The functional-analytic versus the functional-integral approach to quantum Hamiltonians: The one-dimensional hydrogen atom.

TL;DR

This paper compares the functional-analytic and functional-integral methods for constructing quantum Hamiltonians, using the one-dimensional hydrogen atom to clarify spectral properties and the multiplicity of Hamiltonians in modeling quantum systems.

Contribution

It provides a detailed comparison of two approaches for defining Hamiltonians, deriving explicit Green functions for the one-dimensional hydrogen atom, and highlighting the implications of Hamiltonian multiplicity.

Findings

Explicit Green functions for the family of Hamiltonians are derived.

Differences between the approaches are clarified through the example.

Multiplicity of Hamiltonians impacts modeling in quantum wires.

Abstract

The capabilities of the functional-analytic and of the functional-integral approach for the construction of the Hamiltonian as a self-adjoint operator on Hilbert space are compared in the context of non-relativistic quantum mechanics. Differences are worked out by taking the one-dimensional hydrogen atom as an example, that is, a point mass on the Euclidean line subjected to the inverse-distance potential. This particular choice is made with the intent to clarify a long-lasting discussion about its spectral properties. In fact, for the four-parameter family of possible Hamiltonians the corresponding energy-dependent Green functions are derived in closed form. The multiplicity of Hamiltonians should be kept in mind when modelling certain experimental situations as, for instance, in quantum wires.