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Advanced Quantum Theory

Post-graduate lecture series held at University College London in 2020

Below you will find videos of the asynchronous lectures. I have decided to make the material available via this website as it is a great resource.

Advanced Quantum Theory

Advanced Quantum Theory

Advanced Quantum Theory
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Lecture 0 Overview - Advanced Quantum Theory - Prof Carla Faria

Lecture 0 Overview - Advanced Quantum Theory - Prof Carla Faria

17:05
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Lecture 1 - Part 1 - Advanced Quantum Theory - Prof Carla Faria

Lecture 1 - Part 1 - Advanced Quantum Theory - Prof Carla Faria

16:19
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Lecture 1 - Part 2 - Advanced Quantum Theory - Prof Carla Faria

Lecture 1 - Part 2 - Advanced Quantum Theory - Prof Carla Faria

39:48
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Lecture 1:

 

  • Part 1: Three postulates of quantum mechanics

  • Part 2: Characteristics of vector spaces and their generalization to quantum mechanics. I also emphasize why axioms are important as road maps, even if they seem obvious sometimes. 

  • Part 3: Dirac notation, subspaces and operators

  • Part 4: closure relations and vector representations of operators.  There are some blackboard style derivations.

  • Part 5: Hermitian conjugation, change of basis and unitary operators.

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Lecture 2:

 

  • Part 1: Hermitian operators, projectors and spectral decomposition.  There are also some nice examples of how to deal with degenerate eigenstates.

  • Part 2:  Unitary time evolution. It introduces the time evolution operator, looks at specific cases and discusses its properties.

  • Part 3: Continuous variables and what changes with regard to discrete basis sets. It also shows how to describe operators in a continuous basis. 

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Lecture 3:

 

  • Part 1: Position and momentum representations and exemplifies why kets are more general than wavefunctions.

  • Part 2:  Measurement postulate

  • Part 3: Density matrices and density operators and how they can be used to deal with systems in which there is classical/statistical uncertainty. 

  • Part 4: How to deal with compound systems and construct a compound Hilbert space, state vectors, operators, etc in a consistent way. 

  • Part 5: Entangled systems and how to deal with them using density matrices

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Lecture 4:

 

  • Part 1: Density matrices of compound systems and how reduced density matrices can be used to deal with entangled systems.

  • Part 2:  The reduced density matrix is used to study a heuristic model of spontaneous emission, by considering an atom + field compound system.

  • Part 3:  Why the WKB Approximation is important, the key assumptions behind it and how it can be used

  • Part 4: Blackboard style derivations of the key WKB expressions

  • Part 5:  WKB connection formulae. The emphasis is more on how they work an on key ideas than on any rigorous derivation.  For the latter please see Joachain, Merzbacher or Landau and Lifshitz.

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Lecture 5:

 

  • Part 1: Application of the WKB approximation in which the tunneling probability  through a barrier is calculated. The expression derived here is quite general and used in many areas of physics.

  • Part 2:  Preamble for dealing with non-classical light. It explains how to use unitary transformations to go from one  physical picture  to another in a consistent way. Time-dependent transformations and Hamiltonians are included in the derivation. 

  • Part 3:  Example of unitary transformation that is widely used in light-matter interaction: the Goeppert-Mayer transformation.

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Lecture 6:

 

  • Part 1: Three important physical pictures as an application of unitary transformations: Schroedinger, Heisenberg and Interaction pictures.  Detailed explanations of how to go from one picture to the other and of their advantages are provided. 

  • Part 2:  Brief review of the quantum harmonic oscillator and link it to the second quantization of the electromagnetic field

  • Part 3:  Two-level atom as a widely used approximation and provide a whiteboard style derivation of  how the light-matter interaction Hamiltonian looks in the interaction picture. This picture is also highlighted as convenient for applying the rotating wave approximation. 

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Lecture 7:

 

  • Part 1: Key points associated with time dependent perturbation theory are discussed. It contains the general derivations of perturbative transition amplitudes in the interaction picture. 

  • Part 2: In this lecture I analyze the first and second order contributions in detail.

  • Part 3:  In this lecture I apply first-order time-dependent perturbation theory to compute the transition probability due to a constant perturbation.

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Lecture 8:

 

  • Part 1: Application of time-dependent perturbation theory to a sinusoidal perturbation 

  • Part 2: Introductory remarks on open quantum systems and the derivation of the von Neumann equation

  • Part 3: Superoperators and the Kraus decomposition

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Lecture 9:

 

  • Part 1: key ideas behind Markovian evolution

  • Part 2: whiteboard derivation of the Lindblad form of the master equation, with jump operators and effective Hamiltonians.

  • Part 3:  In this lecture I apply first-order time-dependent perturbation theory to compute the transition probability due to a constant perturbation.

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Lecture 10:  The Lindblad form of the master equation is applied to spontaneous emission in a whiteboard-the derivation.  General remarks on  further applications are provided in the end.

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