Introduction
In classical mechanics, a charged particle will move in circular orbits in a magnetic field with frequency \( \omega = eB / m \) where \(m\) is the mass of the particle, \(e\) is its electric charge and \(B\) is the magnetic field strength. There is a continuous spectrum \( E>0 \) of allowed energies.
Quantum mechanics is much more interesting! There is a discrete spectrum of allowed energies,
\( E_n = \hbar \omega (n+\frac{1}{2}) \)
where \( n \) is a non-negative integer, with a large number of degenerate wavefunctions at each energy level. They are known collectively as Landau Levels.
Landau levels are an important and rich topic in mathematical physics. They form the theoretical foundation for one of the most remarkable and surprising experimental results in physics, known as the quantum Hall effect, and are further related to geometry and representation theory through the Borel-Weil-Bott theorem.
This project is a wonderful opportunity to explore these ideas and gain a deeper understanding of quantum mechanics!
Outline
- Term 1: You will begin by developing a firm understanding of the classical and quantum mechanics of charged particles in an electromagnetic field.
- Term 2: Building on this foundation, in the second term you will apply these ideas to an application of Landau levels in physics or mathematics. A physically oriented student could study aspects of the quantum Hall effect, while a more mathematically oriented student may wish to explore connections to the Borel-Weil-Bott theorem.
Prerequisites
Mathematical Physics IICorequisites
Quantum Mechanics IIIResources
Landau Levels - Berkeley Lecture NotesQuantum Hall Effect - Chapter 1, Lecture Notes by David Tong