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D J C MacKay
The energy method: Lagrangian and Hamiltonian dynamics. State-space diagrams. Conservation of angular momentum, energy, phase space volume. Perturbation methods and the simple pendulum. Dimensional analysis. Normal modes: Counting degrees of freedom. Square matrices as linear operators and in quadratic forms. Eigenvectors. Transformation of linear operators and of quadratic forms. Perturbation theory: coupling of normal modes. Weak coupling of nonlinear oscillators. Stability analysis using normal modes. Modes of molecules. Use of symmetries to find normal modes. Beats. Elasticity: Definitions of strain and stress as tensors. Young's modulus, Poisson ratio, Shear and bulk modulus. Relationship between shear, compression and extension. Central forces: Kepler's laws. Planetary orbits. Perturbations of circular orbits. Scattering. Cross sections of hard spheres and inverse square potentials. Predicting cross sections using dimensional analysis. Orbits resulting from other force laws. Orbital transfers. Gravitational slingshot. Tides. Rotating frames and fictitious forces: Centrifugal and Coriolis forces. Central force problems re-expressed as one-dimensional problems. The three-body system, the Lagrange points and Trojan asteroids. Rigid bodies: Relationship between angular velocity vector and angular momentum vector. Precession of gyroscope subjected to a torque. Examples: the Earth; the Earth-Moon system; NMR; levitron. Free precession of a rigid body. Applications of dynamical systems: The driven inverted pendulum. Different ways of driving a playground swing. Harrison's clocks. Chaotic systems.R D E Saunders
The first part of the course is specifically an introduction to the first part of the Systems and Measurement Practical Class. The rest of the course deals with material that a physicist needs to design experiments and analyse data, and also to evaluate other people's results. Examples of physics experiments, especially of modern ones, provide the context. Systems: Impedance and measurement. Filters. Operational amplifiers. Positive and negative feedback with both ideal and non-ideal amplifiers. Random errors: examples, propagation, reduction with repeated sampling. Systematic errors: examples, designs to reduce them (e.g. nulling and differencing), selection effects. Basic data handling: taking and recording data, by hand and electronically. The right plot; error bars. Digitisation, Nyquist's criterion, aliasing. Exclusion of unwanted influences: electrical and vibrational filtering; thermal and electrical shielding; phase-sensitive detection and lock-in amplifiers. Writing a scientific report. Probability distributions: binomial, Poisson and Gaussian; central limit theorem (excluding formal proof); shot noise and Johnson noise. Parameter estimation: likelihood, inference and Bayes' theorem, chi-squared, least-squares, hypothesis testing, sign test.W Allison
Driven oscillations: revision of complex representation; damped and driven oscillations; impedance and impedance matching in circuits; superposition, coherence and beats; transients. Wave motion: revision of the wave equation and harmonic wave motion; plane and spherical waves. Transverse waves on a stretched string: derivation of wave equation, wave impedance, energy propagation; polarization. Reflection and transmission: derivation of reflection and transmission coefficients; reflection and transmission of energy; impedance matching for waves; revision of standing waves in 1, 2 and 3 dimensions. Longitudinal waves: sound waves in a gas; sound waves in solids and liquids; acoustic impedance; the Doppler effect for sound waves. Fourier theory: Fourier series and waveform synthesis; Fourier transforms and wave groups; convolution; amplitude modulation and sidebands. Wave propagation: Huygens' principle; reflection and refraction; Fraunhofer diffraction by slits and gratings. Dispersive waves: dispersion relation; phase and group velocity; propagation of a wave group; water waves; guided waves on a membrane, evanescent waves; total internal reflection. Matter waves: the dispersion relation: the Schrodinger equation; wavepackets and Heisenberg's Uncertainty Principle; applications to simple systems, reflection, tunnelling.S F Gull
Introduction: Maxwell's equations; Lorentz force; revision of grad, div and curl; divergence theorem, Stokes's theorem. Electrostatics: Coulomb's law; E and V; Gauss's theorem; Laplace and Poisson equations; electric dipoles; uniqueness theorem; conducting sphere in E field, method of images; point charge near conducting sphere, line charge near conducting cylinder, capacitance of parallel cylinders; electrostatic energy; force on charged conductor: capacitors and batteries. Isotropic dielectrics; polarisation charges; Gauss's theorem; permittivity and susceptibility; properties of D and E; boundary conditions at dielectric surfaces; relationship between E and P; thin slab in field, dielectric sphere in field; images in semi-infinite dielectrics; local fields inside dielectrics; Clausius-Mossotti equation. Magnetostatics: Forces between current elements; Gauss's theorem; dipoles; magnetic scalar potential; Ampère's theorem; magnetic vector potential. Magnetic media; magnetisation; permeability