Mathematical Physics

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; holomorphic 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.

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This subject surveys the foundations of Thermal Physics and Classical Mechanics and develops the fundamental principles of Quantum Mechanics. Topics in Thermal Physics include the kinetic theory of gases, the classical laws of thermodynamics, temperature, work, heat, chemical thermodynamics and chemical potential, heat engines, refrigerators, Gibbs and Helmholtz free energies and phase changes. Topics in Classical Mechanics include a review of Newton’s Laws, the Principle of Least Action, Lagrange’s equations, Hamilton’s equations and the Legendre transform. These principles will be illustrated by application to the simple harmonic oscillator. Topics in Quantum Physics include the inadequacies of Classical Physics, matter waves and quantum behaviour, one-dimensional quantum systems, expectation values, observables, operators, quantum tunnelling, and the quantization of three-dimensional systems.

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This subject introduces Einstein’s Special Principle of Relativity and develops the fundamental principles of electromagnetism and Maxwell’s equations in differential form. Special relativity topics include the foundations of special relativity, spacetime invariance, simultaneity, and Minkowski diagrams, relativistic kinematics, the Doppler effect, relativistic dynamics, and nuclear reactions. Electromagnetism topics include the electric field (e.g. Gauss’s law in integral and differential form, scalar potential and gradient, Poisson and Laplace equations), the magnetic field (e.g. Ampere’s law in integral and differential forms), Maxwell’s equations in vacuum (integral and differential forms), Maxwell’s equations in matter (polarization, electric displacement, magnetic vector potential), time-varying electric and magnetic fields (Maxwell’s equations in general form, wave equations for E and B, plane electromagnetic wave, Poynting vector). The presentation concludes with the relativistic formulation of the Lorentz force law.

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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.

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This subject studies the fundamental concepts of functions of several variables and vector calculus. It develops the manipulation of partial derivatives and vector differential operators. The gradient vector is used to obtain constrained extrema of functions of several variables. Line, surface and volume integrals are evaluated and related by various integral theorems. Vector differential operators are also studied using curvilinear coordinates.

Functions of several variables topics include limits, continuity, differentiability, the chain rule, Jacobian, Taylor polynomials and Lagrange multipliers. Vector calculus topics include vector fields, flow lines, curvature, torsion, gradient, divergence, curl and Laplacian. Integrals over paths and surfaces topics include line, surface and volume integrals; change of variables; applications including averages, moments of inertia, centre of mass; Green's theorem, Divergence theorem in the plane, Gauss' divergence theorem, Stokes' theorem; and curvilinear coordinates.

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This subject introduces the theory of groups, which is at the core of modern algebra, and which has applications in many parts of mathematics, chemistry, computer science and theoretical physics. It also develops the theory of linear algebra, building on material in earlier subjects and providing both a basis for later mathematics studies and an introduction to topics that have important applications in science and technology.

Topics include: modular arithmetic and RSA cryptography; abstract groups, homomorphisms, normal subgroups, quotient groups, group actions, symmetry groups, permutation groups and matrix groups; theory of general vector spaces, inner products, linear transformations, spectral theorem for normal matrices, Jordan normal form.

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This subject introduces the field of mathematical analysis both with a careful theoretical framework as well as selected applications. Many of the important results are proved rigorously and students are introduced to methods of proof such as mathematical induction and proof by contradiction.

The important distinction between the real numbers and the rational numbers is emphasized and used to motivate rigorous notions of convergence and divergence of sequences, including the Cauchy criterion. These ideas are extended to cover the theory of infinite series, including common tests for convergence and divergence. A similar treatment of continuity and differentiability of functions of a single variable leads to applications such as the Mean Value Theorem and Taylor's theorem. The definitions and properties of the Riemann integral allow rigorous proof of the Fundamental Theorem of Calculus. The convergence properties of sequences and series are explored, with applications to power series representations of elementary functions and their generation by Taylor series. Fourier series are introduced as a way to represent periodic functions.

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Differential equations arise as common models in the physical, mathematical, biological and engineering sciences. This subject covers linear differential equations, both ordinary and partial, using concepts from linear algebra to understand the structure of the general solutions. It balances basic theory with concrete applications. Topics include:
- linear ordinary differential equations and initial-value problems, including systems of first-order linear ordinary differential equations;
- Taylor series solutions of linear ordinary differential equations;
- Laplace transform methods for solving dynamical models with discontinuous inputs;
- boundary-value problems for linear ordinary differential equations and their interpretation in terms of eigenvalues and eigenfunctions;
- Fourier series solutions of certain linear partial differential equations on spatially bounded domains using separation of variables and eigenfunction expansion;
- Fourier transform solutions of certain linear partial differential equations on unbounded spatial domains.

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This subject covers the material presented in MAST20009 Vector Calculus plus additional material designed to provide deeper insight into interesting areas of calculus and has a greater emphasis on mathematical rigour and proof.

This subject studies the fundamental concepts of functions of several variables and vector calculus. It develops the manipulation of partial derivatives and vector differential operators. The gradient vector is used to obtain constrained extrema of functions of several variables. Line, surface and volume integrals are evaluated and related by various integral theorems. Vector differential operators are also studied using curvilinear coordinates.

Functions of several variables topics include: limits, continuity, differentiability, the chain rule, Jacobian, implicit and inverse function theorems, Taylor polynomials and Lagrange multipliers. Vector calculus topics include: vector fields, flow lines, curvature, torsion, gradient, divergence, curl and Laplacian. Integrals over paths and surfaces topics include line, surface and volume integrals; change of variables; applications including moments of inertia, centre of mass; Green's theorem, Divergence theorem in the plane, Gauss' divergence theorem, Stokes' theorem; and curvilinear coordinates. Possible additional topics include differential geometry of surfaces.

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This subject introduces the field of mathematical analysis both with a careful theoretical framework as well as selected applications. Many of the important results are proved rigorously and students are introduced to methods of proof such as mathematical induction and proof by contradiction.

The important distinction between the real numbers and the rational numbers is emphasised and used to motivate rigorous notions of convergence and divergence of sequences, including the Cauchy criterion. Various constructions of the real numbers, for example using Dedekind cuts or by completion, are discussed and shown to be equivalent. These ideas are extended to cover the theory of infinite series, including common tests for convergence and divergence. Compactness of the unit interval is established and various consequences of compactness, such as the Extreme Value Theorem, are discussed. A similar treatment of continuity and differentiability of functions of a single variable leads to applications such as the Mean Value Theorem and Taylor’s theorem. We define the Riemann integral and explore its properties, and we prove the Fundamental Theorem of Calculus. The convergence properties of sequences and series are explored, with applications to power series representations of elementary functions and their generation by Taylor series. Fourier series are introduced as a way to represent periodic functions. Further topics may include: uniform continuity, equicontinuity, the Arzela-Ascoli theorem, and the Stone-Weierstrass theorem.

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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.

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This subject gives an example-oriented overview of various advanced topics that are important for mathematical physics and physics students, as well as being of interest to students of pure and applied mathematics. These topics include:

• Further differential equations: Bessel functions, Legendre polynomials, spherical harmonics and applications such as the Laplace/Schrodinger equation in polar/spherical coordinates;
• Further vector calculus: Differential forms, integration, Stokes’ theorem and applications such as Maxwell’s equations, charge conservation and Dirac monopoles;
• Hilbert spaces: L2 spaces, bounded and unbounded operators, normalisable and non-normalisable eigenfunctions, distributions and applications to quantum theory;
• Group theory: Lie groups and algebras, representations and applications such as quantum spin and particle physics.

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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.

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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.

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