Mathematical Physics

Subjects you could take in this major

• Complex Analysis

  Complex analysis is a core subject in pure and applied mathematics, as well as the physical and engineering sciences. While it is true that physical phenomena are given in terms of real numbers and real variables, it is often too difficult and sometimes not possible, to solve the algebraic and differential equations used to model these phenomena without introducing complex numbers and complex variables and applying the powerful techniques of complex analysis.

  Topics include:the topology of the complex plane; convergence of complex sequences and series; analytic functions, the Cauchy-Riemann equations, harmonic functions and applications; contour integrals and the Cauchy Integral Theorem; singularities, Laurent series, the Residue Theorem, evaluation of integrals using contour integration, conformal mapping; and aspects of the gamma function.

  Handbook entry for MAST30021
• Electrodynamics

  This subject provides an introduction to electrodynamics and a wide range of applications including communications, superconductors, plasmas, novel materials, photonics and astrophysics. Topics include: revision of Maxwell’s equations, strategies for solving boundary value problems for static and time-varying fields, electromagnetic fields in materials (including dielectrics, magnetic materials, conductors, plasmas and metamaterials), electromagnetic waves, derivation of geometric optics from Maxwell’s equations, guided waves, relativistic electrodynamics and the covariant formulation of electrodynamics, radiation by antennas and accelerating charged particles.

  Handbook entry for PHYC30016
• Methods of Mathematical Physics

  This subject builds on, and extends earlier, related undergraduate subjects with topics that are useful to applied mathematics, mathematical physics and physics students, as well as pure mathematics students interested in applied mathematics and mathematical physics. These topics include:

  • Special functions: Spherical harmonics including Legendre polynomials and Bessel functions, including cylindrical, modified and spherical Bessel functions;
  • Integral equations: Classification, Fourier and Laplace transform solutions, separable kernels, singular integral equations, Wiener-Hopf equations, and series solutions;
  • Further vector analysis: Differential forms, covariant derivatives, and integrating p-forms;
  • Further complex analysis: The Schwarz reflection principle, and Riemann-Hilbert problem.

  Handbook entry for MAST30031
• Metric and Hilbert Spaces

  This subject provides a basis for further studies in modern analysis, geometry, topology, differential equations and quantum mechanics.It introduces the idea of a metric space with a general distance function, and the resulting concepts of convergence, continuity, completeness, compactness and connectedness. The subject also introduces Hilbert spaces: infinite dimensional vector spaces (typically function spaces) equipped with an inner product that allows geometric ideas to be used to study these spaces and linear maps between them.

  Topics include: metric and normed spaces, limits of sequences, open and closed sets, continuity, topological properties, compactness, connectedness; Cauchy sequences, completeness, contraction mapping theorem; Hilbert spaces, orthonormal systems, bounded linear operators and functionals, applications.

  Handbook entry for MAST30026
• Quantum Physics

  Quantum mechanics plays a central role in our understanding of fundamental phenomena, primarily in the microscopic domain. It lays the foundation for an understanding of atomic, molecular, condensed matter, nuclear and particle physics.

  Topics covered include:

  • the basic principles of quantum mechanics (probability interpretation; Schrödinger equation; Hermitian operators, eigenstates and observables; symmetrisation, antisymmetrisation and the Pauli exclusion principle; entanglement)
  • wave packets, Fourier transforms and momentum space
  • eigenvalue spectra and delta-function normalisation
  • Heisenberg uncertainty principle
  • matrix theory of spin
  • the Hilbert space or state vector formation using Dirac bra-ket notation
  • the harmonic oscillator
  • the quantisation of angular momentum and the central force problem including the hydrogen atom
  • approximation techniques including perturbation theory and the variational method
  • applications to atomic and other systems.

  Handbook entry for PHYC30018
• Statistical Physics

  Statistical mechanics, the microscopic basis of classical thermodynamics, is developed in this subject. It is one of the core areas of physics, finding wide application in solid state physics, astrophysics, plasma physics and cosmology.

  Using fundamental ideas from quantum physics, a systematic treatment of statistical mechanics is developed for systems in equilibrium. The content of this subject includes ensembles and the basic postulate; the statistical basis of the second and third laws of thermodynamics; canonical, micro-canonical and grand-canonical ensembles and associated statistical and thermodynamic functions; ideal quantum gases; black body radiation; the classical limit and an introduction to real gases and applications to solid state physics.

  Handbook entry for PHYC30017