Exact path integral of the hydrogen atom and the Jacobi's principle of least action

TL;DR

This paper derives an exact path integral formulation for the hydrogen atom's Green's function using Jacobi's principle, demonstrating gauge invariance and connecting classical least action with quantum operator methods.

Contribution

It presents a novel exact path integral approach for the hydrogen atom based on Jacobi's principle, linking classical and quantum formalisms and emphasizing gauge invariance.

Findings

Exact path integral for hydrogen atom derived in parabolic coordinates

Green's function shown to be gauge independent with proper operator ordering

Connection established between Jacobi's principle and quantum path integral formalism

Abstract

The general treatment of a separable Hamiltonian of Liouville-type is well-known in operator formalism. A path integral counterpart is formulated if one starts with the Jacobi's principle of least action, and a path integral evaluation of the Green's function for the hydrogen atom by Duru and Kleinert is recognized as a special case. The Jacobi's principle of least action for given energy is reparametrization invariant, and the separation of variables in operator formalism corresponds to a choice of gauge in path integral. The Green's function is shown to be gauge independent,if the operator ordering is properly taken into account. These properties are illustrated by evaluating an exact path integral of the Green's function for the hydrogen atom in parabolic coordinates.