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arXiv:1012.5355 · Physical Review A

General comparison theorem for eigenvalues of a certain class of Hamiltonians

Claude Semay

Published Feb 16, 2011• 21 citations

Abstract

Using the Hellmann-Feynman theorem, a general comparison theorem is established for an eigenvalue equation of the form $(T+V)|\ensuremath{\psi}\ensuremath{\rangle}=E|\ensuremath{\psi}\ensuremath{\rangle}$, where $T$ 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.

DOI: 10.1103/physreva.83.024101

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Using the Hellmann-Feynman theorem, a general comparison theorem is established for an eigenvalue equation of the form

(T+V)|\ensuremath{\psi}\ensuremath{\rangle}=E|\ensuremath{\psi}\ensuremath{\rangle}

, where

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.

If the Hamiltonian operator depends on a parameter λ, which may be a particle charge or mass, a field strength, or simply a dummy parameter artificially introduced to apply perturbation theory, then the eigenvalues and eigenfunctions of Hˆ will also depend on that parameter. If we differentiate < |(Hˆ − E)| >= 0 with respect to λ 〈 ∂ ∂λ ∣∣∣(Hˆ − E)∣∣∣ 〉 + 〈 ∣∣∣∣∣ ∂(Hˆ − E) ∂λ ∣∣∣∣∣ 〉 + 〈 ∣∣∣(Hˆ − E)∣∣∣ ∂ ∂λ 〉 = 0 (3.5) and take into account equation (3.2), then we obtain the Hellmann-Feynman theorem [25] ∂E ∂λ < | >= 〈 ∣∣∣∣∣ ∂Hˆ ∂λ ∣∣∣∣∣ 〉 , (3.6) or ∂E ∂λ = 〈 ∂Hˆ ∂λ 〉 (3.7) if is normalized to unity. In what follows we show that the general results just derived facilitate the application of pertur- bation theory.

We revisit the celebrated Hellmann-Feynman theorem (HFT) in the parity-time

(\mathcal{PT})

invariant non-Hermitian quantum physics framework. We derive a modified version of HFT by changing the definition of inner product and explicitly show that it holds good for both

\mathcal{PT}

broken and unbroken phases and even at the exceptional point of the theory. The derivation is extremely general and works for even the

\mathcal{PT}

noninvariant Hamiltonian. We consider several examples of discrete and continuum variable systems to test our results. We find that, if the eigenvalue goes through a real-to-complex transition as a function of the Hermiticity breaking parameter, both sides of the modified HFT expression diverge at that point. If that point turns out to be an EP of the

\mathcal{PT}

-invariant quantum theory, then one also sees the divergence at EP. Moreover, we have also demonstrated that using the modified HFT can give rise to a potential numerical advantage for computing the expectation value of many-body operators for interacting many-body Hamiltonian. Finally, we also derive a generalized virial theorem for non-Hermitian systems using the modified HFT, which potentially can be tested in experiments.

Two discrete energy spectra of a particle moving in two different potential wells are compared. We find a general property: If one potential well

{\mathit{V}}_{1}

(r) is less than or equal to the other potential well

{\mathit{V}}_{2}

(r) everywhere in space, i.e.,

{\mathit{V}}_{1}

(r)\ensuremath{\le}

{\mathit{V}}_{2}

(r) for all r, then the nth eigenvalue of the particle in the first well,

{\mathit{V}}_{1}

, is always not greater than the nth eigenvalue of the particle in the second well,

{\mathit{V}}_{2}

.

We point out that two of the most important theorems of Quantum Mechanics, the Ehrenfest theorem and the Hellmann-Feynman theorem, lack in their standard form important information: there are cases where non-Hermitian boundary contributions emerge. These contributions actually appear naturally, in order for the above theorems to be valid and applicable (i.e. in multiply-connected spaces), and this occurs for physical quantities that are not represented by well-defined self-adjoint operators (such as the position operator in a periodic potential, or in general Aharonov-Bohm configurations, either in real or in an arbitrary parameter-space, in the sense of Berry's adiabatic and cyclic procedures). In this short note, we report modifications of these two theorems when such non-Hermiticities appear, and we demonstrate how they resolve certain Quantum Mechanical paradoxes (most of them having been noticed in the past as violations of the so-called Hypervirial theorem in Quantum Chemistry). This resolution of paradoxes (essentially the re-establishment of applicability of the Ehrenfest theorem even in multiply-connected spaces) always proceeds through the appearance of certain generalized currents, in a theoretical picture with interesting structure (where a generalized continuity equation with a sink term shows up naturally).