Variational approach to the Schrödinger equation with a delta-function potential

Abstract

We obtain accurate eigenvalues of the one-dimensional Schr\"{o}dinger equation with a Hamiltonian of the form $H_{g}=H+g\delta (x)$, where $\delta (x)$ is the Dirac delta function. We show that the well known Rayleigh-Ritz variational method is a suitable approach provided that the basis set takes into account the effect of the Dirac delta on the wavefunction. Present analysis may be suitable for an introductory course on quantum mechanics to illustrate the application of the Rayleigh-Ritz variational method to a problem where the boundary conditions play a relevant role and have to be introduced carefully into the trial function. Besides, the examples are suitable for motivating the students to resort to any computer-algebra software in order to calculate the required integrals and solve the secular equations.

Nonlinear localized states near the interface with nonlinear response between the medium with a parabolic index spatial profile and Kerr-type medium

“…It is shown that delta-function potential changes significantly the quantum harmonic oscillator spectral features [35,36]. However, the delta-function potential has often been viewed as an excitation introduced into the oscillator [37][38][39]. The variational method can be applied to investigation of the spectral features of the oscillator with the delta-function potential [40].…”