Introduction

In this chapter, we shall investigate the interaction of a non-relativistic particle of mass $m$ and energy $E$ with various one-dimensional potentials, $V(x)$. Since we are searching for stationary solutions with unique energies, we can write the wavefunction in the form (see Sect. 4.12)

$$\psi(x,t) = \psi(x) {\rm e}^{-{\rm i} E t/\hbar},$$ (300)

where $\psi(x)$ satisfies the time-independent Schrödinger equation:

$$\frac{d^2 \psi}{d x^2} = \frac{2 m}{\hbar^2} \left[V(x)-E\right]\psi.$$ (301)

In general, the solution, $\psi(x)$, to the above equation must be finite, otherwise the probability density $\vert\psi\vert^{ 2}$ would become infinite (which is unphysical). Likewise, the solution must be continuous, otherwise the probability current (155) would become infinite (which is also unphysical).