(or equivalents).
A-level Physics is desirable in order to undertake this unit.
Aims & Learning Objectives:
The aims of this unit are to show how a mathematical model of considerable elegance may be constructed, from a few basic postulates, to describe the seemingly contradictory behaviour of the physical universe and to provide useful information on a wide range of physical problems.
After taking this unit the student should be able to:
* explain the relation between wave functions, operators and experimental observables;
* justify the need for probability distributions to describe physical phenomena;
* set up the Schrödinger equation for simple model systems;
* derive eigenstates of energy, momentum and angular momentum;
* apply approximate methods to more complex systems.
Content:
Quantum mechanical concepts and models: The "state" of a quantum mechanical system. Hilbert space. Observables and operators. Eigenvalues and eigenfunctions. Dirac bra and ket vectors. Basis functions and representations. Probability distributions and expectation values of observables.
Schrödinger's equation: Operators for position, time, momentum and energy. Derivation of time-dependent Schrodinger equation. Correspondence to classical mechanics. Commutation relations and the Uncertainty Principle. Time evolution of states. Stationary states and the time-independent Schrödinger equation.
Motion in one dimension: Free particles. Wave packets and momentum probability density. Time dependence of wave packets. Bound states in square wells. Parity. Reflection and transmission at a step. Tunnelling through a barrier. Linear harmonic oscillator.
Motion in three dimensions: Stationary states of free particles. Central potentials; quantisation of angular momentum. The radial equation. Square well; ground state of the deuteron. Electrons in atoms; the hydrogen atom. Hydrogen-like atoms; the Periodic Table.
Spin angular momentum: Pauli spin matrices. Identical particles. Symmetry relations for bosons and fermions. Pauli's exclusion principle.
Approximate methods for stationary states: Time independent perturbation theory. The variational method. Scattering of particles; the Born approximation.
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