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

A peculiar relation between mass and energy with respect to the Schrödinger equation

Dieter Flamm, Franz F. Schöberl, Hikotaro Uematsu

Published Oct 1, 1987• 7 citations
Publisher

Abstract

Considering energy eigenvalues E(${m}_{1}$,${m}_{2}$) with respect to two particles whose masses are ${m}_{1}$ and ${m}_{2}$, respectively, one would expect that if M>m then E(M,${m}_{2}$)>E(m,${m}_{2}$), but actually the opposite can happen if one solves the Schr\"odinger equation. In other words, the derivative \ensuremath{\partial}E/\ensuremath{\partial}${m}_{1}$ can be negative. In this paper a general condition of this problem is given.

DOI
10.1103/physrevd.36.2176

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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.
The Schrodinger equation is considered for a one-dimensional potential function composed of the Morse oscillator and an infinite barrier at long range. The boundary condition introduced by the barrier implies discretization of the unbound states. The eigenstates and eigenvalues are determined along with the discretization condition and the proper normalization for positive and negative energies. An analysis is presented for a particle in this potential driven by an external field coupled in through a dipole. Analytical formulae are derived for the matrix elements of an exponential and linear dipole function with respect to the eigenstates. Numerical time-dependent solutions to the Schrodinger equation are obtained for a periodic external field.
Two discrete energy spectra of a particle moving in two different potential wells are compared. We find a general property: If one potential well V1{\mathit{V}}_{1}V1​(r) is less than or equal to the other potential well V2{\mathit{V}}_{2}V2​(r) everywhere in space, i.e., V1{\mathit{V}}_{1}V1​(r)\ensuremath{\le}V2{\mathit{V}}_{2}V2​(r) for all r, then the nth eigenvalue of the particle in the first well, V1{\mathit{V}}_{1}V1​, is always not greater than the nth eigenvalue of the particle in the second well, V2{\mathit{V}}_{2}V2​.