and magnetic susceptibility; properties of B and H; boundary conditions at surfaces; methods of calculating B and H, magnetisable sphere in uniform field; electromagnets; magnetic circuits; diamagnetism; paramagnetisn; Langevin function; ferromagnetism; Curie-Weiss law; domains; hysteresis; permanent magnets. Changing EM fields: Electromagnetic induction, Faraday's law; magnetic energy; self-inductance; inductance of long solenoid, coaxial cylinders, parallel cylinders; mutual inductance; transformers; displacement current; Maxwell's equations; electromagnetic waves; plane waves in isotropic media; energy flow; Poynting theorem; radiation pressure and momentum; insulating media; plasmas; conditions above and below plasma frequency; evanescent fields; conducting media; skin effect. Transmission lines; characteristic impedance; coaxial, parallel-wire, strip transmission lines; power flow; terminated lines; matching: reflection and transmission coefficients; impedance of short terminated lines; impedance matching. Waveguides; E and H, s and j; TE and TM modes; geometrical treatment; dispersion relation; waveguide equation; cut-off frequency; characteristic impedance; phase and group speeds; cavity resonators; optical fibres.H P Hughes
Introduction to geometrical optics and wave propagation: Fermat's principle and the classical path in a wave system. Revision of Snell's law. Reflection and refraction at spherical boundaries. Basic properties of thin lenses. Lensmakers' formula. Geometrical optics and imaging instruments: Basic optical instruments, magnifying glass, microscope, telescopes. Brief discussion of aberrations. Wave propagation and diffraction: Revision of Huygens-Fresnel principle, Kirchhoff intregal. Conditions for interference, wavefront and amplitude division. Fraunhofer and Fresnel diffraction: Near- and far-field limits. Fraunhofer integral, revision of amplitude-phase diagrams. Aperture functions. Linear phase variation and relation to Fourier transforms - superposition and convolution. Babinet's principle. N-slit grating, chromatic resolving power, crystal as a 3D grating. Resolution of imaging systems. Quadratic phase variation and Fresnel diffraction. Linear obstructions, Fresnel integrals, Cornu spiral. Circular apertures and obstacles, Fresnel zones, zone plates, Poisson's spot. Image formation: Abbé theory of image formation, optical filtering. Diffraction contrast, dark field, phase contrast and Schlieren techniques. Resolution of telescopes, microscopes. Coherence and interferometric instruments: Interference condition and the concept of coherence. General conditions for interference, "optical stethoscope", spatial and temporal coherence. Power spectra, broadening of spectral lines. Michelson spectral interferometer. Fringe visibility, coherence length. Michelson stellar interferometer, coherence width. Other topics: Fringes of equal inclination and of equal thickness. Multiple beam interference, thin films, Fabry-Perot etalon. Polarisation; linear, elliptical, circular. Fresnel Equations, Brewster angle. Waveplates. If time: Introduction to holography, photonics.M C Payne
Failure of Classical Physics, a brief review: Quantisation of light waves; UV catastrophe, photoelectric effect, Compton scattering. Interference of light at low intensity. Atomic structure, de Broglie matter waves. Wave-Particle duality; the Uncertainty Principle: Free particle in one dimension; plane waves and wave packets; dispersion. Introduction to Operators. The Schrodinger equation. Unbound particles: Particle flux. Reflection from a potential step; total reflection. Reflection from a potential barrier; tunnelling of particles through classically forbidden regions. Potential Wells: Scattering from a potential well. Bound states. Infinite potential well; energy levels. Finite potential well. Particle in a 3D box, separation of variables. Operator Methods: Dirac notation, eigenstates and eigenvalues. The postulates of quantum mechanics. Orthogonality and completeness of eigenstates: particle in a box; degeneracy; discrete and continuous spectra. Compatible and incompatible observables: commuting operators and simultaneous eigenstates; non-commuting operators; generalised uncertainty relations; minimum-uncertainty states. The harmonic oscillator: ladder operators, eigenstates, equipartition. Time dependence: expectation values, Ehrenfest's theorem, stationary states, time evolution operator, time-energy uncertainty relation, conserved quantities. Quantum Mechanics in Three Dimensions: General formulation. Spherically symmetric systems: orbital angular momentum: angular momentum operators, eigenvalues, eigenstates, parity, rotational invariance and angular momentum conservation. The hydrogen atom and three-dimensional harmonic oscillator: quantum numbers and degeneracies. Two-particle systems: separation of centre-of-mass motion; symmetries and conservation laws. Spin and Identical Particles: Quantum mechanics of spin: Stern-Gerlach experiments, combining spin and orbital angular momentum, combining spins. Identical particle symmetry: multiparticle states; fermions and bosons; exchange operator, exclusion principle, symmetry and interacting perticles, counting states. Two-electron system: helium ground and excited states.A Howie
