Course: B7.3 Further Quantum Theory (2021-22)

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Lecturer: Profile: Christopher Beem

Course information

General Prerequisites:

Part A Quantum Theory.

Course Term: Hilary

Course Lecture Information: 16 lectures

Course Weight: 1

Course Level: H

Assessment Type: Written Examination

Course Overview:

This course builds on Part A Quantum Theory. The mathematical foundations of quantum theory are developed more deeply than in the Part A course, and general principles regarding the realization of symmetries in quantum mechanical systems are developed. Systems of several particles are studied, including a consideration of identical particles and particle statistics. Along the way, several simple-but-relevant concepts from the theories of Lie groups, representation theory, and functional analysis are introduced in a hands-on fashion.
In semi-realistic systems, an exact solution to the quantum dynamics is rarely forthcoming. A number of important approximation techniques are also developed. These are employed to address problems such as the determination of energy levels of the Helium atom.

Learning Outcomes:

Student will be able to state the postulates of nonrelativistic quantum theory and explain how they are realized in key examples. They will be able to analyse simple quantum systems that admit exact solutions by exploiting symmetries and algebraic techniques. They will be able to calculate approximations to energy-levels and other properties of more complicated systems using perturbation theory, semi-classical techniques, and variational methods.

Course Synopsis:

Abstract formulation of quantum mechanics in terms of linear operators on Hilbert spaces; Dirac notation; discrete and continuum states; time evolution and the propagator.
Systems of several particles and Hilbert space tensor products; distinguishable and indistinguishable particles; Fermi-Dirac and Bose-Einstein statistics; Pauli exclusion principle; elementary aspects of quantum entanglement.
Symmetries in quantum mechanics as unitary and anti-unitary operators; rotations, angular momentum, and spin; spin-1/2 and projective representations of SO(3); addition of angular momentum; Spin-statistics theorem.
Approximation methods: Rayleigh-Schrödinger perturbation theory; variational methods; WKB approximation and Bohr-Sommerfeld quantization.
Time-dependence in the Heisenberg, Schrödinger, and interaction/Dyson; time-dependent perturbation theory and the Feynman-Dyson expansion.
Elementary scattering theory.

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Section outline

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General

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Select section Lecture Notes

Lecture Notes
Lecture notes for the course will uploaded here. These include a full-course file (that may be updated as the course progresses) as well as (hopefully finalised) disaggregated files for the individual chapters. The individual chapter notes may have hyperlinks that don't work.
- Select activity Lecture Notes
  
  Lecture Notes File
  
  Uploaded 3/03/22, 16:57
- Select activity Prologue
  
  Prologue File
  
  (this material is review and will not be covered in lectures)
  
  Uploaded 16/01/22, 21:45
- Select activity Chapter 1 - Postulates
  
  Chapter 1 - Postulates File
  
  Uploaded 16/01/22, 21:46
- Select activity Chapter 2 - Dirac Formalism and Continuum Normalisation
  
  Chapter 2 - Dirac Formalism and Continuum Normalisation File
  
  Uploaded 20/01/22, 13:48
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  Chapter 3 - Composite Systems and Tensor Products File
  
  Uploaded 19/01/22, 15:13
- Select activity Chapter 4 - Identical Particles and Statistics
  
  Chapter 4 - Identical Particles and Statistics File
  
  Uploaded 23/01/22, 21:37
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  Chapter 5 - Symmetries File
  
  Uploaded 30/01/22, 23:05
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  Chapter 6 - Rotations File
  
  Uploaded 6/02/22, 17:34
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  Chapter 7 - Addition of Angular Momentum File
  
  Uploaded 10/02/22, 23:24
- Select activity Chapter 8 - Time Independent Perturbation Theory
  
  Chapter 8 - Time Independent Perturbation Theory File
  
  Uploaded 13/02/22, 21:12
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  Chapter 9 - Variational Methods File
  
  Uploaded 16/02/22, 10:20
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  Chapter 10 - WKB Approximation File
  
  Uploaded 28/02/22, 22:41
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  Chapter 11 - One Dimensional Scattering File
  
  Uploaded 3/03/22, 16:56
- Select activity Chapter 12 - Epilogue
  
  Chapter 12 - Epilogue File
  
  Uploaded 3/03/22, 16:59
- Select activity Appendix A - Hilbert Spaces
  
  Appendix A - Hilbert Spaces File
  
  Details and further information on Hilbert spaces (includes a good deal of non-examinable material)
  
  Uploaded 17/01/22, 14:10
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Problem Sheets
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  Problem Sheet 1 Assignment
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  Problem Sheet 2 Assignment
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  Problem Sheet 3 Assignment
- Select activity Extra Hydrogen Atom Problem Sheet
  
  Extra Hydrogen Atom Problem Sheet Assignment
  
  An additional problem on the first-order perturbation theory treatment of the Hydrogen atom. This is a classic problem that is well discussed in text books (in particular the text of Shankar), so you can also read up on the solution without solving it on your own.
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  Problem Sheet 4 Assignment
Select section Whiteboards from Lectures

Whiteboards from Lectures
For students watching Panopto recordings of lectures, I am attempting to capture and upload images of the whiteboards from lecture that are legible, in case the writing is difficult to see on the videos. These will be uploaded here. Unfortunately this starts with Lecture 4 and doesn't include Lecture 16; my apologies for any difficulties in reading the writing in the missing lectures.
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  Lecture 4 Folder
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  Lecture 5 Folder
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  Lecture 9 Folder
  
  Apologies; missing some photos.
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