# 1.71: The Wigner Distribution for a Particle in a One‐dimensional Box


The following outlines the calculation of the Wigner distribution for a particle in a one‐dimensional box for the n = 10 state. First the coordinate wave function is Fourier transformed into momentum space. Following that the Wigner function is calculated using the momentum space wave function.

$\Psi(x) :=\sqrt{2} \cdot \sin (10 \cdot \pi \cdot x) \nonumber$

$\Phi(\mathrm{p}) :=\frac{1}{\sqrt{2 \cdot \pi}} \cdot \int_{0}^{1} \exp (-\mathrm{i} \cdot \mathrm{p} \cdot \mathrm{r}) \cdot \Psi(\mathrm{x}) \mathrm{dx} \text { simplify } \rightarrow-\frac{10 \cdot \sqrt{\pi} \cdot\left(\mathrm{e}^{-\mathrm{p} \cdot \mathrm{i}}-1\right)}{100 \cdot \pi^{2}-\mathrm{p}^{2}} \nonumber$

$\mathrm{W}(\mathrm{x}, \mathrm{p}) :=\frac{1}{2 \cdot \pi} \cdot \int_{-\infty}^{\infty} \overline{\Phi\left(\mathrm{p}+\frac{\mathrm{s}}{2}\right)} \cdot \exp (-\mathrm{i} \cdot \mathrm{s} \cdot \mathrm{x}) \cdot \Phi\left(\mathrm{p}-\frac{\mathrm{s}}{2}\right) \mathrm{ds} \nonumber$

$\mathrm{N} :=80 \qquad \mathrm{i} :=0 \ldots \mathrm{N} \qquad \mathrm{x}_{\mathrm{i}} :=\frac{\mathrm{i}}{\mathrm{N}} \\ \mathrm{j} :=0 \ldots \mathrm{N} \qquad \mathrm{p}_{\mathrm{j}} :=-40+\frac{80 \cdot \mathrm{j}}{\mathrm{N}} \qquad \text{Wigner}_{i, j} :=W\left(x_{i}, p_{j}\right) \nonumber$

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