. KC 003 Students will have a solid understanding of the principles of calculus of variations and its application in a range of physics problems. KC 004 Students will have a solid understanding of the definition and application of integral transforms in theoretical physics problems. KC 005 On successful completion of this module, students will have developed strong problem solving skills, and will be able to apply them across a wide range of physics problems. PT

Attributes Developed

C - Cognitive/analytical

K - Subject knowledge

T - Transferable skills

P - Professional/Practical skills

## Methods of Teaching / Learning

The learning and teaching strategy is designed to teach students the practical and problem solving skills required to tackle mathematical physics questions.  As such the teaching strategy places equal importance on lecture content and practical skills. For each topic covered in lectures there is an equivalent tutorial session, in which students will have further learning opportunities to develop their hands-on skills on the topics discussed in the lectures.

The learning and teaching methods include:

• lectures
• tutorial sessions
• hours of self-study

Indicated Lecture Hours (which may also include seminars, tutorials, workshops and other contact time) are approximate and may include in-class tests where one or more of these are an assessment on the module. In-class tests are scheduled/organised separately to taught content and will be published on to student personal timetables, where they apply to taken modules, as soon as they are finalised by central administration. This will usually be after the initial publication of the teaching timetable for the relevant semester.

Upon accessing the reading list, please search for the module using the module code: PHYM039

## Other information

The School of Mathematics and Physics is committed to developing graduates with strengths in Employability, Digital Capabilities, Global and Cultural Capabilities, Sustainability, and Resourcefulness and Resilience. This module is designed to allow students to develop knowledge, skills, and capabilities in the following areas:

Resourcefulness & resilience:

The mathematical content and methodology of the module forces students to develop problem solving skills in Physics and encourages critical thinking and reflection. Further to the more standard assessment in the form of a final year exam, students are also assessed on a coursework component, where they are given 3 to 4 topics to investigate on, with the aim to analyse them more in depth by performing some dedicated research. Students take up this extra challenge very well, and show a high degree of commitment, resourcefulness and resilience.

Employability:

By enhancing the mathematical and problem-solving skills of the students, the module offers large scope for increased employability. The general methodologies reviewed in the module will serve as a solid basis for making predictions in terms of theoretical modelling of different phenomena, not only in the physical sciences, but also in other disciplines. The coursework component of the module also offers scope for independent research work, to be compiled into a professional report, thereby helping student to enhance their written presentation skills.

## Programmes this module appears in

Programme Semester Classification Qualifying conditions
Physics with Quantum Technologies MPhys 2 Optional A weighted aggregate mark of 50% is required to pass the module
Mathematics with Statistics MMath 2 Optional A weighted aggregate mark of 50% is required to pass the module
Physics with Nuclear Astrophysics MPhys 2 Optional A weighted aggregate mark of 50% is required to pass the module
Physics with Astronomy MPhys 2 Optional A weighted aggregate mark of 50% is required to pass the module
Physics MPhys 2 Optional A weighted aggregate mark of 50% is required to pass the module
Physics with Quantum Computing MPhys 2 Optional A weighted aggregate mark of 50% is required to pass the module
Physics MSc 2 Optional A weighted aggregate mark of 50% is required to pass the module
Mathematics MMath 2 Optional A weighted aggregate mark of 50% is required to pass the module
Mathematics and Physics MPhys 2 Optional A weighted aggregate mark of 50% is required to pass the module
Mathematics and Physics MMath 2 Optional A weighted aggregate mark of 50% is required to pass the module
Mathematics MSc 2 Optional A weighted aggregate mark of 50% is required to pass the module

Please note that the information detailed within this record is accurate at the time of publishing and may be subject to change. This record contains information for the most up to date version of the programme / module for the 2024/5 academic year.