This section will introduce the concepts of Schrödinger’s approach to quantum mechanics. Measurement operators for observable quantities, such as energy and momentum, will be applied to a range of problems and the solutions analysed.
Review of Classical Mechanics applied to microscopic systems.
Vector Spaces and Matrices. Matrices as operators. Hermitian matrices, operators, eigenvectors and eigenvalues. Dirac Notation. Delta functions. Functions as vectors. Representation and Basis. Inner product.
Basic principles of quantum mechanics. Wavefunctions, operators and observables. The uncertainty principle. Probability. Eigenvalues and expectation values.
Energy operators:
The Time Independent Schrodinger Equation and it justification from de Broglie. Application to simple systems. Normalisation. Expectation Values.
The Time Independent Schrodinger Equation. Superposition states. Orthonormality. Time evolution of a state.
The momentum operator, eigenvalues and expectation values
The position operator.
The Hamiltonian applied to various potential wells in 1, 2 and 3D, such as finite wells, potential steps, potential barriers and linear ramps. Degeneracy. Barrier penetration, tunnelling, transmission and reflection coefficients.
Commutation of operators, conjugate variables and the uncertainty principle.
The harmonic oscillator, its Hamiltonian, energy eigenfunctions and eigenvalues. Creation and annihilation operators. Coherent states.
The gaussian wavepacket and uncertainty.
Angular momentum. Orbital angular momentum, the Lz and L2 operators, their eigenfunctions and eigenvalues. Spin angular momentum: Pauli matrices, Sz and S2 operators, eigenvalues. Magnetic moments.
An electron in a central potential: the Hydrogen atom. Radial and tangential functions. Orbitals.
Multi particle wavefunctions, indistinguishability. Fermions and Bosons. Introduction to quantum statistics. The Pauli exclusion principle and the exchange interaction.
Time independent perturbation theory to first order.
Lectures supported by tutorials, problem sheets, discussion and self directed learning.
| Module Content & Assessment | |
|---|---|
| Assessment Breakdown | % |
| Other Assessment(s) | 35 |