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4: One-Dimensional Potentials

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  • 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).