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1.27: Bound solutions

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    50131

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    Electrons with energies within the well (\(0<E<V_{0}\)) are bound. The wavefunctions of the bound electrons are localized within the well and so they must be normalizable. Thus, the wavefunction of a bound electron in the classically forbidden region (outside the well) must decay exponentially with distance from the well.

    A possible solution for the bound electrons is then

    \[ \psi(x)=\left\{\begin{array}{lc}
    C e^{\alpha x} & \text { for } x \leq-L / 2 \\
    A \cos (k x)+B \sin (k x) & \text { for }-L / 2 \leq x \leq L / 2 \\
    D e^{-\alpha x} & \text { for } x \geq L / 2
    \end{array}\right. \nonumber \]

    where

    \[ \alpha = \sqrt{\frac{2m(V_{0}-E)}{\hbar^{2}}} \nonumber \]

    and

    \[ k = \sqrt{\frac{2mE}{\hbar^{2}}} \nonumber \]

    and A, B, C and D are constants.

    This page titled 1.27: Bound solutions is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Marc Baldo (MIT OpenCourseWare) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.