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4: One-Dimensional Potentials

Last updated

Apr 1, 2025
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- 3.11: Exercises
- 4.1: Infinite Potential Well

( \newcommand{\kernel}{\mathrm{null}\,}\)

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]) $\begin{matrix} (4.1) & ψ (x, t) = ψ (x) e^{- i E t / ℏ}, \end{matrix}$ where $ψ (x)$ satisfies the time-independent Schrödinger equation: $\begin{matrix} (4.2) & \frac{d^{2} ψ}{d x^{2}} = \frac{2 m}{ℏ^{2}} [V (x) - E] ψ . \end{matrix}$ In general, the solution, $ψ (x)$ , to the previous equation must be finite, otherwise the probability density ${| ψ |}^{2}$ would become infinite (which is unphysical). Likewise, the solution must be continuous, otherwise the probability current ([eprobc]) would become infinite (which is also unphysical).

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