A general comparison theorem

Details A general comparison theorem A general comparison theorem

2010-12-24

arxiv.org/abs/1012.5355v2

Phys. Rev. A 83, 024101 (2011)

Physics

Quantum Physics

Presentation improved, results unchanged. Version to appear in Phys. Rev. A

Scientific paper

10.1103/PhysRevA.83.024101

Using the Hellmann-Feynman theorem, a general comparison theorem is established for an eigenvalue equation of the form $(T+V)|\psi> = E|\psi>$, where $T$ is a kinetic part which depends only on momentums 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 $E^{(1)}_{\{\alpha\}}$ and $E^{(2)}_{\{\alpha\}}$, where ${\{\alpha\}}$ represents a set of quantum numbers. We prove that, if $V^{(1)} \le V^{(2)}$ ($T^{(1)} \le T^{(2)}$) for all position (momentum) variables, then the corresponding eigenvalues are ordered $E^{(1)}_{\{\alpha\}} \le E^{(2)}_{\{\alpha\}}$. Some analytical applications are given.

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