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I've singled it down to some choices:

Hamiltonian Algebra and Relativity Symmetries - Huge description, ask if more information is needed.

The Propagator and Sums Over Paths - "Knowing the propagator means that we have solved the time-dependent SchrÄodinger equation for arbitrary initial conditions and its calculation is the starting point for many approaches to quantum mechanics.

This project will involve understanding and calculating the propagator for the special cases of a free particle and a harmonic oscillator and understanding the general relationship between K(x; x0; t) and paths connecting x0 to x in time t."

Free Point Particle in Special Relativity

"In special relativity, a free point particle moves on a straight line in four-dimensional Minkowski

spacetime. Compare variational principles from which this motion can be obtained."

Tunneling of a relativistic particle through a potential barrier -

"The tunnelling probability through a potential barrier for a non-relativistic particle decreases

exponentially with a length of the barrier. This result can be derived from a solution of the

one-dimensional Schroedinger equation. In contrast, the situation is di®erent for a relativistic particle,

which can penetrate through a barrier with the probability one (Klein paradox). The aim of this

project is to calculate transmission and re°ection probabilities of a relativistic particle using analytical

solutions of the Dirac equation."

Special Relativity in a Periodic Universe

"How does special relativity work in a universe in which a spatial dimension is periodic? Are all inertial

observers still equivalent?"

The Bohr-Sommerfeld Atom

"In the early days of Quantum Mechanics, Bohr \explained" the spectral lines of hydrogen by assuming

electrons orbited nuclei in circular orbits with orbital angular momentum only in multiples of Planck's

constant.

In this project your assignment is to investigate Bohr's model , taking into account elliptic orbits, ¯rst

for the non-relativistic situation, and then taking some aspects of special relativity into account, such

as the varying mass of the electron (if that is relevant). What are the predicted modifcation to the

energy levels of hydrogen?"

I can also do some projects on Knots/Graphs, Alexander Polynomials and Hopf Algebras.

Does anybody have any advice? Which project would best prepare me for an MSc/PhD in high energy physics? Thank you.