Basic Thermodynamic Concepts: Definitions. Thermodynamic equilibrium and the idea of temperature. Functions of state. Heat as a form of energy. The First Law of Thermodynamics. Revision of CV, Cp and pVg. A word on real gases. The Second Law of Thermodynamics: Reversible and irreversible changes. Carnot's reversible engine. Theorems on reversible cycles. The thermodynamic temperature scale. Efficiency of heat engines and heat pumps. Physical relations deduced directly from the Carnot cycle: Stefan's Law, Clausius-Clapeyron, reversible cells. Entropy: Entropy is a function of state. The law of increase of entropy. Entropy of an ideal gas. Physical relations deduced from entropy function. Joule expansion. What is entropy? Foundations of Statistical Mechanics: Quantum states of the simple harmonic oscillator, the paramagnetic salt and the ideal gas. Macrostates versus microstates of large assemblies of quantum objects. The principle of equal a priori probability. Statistical definition of temperature and entropy. The Boltzmann distribution. Heat, work and entropy in statistical mechanics. Some applications of statistical mechanics: The Maxwellian velocity distribution. The heat capacities of diatomic gases. The heat capacity of a vacuum - black body radiation. The heat capacity of a solid (Einstein theory). Paramagnetism and ferromagnetism (Weiss theory). Rubber elasticity. The electron gas - Fermi energy and distribution function (detailed balance argument). Back to Basics: Entropy at absolute zero - the Third Law. Does entropy measure disorder? A little information theory (non-examinable). Gibbs' Entropy. Time's arrow (time permitting).R H Friend
Structure of materials: Interatomic forces, survey of crystallography, amorphous/glassy solids, liquid crystals, polymers. Elastic and thermal properties of solids: Monatomic and diatomic linear chains. Boundary conditions, phonon density of states. Heat capacity, Debye model. Thermal expansion, Grüneisen parameter, thermal conductivity of insulators, scattering mechanisms: phonon-phonon (normal and Umklapp), with defects, with boundaries. Phase transitions: Van der Waals' equation, imperfect gases, law of corresponding states, nature of liquids and solids. Phase diagrams, critical points, triple points, latent heats, vapour pressure. Mean-field description of phase transitions.. Structural excitations, complex structures: Vacancies, interstitials, stacking faults, grain boundaries, dislocations, strength of solids, disclinations in liquid crystals, biological structures.P B Littlewood
These classes are designed to help you develop mathematical skills in solving problems, and so strengthen your knowledge of key material from IB coursework. This is valuable regardless of whether you intend to pursue theoretical topics in Part II (for which attendance at these courses is assumed). Although primarily intended for those taking Mathematics as well as Advanced Physics in IB, most of of the classes have also been found helpful by those taking the Mathematical Concepts in Physics course. Vectors and Vector Fields: Simplification of problems using vectors. Vector area. Divergence theorem and Stokes’ theorem. Derivation of physical law from postulates. Fourier Series and Eigenfunction Expansions: Eigenfunction analysis: the stretched string of changing length. Systems with few degrees of freedom: the problem with beer. Fourier analysis: the diffusion equation. Waves and Fourier Transforms: Fourier transforms: a problem of diffusion. Propagation of a wavepacket with dispersion. Spherical Harmonics in Electromagnetism and Fluid Mechanics: Electrostatics; expansion in multipoles. Polarization of a sphere. Field inside a spherical capacitor. Fluid Mechanics; motion of a sphere under water. Introduction to Tensors: Tensors in electrostatics. Suffix notation, summation convention. The outer product. Use of principal axes. The force on a dipole in a field. The Structure of Electromagnetic Theory: Charges and currents in vacuo. Polarisation and Magnetisation. Fields and their energy. Variational Calculus in Physics: Functionals. Euler's equation. Analytical dynamics. Method of undetermined multipliers. The mathematical shepherd. Further Applications of Vector and Tensor Calculus: The antisymmetric tensor. Rigid body rotations, principal axes, precession. Stress and strain tensors. Transformation laws. Surgery /Consultation Session. A chance to get help with any remaining difficulties from the earlier classes. Mathematical Methods Applied to Quantum Mechanics. Methods of earlier sheets illustrated in the context of Quantum Mechanics - Fourier Series, quantum waves and their dispersion, variational basis of Schrodinger's equation and the connection with the eigenvalue problem.S Withington
Vectors, fields and vector differential operators: grad, div and curl. Stokes' theorem, Green's theorem. Vector differential operators in Cartesian and non-Cartesian coordinate systems. Fourier series: Fourier series of periodic functions. Complex form of series. Even and odd functions. Gibbs' phenomenon. Integral transforms: Fourier transforms. Convolution. Parseval's theorem. Solution of linear differential equations using Fourier transforms. Differential equations: Laplace's equation, Poisson's equation, the diffusion equation, the wave equation, the Helmholtz equation, Schrodinger's equation. Boundary conditions. Separation of variables. Series solutions. Orthogonal functions. Sturm-Liouville theory. Matrices and tensors: Matrix types. Eigenvalues and eigenvectors. Diagonalization. Cartesian tensors. Vector differential operators in tensor notation. A brief introduction to functions of a complex variable: Functions as mappings. Residues and contour integration.R D E Saunders and others
