1.69: The Wigner Distribution Function for the Harmonic Oscillator

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Given the quantum number this Mathcad file calculates the Wigner distribution function for the specified harmonic oscillator eigenstate using the coordinate wave function.

Quantum number: \( n :=4\)

Harmonic oscillator coordinate eigenstate:

\[
\Psi(\mathrm{n}, \mathrm{x}) :=\frac{1}{\sqrt{2^{\mathrm{n}} \cdot \mathrm{n} ! \sqrt{\pi}}} \cdot \operatorname{Her}(\mathrm{n}, \mathrm{x}) \cdot \exp \left(-\frac{\mathrm{x}^{2}}{2}\right)
\nonumber \]

Display coordinate wave function and distribution function:

Calculate Wigner distribution:

\[
\mathrm{W}(\mathrm{n}, \mathrm{x}, \mathrm{p}) :=\frac{1}{\pi^{\frac{3}{2}}} \cdot \int_{-\infty}^{\infty} \Psi\left(\mathrm{n}, \mathrm{x}+\frac{\mathrm{s}}{2}\right) \cdot \exp (\mathrm{i} \cdot \mathrm{s} \cdot \mathrm{p}) \cdot \Psi\left(\mathrm{n}, \mathrm{x}-\frac{\mathrm{s}}{2}\right) \mathrm{ds}
\nonumber \]

Display Wigner distribution:

\[
\mathrm{N} :=80 \qquad \mathrm{i} :=0 \ldots \mathrm{N} \qquad \mathrm{x}_{\mathrm{i}}=-4+\frac{8 \cdot \mathrm{i}}{\mathrm{N}} \\ \mathrm{j} :=0 \ldots \mathrm{N} \qquad \mathrm{p}_{\mathrm{j}}=-5+\frac{10 \cdot \mathrm{j}}{\mathrm{N}} \qquad \text{Wigner}_{i,j} : = W(n, x_{i}, p_{j})
\nonumber \]

Calculate the momentum distribution function using the Wigner function:

\[
\rho(\mathrm{p}) :=\int_{-\infty}^{\infty} \mathrm{W}(\mathrm{n}, \mathrm{x}, \mathrm{p}) \mathrm{dx} \qquad \mathrm{p} :=-5,-4.95 \ldots5
\nonumber \]