One-Dimensional Infinite Square Well

One-dimensional Infinite Square Well

Energy

$$ \large E_n = \frac{n^2 \hbar^2 \pi^2}{2 m L^2} $$
  • GRE Physics GR0177 Problem 044

Wave Function

$$ \large \psi (x) = A \sin \left( \frac{n \pi}{L}x \right) = \sqrt{\frac{2}{L}} \sin \left( \frac{n \pi}{L}x \right) $$

Normalization

$$ \begin{split} \int^L_0 \psi^* \psi dx & = \large \int^L_0 A^2 \sin^2 \left( \frac{n \pi}{L} x \right) dx \quad & = \ 1 \\ & = A^2 \left[ \frac{x}{2} - \frac{\sin \left( \frac{2 n \pi}{L} x\right)}{4 \pi n/L} \right] & = \frac{A^2}{2}L \end{split} $$
  • GRE Physics GR0877 Problem 085

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