Path integrals and the quantum Routhian

Abstract

In this thesis, we consider the use of the Routhian in quantum mechanics, which is an object halfway between a Lagrangian and a Hamiltonian expressing the dynamics of a system in terms of conserved momentum and non-cyclic coordinates. Starting from the phase space path integral, we derive an expression for the quantum mechanical propagator of a system written in terms of its Routhian. We then go on to show how this formalism can provide calculational simplifications in simple situations such as a free particle on a line or a circle, and we demonstrate that for a particle in a constant magnetic field, by using conserved momentum it is possible to obtain a positive definite measure after Wick rotating to imaginary time. By doing this, we are able to obtain the quantum corrections to the partition function. Finally, we attempt to develop a general method for approximating the partition function for a particle on a sphere if there is a conserved azimuthal momentum, and consider in detail the free particle on a sphere. We reduce the problem to having to solve a complicated differential equation, obtaining an answer very close to the exact result in the simplest approximation.