Description

There are different ways to quantise a classical system. This should not be surprising as there are different, but equivalent, ways to describe classical systems, notably Hamiltonian or Lagrangian mechanics. Usually introductions to quantum mechanics (QM) take the canonical quantisation approach. This is what you will have seen in Mathematical Physics II and will develop further in Quantum Mechanics III (or equivalent modules in Physics). This approach is based on classical Hamiltonian mechanics: Poisson brackets are replaced by commutators and the Hamiltonian appears in the Schroedinger equation describing the time-evolution of the system. In this project you will explore an alternative, but equivalent, approach developed by Feynman. This Path Integral (PI) method is related to classical Lagrangian mechanics.

What is the PI?

Essentially it is a generalisation of the concept of wave-particle duality seen in the 2-slit interference pattern. The basic idea is that a particle (e.g. a photon which is a particle of light) can travel from a source to a point on a screen, but between there is a barrier with two small gaps (the slits). Classically the particle would have to travel via a specific path. In QM the particle also behaves like a wave which of course spreads out from the source, passes through both slits and then the wave from each slit can interfere constructively or destructively to give the pattern seen on the screen. Alternatively we can say that the particle takes all possible paths from the source to the screen, but to reproduce the observed pattern it must be possible for different paths to cancel (destructive interference) so each path must be given a weight which need not be positive - in fact each path is given a phase, i.e. a complex number on the unit circle in the Argand diagram. Summing (or integrating) over the possible paths (which do not typically correspond to the classical motion of the particle) then gives the correct interference pattern. More generally the PI gives us the QM amplitude for a particle to start at one position and end after some fixed time at another position - this is entirely equivalent to solving Schroedinger's equation for time-evolution.

What will I do in this project?

The project will start by learning about the PI, to make precise what was outlined above. You can then calculate the PI in specific examples, such as the Simple Harmonic Oscillator. You can also show that the PI approach is equivalent to canonical quantisation. One feature of the PI is that it gives a direct connection to Lagrangian mechanics. In fact you can start with PI QM and then derive the principle of stationary action (giving the Euler-Lagrange equations) as a classical limit. This also demonstrates how the classical path is selected from the sum over all paths between two fixed points. There are then many different applications so you have several choices of direction in the project.

The PI gives a geometric description of QM and in some situations interesting effects arise from topology. E.g. imagine a two-dimensional plane with a hole cut out. The paths a particle takes moving between two points can be classified in terms of how many times they wind around the hole. In some systems this winding number has an import effect and leads to effects such as the Aharanov-Bohm effect and interesting types of particles known as anyons whose wavefunctions gain important non-trivial phases when moved around each other. PIs are also useful in Statistical Mechanics so you could explore this connection (especially if you are taking Statistical Mechanics III or have seen this subject in another module.) Usually we cannot solve a system exactly in QM, but often we can use an approximation method known as perturbation theory. This can be done using the PI approach and is closely related to method used in Quantum Field Theory (QFT). (You could explore QFT a bit if you are really interested.) There are some effects not captured by standard perturbation theory, and these are called non-perturbative effects. These so-called instantons are included in the exact PI, and their effects can be described using different approximation techniques.

Prerequisites

Mathematical Physics II or equivalent Physics module(s) covering Lagrangian and Hamiltonian dynamics, and some quantum mechanics.

Corequisites

Quantum Mechanics III or equivalent Physics module(s) covering Dirac notation and solving simple systems such as the Simple Harmonic Oscillator.

Resources

A good place to start is the Wikipedia page: Path integral formulation

Many textbooks on QM, and often introductory QFT books cover PI QM. One good introduction is Quantum Mechanics and Path Integrals by Feynman and Hibbs. You can read some of it at Google Books.

There are also many online resources and reviews. One useful (although slightly advanced so don't expect to immediately follow everything) review is Path Integral Methods and Applications by MacKenzie.