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Mathematics IV and Quantum Physics:
Mathematics IV
Introduction to the physical significance and properties of the gradient, divergence and curl operators and to Ñ2 for use in equations of the mathematical physics.
Review of the solution of first-order ordinary differential equations: use of integrating factors.
Homogeneous and inhomogeneous ordinary second order differential equations: the source term and its physical interpretation, arbitrary constants of solution and boundary conditions.
The solution of equations with constant coefficients: the complementary function, the particular integral; the general solution, development of the D operator technique of solution, the characteristic equation, detailed solution of second order equations with constant coefficients.
Equations with functional coefficients: equations of Cauchy form, solution by series, the method of Frobenius, indicial equations; recurrence relations, convergence, the method of variation of parameters.
Introduction to equations of more than one variable: The equations of mathematical physics, Laplace's equation, the wave equation, the diffusion equation, Poisson's equation; coordinate systems, Ñ2 in Cartesian systems, arbitrary functions of solution and boundary conditions.
Discussion of the method of separable solutions: introduction to separable solutions in Cartesian coordinates and in time, the use of Fourier series.
Quantum Physics
The Schrödinger equation for a point particle (electron) moving in 1-D in a potential field V(x). Ψ as a position probability amplitude. Calculations of &ÈΨ&È2 for various complex forms of Ψ.
Probability interpretation. Probability density. de Broglie waves and the invention of the Schrödinger equation. Uncertainty principle. Connection between Schrödinger equation and classical expression for total energy of a particle.
Contrast between Newtonian and Schrödinger description of time evolution of state. Comparison of role of
Newton
's Laws and Schrödinger equation.
Calculation of average values of functions of x using probabilistic interpretation of ≥Ψ≥2. Definition of <f(x)>,Δx.
Basic ideas about linear operators. Algebra of operators. The commutator and its properties. The momentum operator and its eigenfunctions and eigenvalues. The Hamiltonian operator. Energy as eigenvalue of the Hamiltonian. The Schrödinger equation for complex systems with a classical Hamiltonian. Interpretation of Ψ for systems with many degrees of freedom. Physical interpretation of eigenfunctions and eigenvalues in terms of measurements.
The one-dimensional box (or infinite square well). Dependence of the eigenvalue spectrum on the size of the box. Degeneracy. Eigenfunctions of the position operator. Connection between eigenfunctions of the Hamiltonian and solutions of the Schrödinger equation. Stationary states. General solution of the Schrödinger equation as a superposition of energy eigenstates. Physical meaning of the expansion coefficients. Simple examples of time dependent probability amplitudes, e.g., particle initially in one half of a box. Completeness. Expectation value in terms of expansion coefficients.
Energy eigenfunctions for step potentials: Classically forbidden regions. Comparison with Newtonian predictions. Reflections and transmission coefficients. Applications to neutron scattering from a nucleus. Penetration in depth. The barrier potential. Tunnelling. Solutions with definite parity. Approximation for tunnelling probability for a barrier of arbitrary shape. a - decay.
Energy eigenfunctions for step potentials: Classically forbidden regions. Comparison with Newtonian predictions. Reflections and transmission coefficients. Applications to neutron scattering from a nucleus. Penetration depth. The barrier potential. Tunnelling. Solutions with definite parity. Approximation for tunnelling probability for a wide barrier. Ramsauer effect and neutron size resonances. Comparison with optics. α-decay.
Approximate formula for tunnelling probability for a barrier of arbitrary shape and for a Coulomb barrier. Deuteron fusion at room temperature and in the sun.
The square well potential. Bound states. Parity. Dependence of number of bound states on depth and width of the hole.
General expansion of Ψ in complete sets of eigenfunctions and physical interpretation of the coefficients. Expression for expectation value in terms of coefficients. Comparison with Fourier series. Orthogonality of eigenfunctions. Formula for expansion coefficients as an overlap integral. Deduction of formula for expectation value in terms of Ψ. Applications to simple Ψ. The Correspondence Principle. Newtonian mechanics as a limiting case of quantum mechanics. Condition for validity of Newtonian description.
Computational Mathematics and Computational Modelling:
Computational Mathematics
Discussion of algorithms for the solution of ordinary differential equations: discussion of the methods of Euler (simple, modified, improved) and Runge-Kutta and numerical studies of their accuracy for solution of first order equations. Second order equations expressed as a pair of coupled first order equations, solution of second order differential equations using the methods of Euler and Runge-Kutta, treatment of one point and two point boundary conditions. Elementary discussion of finite difference methods for the solution of partial differential equations: application to the solution of
Laplace
's equation in 2D (Cartesian coordinates), the treatment of Dirichlet (constant) and Neumann (derivative) type boundary conditions.
Computational Modelling
A six half-day computational modelling project on a topic of particular relevance to the chosen field of study or degree course. The precise nature of the projects will vary from year to year, and the student will have a degree of choice. Typical project topics include Waves in an Annular Drum, Schrödinger Equation for the Harmonic Oscillator, Three-Body Interactions, Fourier Analysis of an EEG, Neural Networks, Chaotic Billiards and The Travelling Salesman Problem.
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