Module manager: Vincent Caudrelier
Email: v.caudrelier@leeds.ac.uk
Taught: Semester 2 (Jan to Jun) View Timetable
Year running 2026/27
MATH2130 Further Linear Algebra and Discrete Mathematics, AND MATH2350 Vector Calculus and Partial Differential Equations
| MATH2130 | Further Linear Algebra and Discrete Mathematics |
| MATH2350 | Vector Calculus and Partial Differential Equations |
(Advanced) Hamiltonian Systems MATH3355/5356M and (Advanced) Quantum Mechanics MATH3385/5386M
This module is not approved as an Elective
The Hamiltonian formulation of (conservative) dynamical systems is, together with the Lagrangian formulation, the framework in which the vast majority of physical systems and laws of nature are studied, both classically and quantum mechanically. Historically, the Hamiltonian formalism has been the stepping stone for the transition between classical mechanics to quantum mechanics at the beginning of last century, leading to groundbreaking discoveries that underpin much of today’s society.
This module will introduce the Hamiltonian formalism through its geometric and algebraic aspects: phase space and observables, symplectic forms, Hamiltonian vector fields, Lie bracket, Poisson brackets, canonical coordinates, canonical transformations. A strong emphasis will be placed on the role and structure of symmetries and constants of motion, whose interplay is embodied in a Hamiltonian version of the famous Noether theorem. The conditions under which a Hamiltonian system is Liouville integrable will be discussed. Building on this, the module will then introduce the notion of canonical or Dirac quantisation which postulates how the fundamental concepts of the Hamiltonian formalism are promoted to quantum mechanics. The most important ones are: the phase space is traded for a Hilbert space, the observables are operators on the Hilbert space, the Poisson bracket is replaced by the commutator, and (certain) canonical transformations become unitary transformations. The need for Quantum Mechanics will be motivated by historic developments. Fundamental examples such as the quantum harmonic oscillator and a single quantum particle in various one-dimensional potentials will be discussed in detail.
On successful completion of the module students will be able to: a) Read and understand a ‘brief’ and propose an adequate solution to a specific problem. b) Use clear, concise and unambiguous language in communication. c) Assess their own strengths and weaknesses d) Use and research appropriate sources to aid e) Manage workload and deadlines, develop time management skills through prioritisation. f) Maintain and uphold academic integrity and professional ethics.
1. Historical review: from Newton to Lagrange to Hamilton 2. Hamiltonian formalism: Hamilton’s equations, Hamiltonian vector fields, canonical Poisson brackets 3. Canonical transformations, symmetries and constants of motion, Noether’s theorem 4. Vector fields and Lie bracket 5. Generalised Hamiltonian systems and Poisson brackets. Integrable systems 6. Postulates of Quantum Mechanics and canonical quantisation, Schrödinger equation and wavefunction 7. Hilbert spaces, quantum states, Hermitian and unitary operators. Expectation values 8. Examples e.g. particle in one-dimensional potentials, quantum harmonic oscillator 9. Additional topics building the above may be covered as time allows. Further details of possible topics will be delivered closer to the time that the module runs.
| Delivery type | Number | Length hours | Student hours |
|---|---|---|---|
| Lectures | 33 | 1 | 33 |
| Private study hours | 117 | ||
| Total Contact hours | 33 | ||
| Total hours (100hr per 10 credits) | 150 | ||
Formative feedback will be provided on regular example sets or other similar learning activity.
Check the module area in Minerva for your reading list
Last updated: 12/05/2026
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