Best 3rd year project for High Energy Physics

Main Question or Discussion Point

I have to choose my disertation for next year and I've been given a selection of topics. I'm interested in doing an MSc/PhD in high energy physics but my school doesn't have anybody working in that area (only quantum gravity).

I've singled it down to some choices:

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.

I'm a bit confused, are you in undergrad or in grad school? If the latter, I was wondering about the Klein paradox thing. If you're using analytic solutions to the Dirac equation, chances are that someone has already done it. Correct me if I'm misunderstanding, but this sounds like the kind of thing you'd do for a 2nd semester graduate quantum homework problem rather than an MSc dissertation (in fact, that was one of my homeworks last year). But if you're an undergrad I suppose that might be an appropriate project.

Apologies for the confusion, this is for a 3rd year undergraduate project. It's not meant to be original research, just working on an extended problem indepedently and writing up the results. Obviously you can use books/papers that have already done similar things, but the idea is that you research this for yourself.

As you say, the analytical solutions of the dirac equation for this project have already been worked out, but I was wondering if learning how to solve the dirac equation in quantum tunnelling situations might be helpful, or whether something else is more helpful.

Thanks again.

I'm not really sure if the topic will actually matter that much for the application (they seem to be just small projects anyway, it's not like taking a course). But if you ask, then the subject closest to HEP is "The Propagator and Sums Over Paths", since this is exactly what you do in Quantum Field Theory (or to be more precise you generalize this procedure to quantum fields), so this might be useful. Note however that this is a very standard problem, which is covered in almost every book on QFT and perhaps also in some books on QM, so from this point of view it's not so interesting. But actually all the problems seem to be more or less quite standard in nature. I think you'd be OK with any of them.

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Martin, thank you for your insight. I had kinda overlooked that project. With regards to how big the project is, it's worth 15/120 credits, and I'm expected to do 100 hours of work towards it, so it's non-trivial, and I hope the things I learn will both complement and extend my knowledge of quantum mechanics at this stage so as to prepare me well for further study in the direction of QFT.

If you feel up to it I would seriously consider the one about Hopf Algebras, what exactly would be the topic?
They show up quite naturally in the context of quantum field theories. (but only if you are doing QFT on a somewhat abstract level)

A Hopf algebra is a generalisation of the idea of a group. In essence it is a vector space with a
multiplication law together with a multiplication law on the dual of the vector space. The project will
look at some particularly simple but non-trivial examples of Hopf algebras (involutory Hopf algebras).
The aims of the project are to ¯nd out how far the classi¯cation of these Hopf algebras has progressed
in the literature, and to perform some simple calculations with the Hopf algebras which will show how
one can easily distinguish inequivalent Hopf algebras.
Hopf algebras have recently been used both as a sort of symmetry group for quantum systems, and as
a tool in the classi¯cation of three-dimensional manifolds

When you say feel as if I would be up for it, what do you mean?