4: One-Dimensional Potentials

Last updated
Save as PDF

Page ID: 15749

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

In this chapter, we shall investigate the interaction of a non-relativistic particle of mass \(m\) and energy \(E\) with various one-dimensional potentials, \(V(x)\). Because we are searching for stationary solutions with unique energies, we can write the wavefunction in the form (see Section [sstat]) \[\psi(x,t) = \psi(x)\,{\rm e}^{-{\rm i}\,E\,t/\hbar},\] where \(\psi(x)\) satisfies the time-independent Schrödinger equation: \[\label{e5.2} \frac{d^{\,2} \psi}{d x^{\,2}} = \frac{2\,m}{\hbar^{\,2}} \left[V(x)-E\right]\psi.\] In general, the solution, \(\psi(x)\), to the previous equation must be finite, otherwise the probability density \(|\psi|^{\,2}\) would become infinite (which is unphysical). Likewise, the solution must be continuous, otherwise the probability current ([eprobc]) would become infinite (which is also unphysical).