These practicals follow a "systems" approach - one that features concepts common to a wide range of objects, from bus suspensions to telescopes. We look at properties like gain, non-linearity and feedback, and how noise, offsets and systematic errors can be coped with. A Head-of-Class write-up will be submitted, based on one of the experiments. Basics: Using an oscilloscope; measuring input and output impedances, frequency response and phase shift; hooking up an operational amplifier. Operational amplifiers using negative feedback: Constructing amplifiers, differentiators and integrators; investigating competing feedback loops and resonance. Hysteresis: Investigation of the non-linear phenomenon of hysteresis in three magnetic materials. Data sampling and Fourier methods: Designing, constructing and using apparatus (including a PC) to test the validity of formulae developed by Lord Rayleigh to predict the frequencies of loaded tuning forks. Signals and noise in an optical link: Constructing an optical fibre link, and extraction of the signal from the noise by phase-sensitive detection. Field-effect transistor: Investigating this non-linear device and using it to construct frequency-doubling and frequency-mixing systems.R.J. Butcher and others
The aim of this class is to provide practical demonstrations and investigations of many aspects of wave motion covered in the lectures. The class draws on material from the lecture courses on waves, electromagnetism, and optics. A wide variety of different apparatus is used reflecting the technology required to study electromagnetic waves of different wavelengths from microwaves to the optical and of material waves. Despite this apparent diversity much of the basic physics of wave motion is common to all the experiments. There is one introductory class and six assessed classes; a Head of Class write-up is also done based on one of the assessed practicals. The assessed practicals are performed in weeks 3 - 8. For these latter six practicals each class is divided into six groups which perform the experiments in a different rotation there being insufficient apparatus for all groups to do the same practical simultaneously. 0. Geometrical optics and demonstrations The first practical is not assessed and is designed to introduce some of the basic apparatus and concepts of geometric optics. This material will be familiar to most of the class, but only from school physics. Additionally, it will be possible to look at one or more demonstrations relating to geometrical optics and other topics which will be covered in the optics lecture course. 1. Fraunhofer diffraction Fraunhofer diffraction is investigated experimentally using a laser and a variety of different slits. Quantitative analysis of the experiment can be used as a sensitive test of Fraunhofer theory. Careful analysis will reveal some unexpected results. The experiments also provide a visualisation of Fourier Transforms and help to develop intuition for this important bit of mathematics. 2. Fresnel diffraction The experiment on Fresnel diffraction uses similar apparatus to the Fraunhofer experiment and also permits a quantitative verification of Fresnel theory as discussed in the optics lecture course. Additionally it is possible to study off-axis effects which are difficult to analyse theoretically. 3. Interferometers and spectroscopy Interferometry is a very powerful tool for spectroscopy and the aim of this practical is to demonstrate that extremely accurate measurements can be made. Two types of interferometer are investigated, a Michelson interferometer and a Fabry-Perot interferometer. A variety of experiments and effects can be demonstrated. This class also introduces the student to the practical problems of setting up and calibrating instrumentation. 4. Microwaves and waveguides A solid-state device, a Gunn diode, is used to generate microwaves which can then be used to investigate a wide variety of electromagnetic phenomena. Propagation in wave guides, free space and materials can be investigated. This practical is closely related to the material in the electromagnetism lecture course and also introduces a number of concepts which are of practical importance. 5. Ultrasonic waves This experiment is designed to investigate the propagation of ultrasonic sound waves in air and other fluids. Not only is it possible to verify the standard wave-like behaviour of ultrasound (diffraction, reflection etc.), but also the experiment demonstrates how it is possible to probe the kinetic properties of materials using ultrasound as a probe. 6. Wave tank A wave tank is used to investigate the propagation of waves in the interface layer between two fluids. The experiment is particularly interesting because of the dispersive nature of the system. The propagation and spectral structure of wave packets is also studied.