Comprehensive Course on Quantum Physics: Particles, Atoms, Molecules, Solids, and Fields, Study notes of Quantum Physics

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Advanced Quantum Physics

Aim of the course

Building upon the foundations of wave mechanics, this course will introduce and develop the broad field of quantum physics including: Quantum mechanics of point particles Approximation methods Basic foundations of atomic, molecular, and solid state physics Basic elements of quantum field theory Scattering theory Relativistic quantum mechanics Although these topics underpin a variety of subject areas from high energy, quantum condensed matter, and ultracold atomic physics to quantum optics and quantum information processing, our focus is on development of basic conceptual principles and technical fluency.

Further prerequisites...

Quantum physics is an inherently mathematical subject – it is therefore inevitable that the course will lean upon some challenging concepts from mathematics: e.g. operator methods, elements of Sturm-Liouville theory (eigenfunction equations, etc,), variational methods (Euler-Lagrange equations and Lagrangian methods – a bit), Green functions (a very little bit – sorry), Fourier analysis, etc. Fortunately/unfortunately ∗ ( ∗ delete as appropriate) such mathematical principles remain an integral part of the subject and seem unavoidable. Since there has been a change of lecturer, a change of style, and partially a change of material, I would welcome feedback on accessibility of the more mathematical parts of the course!

Synopsis: (mostly revision) Lectures 1-4ish

(^1) Foundations of quantum physics: Historical background; wave mechanics to Schr¨odinger equation. (^2) Quantum mechanics in one dimension: Unbound particles: potential step, barriers and tunneling; bound states: rectangular well, δ-function well; Kronig-Penney model. (^3) Operator methods: Uncertainty principle; time evolution operator; Ehrenfest’s theorem; symmetries in quantum mechanics; Heisenberg representation; quantum harmonic oscillator; coherent states. (^4) Quantum mechanics in more than one dimension: Rigid rotor; angular momentum; raising and lowering operators; representations; central potential; atomic hydrogen.

Synopsis: Lectures 11-

(^9) Identical particles: Particle indistinguishability and quantum statistics; space and spin wavefunctions; consequences of particle statistics; ideal quantum gases; degeneracy pressure in neutron stars; Bose-Einstein condensation in ultracold atomic gases. (^10) Atomic structure: Relativistic corrections – spin-orbit coupling; Darwin structure; Lamb shift; hyperfine structure. Multi-electron atoms; Helium; Hartree approximation and beyond; Hund’s rule; periodic table; coupling schemes LS and jj; atomic spectra; Zeeman effect. (^11) Molecular structure: Born-Oppenheimer approximation; H

2 ion; H^2 molecule; ionic and covalent bonding; solids; molecular spectra; rotation and vibrational transitions.

Synopsis: Lectures 16-

(^12) Field theory: from phonons to photons: From particles to fields: classical field theory of harmonic atomic chain; quantization of atomic chain; phonons. Classical theory of the EM field; waveguide; quantization of the EM field and photons. (^13) Time-dependent perturbation theory: Rabi oscillations in two level systems; perturbation series; sudden approximation; harmonic perturbations and Fermi’s Golden rule. (^14) Radiative transitions: Light-matter interaction; spontaneous emission; absorption and stimulated emission; Einstein’s A and B coefficents; dipole approximation; selection rules; † lasers.

What’s missing?

“Philosophy” of quantum mechanics (e.g. nothing on EPR paradoxes, Bell’s inequality, etc.) Specializations and applications (covered later in Lent and Part III) (e.g. nothing detailed on quantum information processing, etc.)

Handouts and lecture notes

Both lecture notes and overheads will be available (in pdf format) from the course webpage: www.tcm.phy.cam.ac.uk/~bds10/aqp.html But try to take notes too. The lecture notes are extensive (apologies!) and, as with textbooks, include more material than will covered in lectures or examined. Unlike textbooks, the lecture notes may contain (many?) typos – corrections welcome! For the most part, non-examinable material will be listed as “Info blocks” in lecture notes. Generally, the examinable material will be limited to what is taught in class, i.e. the overheads.

A few (random but recommended) books

B. H. Bransden and C. J. Joachain, Quantum Mechanics, (2nd edition, Pearson, 2000). Classic text covers core elements of advanced quantum mechanics; strong on atomic physics. S. Gasiorowicz, Quantum Physics, (2nd edn. Wiley 1996, 3rd edition, Wiley, 2003). Excellent text covers material at approximately right level; but published text omits some topics which we address. K. Konishi and G. Paffuti, Quantum Mechanics: A New Introduction, (OUP, 2009). This is a new text which includes some entertaining new topics within an old field. L. D. Landau and L. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, Volume 3, (Butterworth-Heinemann, 3rd edition, 1981). Classic text which covers core topics at a level that reaches beyond the ambitions of this course. F. Schwabl, Quantum Mechanics, (Springer, 4th edition, 2007). Best text for majority of course.

Books

...but, in general, there are a very large number of excellent textbooks in quantum mechanics. It is a good idea to spend some time in the library to find the text(s) that suit you best. It is also useful to look at topics from several different angles.

Lecture 1

Foundations of quantum physics

Foundations of quantum physics: outline

(^1) Historically, origins of quantum mechanics can be traced to failures of 19th Century classical physics: Black-body radiation Photoelectric effect Compton scattering Atomic spectra: Bohr model Electron diffraction: de Broglie hypothesis (^2) Wave mechanics and the Schr¨odinger equation (^3) Postulates of quantum mechanics

Black-body radiation: Planck’s resolution

Planck: for each mode, ν, energy is quantized in units of hν, where h denotes the Planck constant. Energy of each mode, ν, 〈ε(ν)〉 =

n= n hν e −nhν/kBT ∑ ∞ n= e −nhν/kBT

hν e hν/kBT − 1 Leads to Planck distribution: ρ(ν, T ) = 8 πν 2 c 3 〈ε(ν)〉 = 8 πhν 3 c 3

e hν/kBT − 1 recovers Rayleigh-Jeans law as h → 0 and resolves UV catastrophe. Parallel theory developed to explain low-temperature specific heat of solids by Debye and Einstein.

Photoelectric effect

When metal exposed to EM radiation, above a certain threshold frequency, light is absorbed and electrons emitted. von Lenard (1902) observed that energy of electrons increased with light frequency (as opposed to intensity). Einstein (1905) proposed that light composed of discrete quanta (photons): k.e.max = hν − W Einstein’s hypothesis famously confirmed by Millikan in 1916