Quantum Mechanics describes the behaviour of matter on sub-microscopic scales and, together with relativity, is one of the two foundations of modern physics. Quantum systems are often described as having both wave-like and particle-like aspects to their behaviour, and are famous for producing results that defy common-sense intuition based on observations at everyday scales. In this module we will introduce Schrödinger's wave equation and use it to investigate the behaviour of simple quantum systems, from a free particle through to single-electron atoms. We will discuss the wavefunction, which describes the state of a system, how to interpret it, and how making a measurement changes the wavefunction. We will illustrate some of the non-intuitive behaviour of quantum systems, show how it arises, and how, in the limit of large energies, it tends towards classical behaviour. We will discuss how mathematical operators are used to represent physical quantities, and see where the Uncertainty Principle comes from. We will introduce the quantum treatment of angular momentum and show how an additional property of the electron (spin) is required to describe atomic states. We will consider the special properties of quantum states consisting of more than one electron, and show how the existence of complex chemistry depends on these.

Learning Outcomes

By the end of the module the student should be able to:

Perform approximate calculations using the de Broglie relation and the Heisenberg Uncertainty Principle;

Normalise a wavefunction;

Use wavefunctions to calculate expectation values and the probabilities of different outcomes of measurements;

Show how measurement changes the wavefunction;

Be familiar with the use of hermitian operators to represent physical quantities in quantum mechanics and the properties of their eigenvalues and eigenfunctions;

Explain the physical significance of each element of an eigenvalue equation;

Be familiar with the time-dependent and time-independent Schrödinger equations;

Solve the time-independent Schrödinger equation for simple 1-D and 3-D potential problems;

Describe the main features of the solutions for a range of problems;

Evaluate the commutator of two operators and explain its physical significance;

Be aware of how the Pauli exclusion principle arises and be able to apply it to multi-electron systems;

Describe the properties of angular momentum in quantum mechanics;

Relate the quantum numbers of atomic electrons to physical variables and know how their different values are related;

Explain why the concept of electron spin is required to explain experimental observations.