The effect of a $\delta$ distribution potential on a quantum mechanical particle in a box

Abstract

We study the effect of a $\delta$ distribution potential placed at $x_0\ge 0$ and multiplied by a parameter $\alpha$ on a quantum mechanical particle in an infinite square well over the segment $\left[-\frac{L}{2},\frac{L}{2}\right]$. We obtain the limit of the eigenfunctions of the time independent Schrödinger equation as $\alpha \nearrow +\infty$ and as $\alpha \searrow -\infty$. We see how each solution of the Schrödinger equation corresponding to $\alpha =0$ changes as $\alpha$ runs through the real line. When $x_0$ is a rational multiple of $L$, there exist solutions of the Schrödinger equation which vanish at $x_0$ and are unaffected by the value of $\alpha$. We show that each one of these has an energy that coincides with the energy of a certain limiting eigenfunction obtained by taking $|\alpha |\to \infty$. The expectation value of the position of a particle with wave function equal to the limiting eigenfunction is $x_0$.

34 pages, 26 figures