Results in Physics

The Hellmann–Feynman theorem, the comparison theorem, and the envelope theory

Claude Semay

Published Jan 1, 2015• 1 citation

Abstract

The envelope theory is a convenient method to compute approximate solutions for bound state equations in quantum mechanics. It is shown that these approximate solutions obey a kind of Hellmann–Feynman theorem, and that the comparison theorem can be applied to these approximate solutions for two ordered Hamiltonians.

DOI: 10.1016/j.rinp.2015.11.004

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Results in Physics

The Hellmann–Feynman theorem, the comparison theorem, and the envelope theory

Claude Semay

Published Jan 1, 2015• 1 citation

Publisher

Abstract

DOI: 10.1016/j.rinp.2015.11.004

Related papers

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T

is a kinetic part which depends only on momenta and

V

is a potential which depends only on positions. We assume that

{H}^{(1)}=T+{V}^{(1)}

and

{H}^{(2)}=T+{V}^{(2)}

(

{H}^{(1)}={T}^{(1)}+V

and

{H}^{(2)}={T}^{(2)}+V

) support both discrete eigenvalues ${E}_{{\ensuremath{\alpha}}}^{(1)}$ and ${E}_{{\ensuremath{\alpha}}}^{(2)}$, where ${\ensuremath{\alpha}}$ represents a set of quantum numbers. We prove that, if ${V}^{(1)}\ensuremath{\leqslant}{V}^{(2)}$ (${T}^{(1)}\ensuremath{\leqslant}{T}^{(2)}$) for all position (momentum) variables, then the corresponding eigenvalues are ordered as ${E}_{{\ensuremath{\alpha}}}^{(1)}\ensuremath{\leqslant}{E}_{{\ensuremath{\alpha}}}^{(2)}$. Some analytical applications are given.