Undergraduate Course: Introduction to Mathematical Physics (PHYS08062)
Course Outline
| School | School of Physics and Astronomy |
College | College of Science and Engineering |
| Credit level (Normal year taken) | SCQF Level 8 (Year 2 Undergraduate) |
Availability | Available to all students |
| SCQF Credits | 20 |
ECTS Credits | 10 |
| Summary | The course teaches the principles of Newtonian mechanics, special relativity and quantum physics, alongside the necessary mathematical tools of differential equations, linear algebra, geometry and symmetry. It focuses on deriving results from first principles and aims at strengthening the student's problem-solving skills. The course consists of lectures to present new material, and workshops to develop understanding, familiarity and fluency. It provides a suitable preparation for Junior Honours Mathematical Physics courses, in particular Lagrangian Dynamics, Electromagnetism and Relativity, and Principles of Quantum Mechanics. |
| Course description |
The course will cover the following syllabus:
Introduction to mathematical reasoning (8 lectures)
- basic logic, mathematical proofs, including proof by contradiction and induction.
- differential equations, existence and uniqueness theorem, separable equations and substitution, first order linear equations, existence and uniqueness, integrating factors.
- several variable calculus: partial derivatives, change of variables, polar, cylindrical and spherical polar co-ordinates, Jacobians.
- linear algebra: expansion in a basis, orthogonality and completeness, inner and outer products, real and complex spaces, orthogonal and unitary transformations, diagonalization of real symmetric and Hermitian matrices.
Classical Dynamics (12 lectures)
- Introduction to dynamics: Classical observables, Newton's laws, kinetic and potential energy, energy conservation, momentum conservation, translational symmetry.
- Simple harmonic motion, equation of motion, kinetic and potential energy, period. Principle of superposition. Simple pendulum. Oscillatory motion in a general one-dimensional potential. Forced damped harmonic oscillator.
- Hamiltonian formulation, Hamilton¿s equations, free particles, particle in a potential, simple harmonic oscillator. Phase space.
- Coupled oscillators, normal modes, eigenvalues and eigenfunctions, transverse and longitudinal oscillations, coupled pendulums. Plane waves, complex representations.
- Dynamics in two and three dims: Newton's Laws in vector form, conservative forces, momentum and energy conservation, translational symmetry. Angular momentum conservation and rotational symmetry.
- Central forces and motion in a plane, effective potential, closed and open orbits. The orbit equation and its solutions. Kepler's laws.
Special Relativity (8 lectures)
- Definition of inertial reference frames and invariance of speed of light. Michelson Morley experiment. Observables. Synchronicity of moving clocks.
- Lorentz transformations. Time dilation and Lorentz contraction. Addition of velocities.
- Minkowski space, the light cone, worldlines, timelike, lightlike and spacelike separations, order of events, relativistic waves, frequency and wave number, Doppler effect.
- Momentum and relation to mass and energy as a relativistic property. Kinematics of relativistic scattering with massive and massless particles.
Quantum (12 lectures)
- Planck's radiation formula, Photoelectric effect, Einstein's photon theory, Radioactive decay, atomic spectra, Bohr atom. Correspondence principle.
- Compton effect, De Broglie hypothesis, electrons and photons as waves and particles. Double slit experiment.
- Hilbert space as a complex vector space, inner product, superposition of states, Born rule, probability.
- Hermitian operators, eigenvectors and eigenvalues, Hamiltonian as energy operator, time translation as a unitary operation.
- x and p as Hermitian operators, Schrodinger and Heisenberg eqns for free massive particle, equivalence to Hamilton's equations. Wave solutions. Heisenberg uncertainty.
- Heisenberg eqns for harmonic oscillator, creation and annihilation, discrete eigenvalues, interpretation in terms of waves or particles. Transition amplitudes.
The course will be delivered through four hours of lecture per week, and four hours of workshops. It will be assessed through a 3hr examination paper.
The syllabus should be read in the context of a Pre-Hon Y2 course, building on foundation laid in Pre-Hons Y1 introducing the key topics listed at appropriate level and depth. The topics should not be seen as independent and complete introductions but as part of a coherent picture. The mathematical methods, for clarity listed in separate section, are taught throughout the course in the context of the physics topics. The syllabus is designed to set the context and naturally lead into the Sem 2 course Introductory Fields and Waves.
Student Learning Experience: the learning is intended to primarily take place in the problem solving workshops where students are encouraged to discuss topics in groups and engage with workshop activities designed to enhance the understanding of mathematics and mathematical methods in the context of theoretical physics, and support development of independent problem solving skill.
|
Information for Visiting Students
| Pre-requisites | Students MUST have passed: Physics 1A AND Mathematics for Physics 2 or their equivalent. |
| High Demand Course? |
Yes |
Course Delivery Information
|
| Academic year 2026/27, Available to all students (SV1)
|
Quota: 50 |
| Course Start |
Semester 1 |
Timetable |
Timetable |
| Learning and Teaching activities (Further Info) |
Total Hours:
200
(
Lecture Hours 44,
Seminar/Tutorial Hours 40,
Programme Level Learning and Teaching Hours 4,
Directed Learning and Independent Learning Hours
112 )
|
| Assessment (Further Info) |
Written Exam
80 %,
Coursework
20 %,
Practical Exam
0 %
|
| Additional Information (Assessment) |
20% Coursework, 80% written Exam, must pass. |
| Feedback |
Feedback/feedforward will be provided to students throughout the course in several ways including one-to-one discussions in workshops, in-lecture response to questions, written feedback on returned class tests and post-exam discussion sessions. |
| No Exam Information |
Learning Outcomes
On completion of this course, the student will be able to:
- Understand the foundational principles of classical mechanics, both nonrelativistic and relativistic, and quantum mechanics, and how they relate to broader physical principles.
- Be able to apply mathematical reasoning in calculus, differential equations, linear algebra and elementary geometry to develop the consequences of these basic principles.
- Devise and implement a systematic strategy for solving a simple problem by breaking it down into its constituent parts.
- Use the experience, intuition and mathematical tools learned from solving physics problems to solve a wider range of unseen problems.
- Resolve conceptual and technical difficulties by locating and integrating relevant information from a diverse range of sources.
|
Reading List
RD Gregory, Classical Mechanics (Cambridge) - first choice
KF Riley and MP Hobson, Essential Mathematical Methods for the Physical Sciences (CUP)
GR Fowles and GL Cassiday, Analytical Mechanics (Saunders)
TWB Kibble, FH Berkshire, Classical Mechanics (Imperial College Press)
WD McComb, Dynamics and Relativity (Oxford)
KF Riley and MP Hobson, Essential Mathematical Methods for the Physical Sciences (CUP)
|
Additional Information
| Graduate Attributes and Skills |
Not entered |
| Keywords | Not entered |
Contacts
| Course organiser | Dr Kristel Torokoff
Tel: (0131 6)50 5270
Email: kristel.torokoff@ed.ac.uk |
Course secretary | Mr Kieran Brodie
Tel:
Email: v1kbrodi@exseed.ed.ac.uk |
|
|
University Homepage