# PX262 Quantum Mechanics and its Applications

## Lecturers: Gavin Bell and Nicholas d'Ambrumenil

## Weighting: 15 CATS

The first part of this year's module uses ideas, introduced in the first year module, to explore atomic structure. The module also covers the mathematical tools needed in quantum mechanics and outlines the fundamental postulates that form the basis of the theory. The module discusses the time-independent and the time-dependent Schrödinger equations for spherically symmetric and harmonic potentials, angular momentum and hydrogenic atoms.

The second half of the module looks at many-particle systems and aspects of the standard model of particle physics. The module introduces the quantum mechanics of free fermions and discusses how it accounts for the conductivity and heat capacity of metals and the state of electrons in white dwarf stars. Introducing the effect of the ionic lattice and the scattering of electrons off ions then leads to a description of the properties of semiconductors and insulators. The standard model of particle physics is a quantum field theory and beyond simple quantum mechanics. However, using ideas from quantum mechanics, it is possible to explain a number of aspects of the standard model such as antiparticles and particle oscillations.

**Aims:**

To introduce the mathematical structure of quantum mechanics and to explain how to compute expectation values for observable quantities of a system. To show how quantum theory accounts for properties of atoms, elementary particles, nuclei and solids.

**Objectives:**

To develop the foundations of quantum mechanics. At the end of the module you should:

- know the origin of the
*n,l,m*and*s*quantum numbers and be able to use the Pauli exclusion principle to explain the periodic table. - understand the significance of Hermitian operators and eigenvalue equations and be able to use the correspondence principle to find the form of a quantum mechanical operator.
- be able to use quantum mechanics to derive a description of the electron states of the hydrogen atom.
- be familiar with the free-electron model of a metal
- be aware of the different crystal lattice types and how waves in a crystal are scattered by the ions
- be able to describe the elements of the standard model and to apply simple ideas from quantum theory to explain phenomena observed in particles and nuclei

**Syllabus:**

Revision of wavefunctions, probability densities and the Schrödinger equation in 1 dimension. The hydrogen atom: orbital angular momentum, quantum numbers, probability distributions. Atomic spectra and Zeeman effect. Electron spin: Stern-Gerlach, spin quantum numbers, spin-orbit coupling, exclusion principle and periodic table. X-ray spectra.

**Formal Quantum Mechanics**

The first postulate - the wavefunction to describe the state of a system; the principle of superposition of states; Operators and their rôle in quantum mechanics; the correspondence principle; measurement, Hermitian operators and eigenvalue equations; the uncertainty principle - compatibility of measurements and commuting of operators; the time dependent Schrödinger equation.

The quantum harmonic oscillator, creation and annihilation operators.

**Angular momentum**

The angular momentum operators and their commutators; the eigenvalues of the angular momentum operators, the l and m quantum numbers; the eigenfunctions of the angular momentum operators, the Spherical Harmonics. The hydrogen atom revisited.

**Models of Matter**

Statement of the many body problem. Why do molecules, nuclei and solids form? The **free fermion model** model, the Fermi surface, density of states. Fermi-Dirac distribution. Heat capacity, magnetic susceptibility, Pauli paramagnetism, ferromagnetism. Current in quantum mechanics and conductivity in a metal. Fermion degeneracy in white dwarf and neutron stars, gravitational collapse. The **liquid drop model** of the nucleus, energy release in fission. The **crystal lattice**: Lattices as repeated cells, unit cell. Lattice types in 3D. Reciprocal lattice vectors, relation to material on Fourier series. Planes and indices. X-ray diffraction. The **nearly free electron model**, scattering of electron waves by a periodic lattice and band structure. Insulators and semiconductors. doping. Semiconductor devices, e.g. diode, LED.

**The Standard Model**

The constituents of the standard model and the use of natural units. Klein-Gordon and Dirac equations. Solution to Dirac for particle it is rest frame and for particle with zero rest mass. Antiparticles and the origin of spin, W± exchange and Fermi's contact interaction.

**Constructing Models**

Relation between quantum mechanics and linear algebra. Dirac's bra-ket notation. Modelling the ammonia clock. Neutrino oscillations, kaon decay.

**Commitment:** about 40 Lectures + problems classes

**Assessment:** 2 hour examination (85%) + assessed work (15%).

**Recommended Text:** H D Young and R A Freedman, University Physics, Pearson, AIM Rae, *Quantum Mechanics*, IOP

**Other useful books:** P.C.W. Davies and D.S. Betts, *Quantum Mechanics*, Chapman and Hall 1994; F. Mandl, *Quantum Mechanics*, John Wiley 1992, S McMurry, *Quantum Mechanics*, Addison-Wesley.

This module has a home page.

**Leads from:** PX101 Quantum Phenomena

**Leads to:** PX382 Quantum Physics of Atoms, PX395 The Standard Model, PX385 Condensed Matter Physics