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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Comment on ‘‘A peculiar relation between mass and energy with respect to the Schrödinger equation’’

S.B. Elegba, M. A. Rashid

Published Nov 1, 1988• 2 citations
Publisher

Abstract

The energy eigenvalue E(m,m') for a system of two particles of masses m and m' may not be a nondecreasing function of m. This problem has recently been analyzed analytically by Flamm, Sch\"oberl, and Uematsu. We prove the main result of this paper using directly the Feynman-Hellmann theorem and the virial theorem without considering modification of the dependent variable which results in the modification of the normalization of the wave function. Our analysis seems to be manifest.

DOI
10.1103/physrevd.38.2911

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Reply to “Comment on ‘Solution of the Schrödinger equation for the time-dependent linear potential’ ”

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In this reply I discuss further the misconception in my paper [Phys. Rev. A 63, 034102 (2001)] related to the solution of the Schr\"odinger equation for a particle in a time-dependent linear potential.

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The energy eigenvalue E(m,m') for a system of two particles of masses m and m' may not be a nondecreasing function of m. This problem has recently been analyzed analytically by Flamm, Sch"oberl, and Uematsu. We prove the main result of this paper using directly the Feynman-Hellmann theorem and the virial theorem without considering modification of the dependent variable which results in the modification of the normalization of the wave function. Our analysis seems to be manifest.
Considering energy eigenvalues E(m1{m}_{1}m1​,m2{m}_{2}m2​) with respect to two particles whose masses are m1{m}_{1}m1​ and m2{m}_{2}m2​, respectively, one would expect that if M>m then E(M,m2{m}_{2}m2​)>E(m,m2{m}_{2}m2​), but actually the opposite can happen if one solves the Schr\"odinger equation. In other words, the derivative \ensuremath{\partial}E/\ensuremath{\partial}m1{m}_{1}m1​ can be negative. In this paper a general condition of this problem is 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.