The functional-analytic versus the functional-integral approach to quantum Hamiltonians: The one-dimensional hydrogen atom.
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.
Keywords
Cite
@article{arxiv.cond-mat/9501081,
title = {The functional-analytic versus the functional-integral approach to quantum Hamiltonians: The one-dimensional hydrogen atom.},
author = {W. Fischer and H. Leschke and P. Mueller},
journal= {arXiv preprint arXiv:cond-mat/9501081},
year = {2016}
}
Comments
14 pages, RevTeX 3.0, AMSfonts, 1 Figure available upon request from the Authors, to appear in J. Math. Phys. May 1995. Changes: Eq. (3) is rewritten and it is clarified that no weak derivatives are used