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Outline:

This module aims to familiarise students with the fundamental concepts and mathematical frameworks of modern quantum science. The module begins presenting a more formal and mathematical postulates approach to quantum mechanics than students have encountered in previous modules and then presents advanced concepts in wave-function quantum mechanics such as tunnelling in non-idealised potentials described via WKB approximation. The module then introduces students to formal mathematical approaches to study quantum dynamics of both isolated quantum systems and quantum systems in interaction with an environment. The advanced techniques the students are expected to learn in this part of the module include treatments of quantum dynamics via Suzuki-Trotter approximations, description of dynamics in the Interaction and Heisenberg pictures, time-dependent perturbation theory, and dynamics of open quantum systems via Markovian master equations.

Aims:

• Review and extend the basics of quantum mechanics in more formal mathematical terms
• Explore the Wentzel, Kramers, Brillouin (WKB) approximation (important for dealing with tunneling and transmission in potentials of arbitrary shapes)
• Describe light in quantum mechanics/use the Jaynes-Cummings model to describe atom-light interactions
• Explore advanced topics in the dynamics of quantum systems such as time-dependent perturbation theory, and open quantum systems
• Provide techniques and terminology which can be applied in specialist modules and research projects

Teaching and Learning Methodology:

This module is delivered via weekly lectures supplemented by a series of workshops and additional discussion. In addition to timetabled lecture hours, it is expected that students engage in self-study in order to master the material. This can take the form, for example, of practicing example questions and further reading in textbooks and online.

Indicative Topics:

Quantum mechanics from the ground up:

• Abstract vector spaces, norm, inner product, basis, linear functionals, operators, column vectors and representations of abstract vectors and operators, Dirac notation, Hermitian an unitary operators, projectors.
• Expectation values. Postulates of quantum mechanics, representations of continuous variables, position and momentum.
• Compound systems, tensor product, entanglement.
• Statistical state preparation and mixed states, density operator formalism, density operators to describe subsystems of entangled states.

• Advanced wave mechanics, WKB approximation: WKB ansatz and derivation of wave functions. The failure of these wave functions at classical turning points. Connection formulae. Applications.
• Advanced topics in time dependence 1 – Unitary evolution:
• Unitary evolution under the Schroedinger equation
• Split operator method and Tsuzuki-Trotter decomposition
• Heisenberg picture, Schroedinger picture, interaction picture
• Two-level atom and dipole approximation
• Rotating wave approximation and Jaynes-Cummings model
• Advanced topics in time dependence 2 – Time dependent perturbation theory:
• Dirac’s method as an application of the interaction picture
• Time-dependent perturbation theory (first & higher order)
• Fermi’s golden rule & applications
• Advanced topics in time dependence 3 – Open Quantum Systems:

• Von Neumann equation for density matrices.
• Interaction with the environment.
• Evolution of a subsystem.
• Markov approximation.
• Abstract approach to non-unitary evolution.
• Completely positive maps.
• Kraus operators.
• Master equations.
• Lindblad form, derivation from Kraus operator ansatz.
• Quantum trajectories and jump operators.
• Applications.