Physics

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 students to the use of computational techniques in the investigation of problems in physics and develops students' skills in experimental physics within areas of optics, acoustics, electromagnetism, classical nuclear and quantum physics. Students will develop programming skills and learn a range of numerical methods commonly used in physics research.

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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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The subject offers a range of projects in modules that offer experience in laboratory techniques and computational methods; the relative weights are indicated in the module descriptions. Students must select four projects with a combined weighting that contains at least 25% Computational Physics and 25% Laboratory Physics. The laboratory projects include nuclear physics, particle physics, diffraction, electronics, atomic physics, optical physics and astronomy. The computational projects are designed to develop programming skills and to introduce a range of numerical methods commonly used in physics research will be based on model problems in physics; these may include electronic structure theory, molecular vibrations, stellar structure, quantum spin systems, large-scale magnetic systems and gravitational lensing by point masses. Some projects may be offered that merge laboratory and computational work with approximately equal weighting.

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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 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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Fourier series and Fourier transforms are introduced as a means of representing and analysing functions of physical significance. The mathematical principles of Fourier theory are developed within the physical context of Fourier optics, diffraction theory, quantum mechanics and signal processing.

The formulation of Classical Newtonian and Lagrangian mechanics is then discussed in the context of the symmetries of nature, conservation laws, Hamilton's equations and integration of the equations of motion. These principles are applied to the description of physical and mechanical systems and includes a detailed discussion of rotational and oscillatory motion, mechanical stability, collisions, scattering, diffusion and continuum mechanics.

The emphasis in this subject will be to the development of techniques for solving problems involving a wide range of physical systems, including the formulation of appropriate mathematical and computational models and the identification of approximate solutions and limiting cases. Particular emphasis will be placed on the development of techniques that have wide applicability. Illustrative examples of these underlying principles will be drawn from classical and quantum mechanics, electromagnetism and optics, electronics, geophysics, astrophysics, physical chemistry and physical biosciences.

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The subject provides an introduction to the unified picture of elementary particles and atomic nuclei - how the elementary quarks combine to form strongly interacting particles, and how two of these, the proton and neutron combine to form atomic nuclei; how quarks and their composites interact with the leptons and with each other; how we study these systems experimentally; and the exciting unanswered questions in this field of physics.

Topics covered will be selected from: quarks and leptons; strong, electromagnetic and weak interactions; symmetries and conservation laws; structure, models and properties of hadrons; structure, models and properties of nuclei; scattering and decay processes; accelerators; detectors; fission and fusion reactors; applications of nuclear and particle physics techniques; and other topics in sub-atomic physics of contemporary interest.

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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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This subject provides an introduction to astrophysics discussing the basic structure of stars, our galaxy, and the universe and introducing the most recent research questions.

Topics covered include:

• structure and evolution of stars, degenerate stars, black holes, the structure of the Milky Way and other cosmic objects, emission processes in astrophysics, high energy astrophysics, relativistic cosmology and cosmological models

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This subject will introduce topics in Theoretical Physics, including:

• Classical Field Theory: Action Principles and Noether’s Theorem. Electrodynamics and wave equations from action principle. Applications selected from: waves in media, dispersion relations, solitons, Kramers-Kronig relations, the Optical Theorem.
• Fluid Mechanics: Euler’s equation, the continuity equation and the Navier-Stokes equation. Rotational and irrotational flows. Turbulence and Reynolds’ numbers, and the onset of chaos. The Rayleigh-Taylor and other instabilities. Shock waves. The Kutta-Joukowski theorem and aerodynamics.
• Symmetry: Crystallography, point groups and space groups. Quasicrystals. Vibrational modes of molecules and molecular electronic structure. Crystal field theory. The uses of SU(2) in quantum mechanics. The SO(4) solution of the hydrogen atom. Relativistic invariance: the Lorentz and Poincare groups.

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The subject will derive the fundamentals of modern optics and apply them to optical systems. We will begin with a matrix approach to geometric ray optics and progress to Gaussian beams with particular emphasis on laser beams and optical resonators. We will review the polarization of light using Jones matrices and Mueller calculus. Interference concepts will be developed and applied to interferometers, thin films and Fabry-Perot cavities. These concepts will be used to explain lasers, from Einstein concepts and population inversion to laser gain and longitudinal mode structure, for three-level and four-level systems, and extended to cover laser dynamics, Q-switched and mode-locked systems, and femtosecond combs.

Fibre optics and applications will include microstructured fibres, coupling, dispersion, fibre amplifiers and lasers. Non-linear optics will be introduced, including coupled-wave theory, harmonic generation, parametric amplification, Pockel and Kerr effects, four-wave mixing and phase conjugation. We will also review Raman, Mie and Brillouin scattering.

Fresnel and Fraunhofer diffraction theory and the angular spectrum representation of wavefields will be reviewed with emphasis on optical imaging. We will also describe modern optical microscopy from phase imaging to optical coherence tomography and super-resolution methods including STEM, STED, SLIM and TIRF.

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This subject will introduce basic concepts in Condensed Matter Physics, the physics of solids and liquids, from a theoretical and experimental perspective. In particular, it will address the most fundamental concepts and techniques which are required to gain a basic understanding of materials. These concepts and techniques include crystal structure, reciprocal space, adiabatic approximation, free electrons, electrons in a periodic potential, insulators, conductors, semi-conductors and mean-field theory. The subject further aims to introduce some of the most basic experimental techniques in solid state physics and material research. Finally, this subject will provide a phenomenological introduction to one of the most fascinating states of matter: superconductors.

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