Introduction

In this section, 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 wave-function in the form (see Sect. 4.12)

$$\psi(x,t) = \psi(x)\,{\rm e}^{-{\rm i}\,E\,t/\hbar},$$ (282)

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

$$\frac{d^2 \psi}{d x^2} = \frac{2\,m}{\hbar^2} \left[V(x)-E\right]\psi.$$ (283)

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 flux (137) would become infinite (which is also unphysical).