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# 4: One-Dimensional Potentials

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).