arXiv:1206.5960 · Results in Physics

Bounds for Hamiltonians with arbitrary kinetic parts

Claude Semay

Published Jan 1, 2012• 2 citations

Abstract

A method is presented to compute approximate solutions for eigenequations in quantum mechanics with an arbitrary kinetic part. In some cases, the approximate eigenvalues can be analytically determined and they can be lower or upper bounds. A semiclassical interpretation of the generic formula obtained for the eigenvalues supports a new definition of the effective particle mass used in solid state physics. An analytical toy model with a Gaussian dependence in the momentum is studied in order to check the validity of the method.

DOI: 10.1016/j.rinp.2012.09.004

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arXiv:1206.5960 · Results in Physics

Bounds for Hamiltonians with arbitrary kinetic parts

Claude Semay

Published Jan 1, 2012• 2 citations

arXiv HTML PDF

Abstract

DOI: 10.1016/j.rinp.2012.09.004

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.

p_0

and of the potential energy computed at a mean distance

r_0

. The quantities

p_0

and

r_0

are linked by a simple relation depending on the quantum numbers of the state considered and are determined by an equation which is linked to the generalized virial theorem. The (anti)variational character of the method is discussed, as well as its connection with the perturbation theory. For a nonrelativistic kinematics, general results are obtained for the structure of critical coupling constants for potentials with a finite number of bound states.