# PX441 Quantum Theory of Interacting Particles

## Lecturer: Dirk Gericke

## Weighting: 15 CATS

At the quantum level, systems of identical particles are always interacting particle systems even if there is no direct interaction between particles. This is because the wavefunction has to satisfy a symmetry condition imposed by quantum statistics - Pauli statistics for fermions and Bose-Einstein statistics for bosons. In atoms and molecules this leads to the familiar rule about never occupying a single-particle level with more than one electron.

This module looks at phenomena found in such interacting particle systems. It will start by looking at the properties of interacting electrons in molecules using ideas you have met before. In parallel we introduce the idea of a quantum field. This is where the wavefunctions of matter/light themselves are quantized (made into operators). Although not strictly necessary for the description of atoms and molecules or systems with fixed particle number, the theory of quantum fields automatically builds the correct fermionic/bosonic statistics into the description of a many-particle system and is the natural theory for discussing the properties of such systems. We will go on to look at Landau's theory of the Fermi liquid, the effect of dimensionality on the stability of ordered states and how, in ultracold trapped atomic gases, the properties can be controlled by tuning the interaction itself.

**Aim:**

To cover some theoretical models and mathematical methods important in theoretical physics.

**Objectives:**

At the end of this module you should:

- understand interactions between electrons in atoms and molecules
- be able to work with quantum fields
- be prepared for postgraduate research in quantum theory of many-body systems

**Syllabus:**

**Brief review of quantum mechanics and atomic physics**. Periodic Table. Molecules- time scales, Born-Oppenheimer method. Application of Rayleigh-Ritz variational method to electrons in H_{2}^{+}molecular ion and H_{2}molecule. Bonding in molecules - "pairing" of electrons, spatially directed orbitals. Solids and metallic bonding.**Introduction to 2nd Quantisation:**creation and annihilation operators, harmonic oscillator. Bosons and Fermions.**Description of the N-electron state:**electronic structure and Bloch’s theorem; electron exchange and correlation, Hartree-Fock and density functional theories;

**Landau Fermi liquid theory:**notion of the quasiparticle, low temperature properties. Landau Fermi liquid parameters, the Stoner criterion. Application to normal^{3}He.**Magnetism and Strong Correlations:**Mott transition. Spin degrees of freedom, spin waves as bosons, Holstein-Primakov. The Heisenberg ferromagnet and antiferromagnet. Canonical transformations. Relevance to (insulating) La_{2}CuO_{4}.**Bose-Einstein Condensation:**Introduction to ultra-cold atom systems. Order parameter and simple theories of the BEC state**Superconductivity:**Experimental phenomenology. Order parameter. The pairing instability, BCS hamiltonian. Predictions of BCS. Phase as a conjugate variable to number, (*if time allows*Josephson effect).

**Commitment:** 30 Lectures (equivalent)

**Assessment:** 2 hour examination

**Recommended Text: **

G Baym,*Lectures on quantum mechanics*, Addison-Wesley, 1969

E K U Gross, E Runge and O Heinonen, *Many Particle Theory*, Hilger, 1991.

R D Mattuck, *A guide to Feynman diagrams in the many-body problem* New York, 1992.

C. J. Pethick & H. Smith, *Bose-Einstein Condensation in Dilute Gases*, Cambridge University Press (2002)

L. Pitaevskii & S. Stringari, *Bose-Einstein Condensation*, Clarendon Press, Oxford (2003)

A. Griffin, D. W. Snoke & S. Stringari (Eds), *Bose Einstein Condensation*, Cambridge University Press (1995)

A. Altland & B. D. Simons, *Condensed Matter Field Theory*, Cambridge University Press (2002)