2.55: The Wigner Distribution for the 1s State of the 1D Hydrogen Atom

This tutorial presents three pictures of the 1s state of the one‐dimensional hydrogen atom using its position, momentum and phase‐space representations.

The energy operator for the one‐dimensional hydrogen atom in atomic units is:

\[ \frac{-1}{2} \frac{d^2}{dx^2} \blacksquare - \frac{1}{x} \blacksquare \nonumber \]

The ground state eigenstate is:

\[ \begin{matrix} \Psi (x) = 2 x \text{exp} (-x) & \int_0^{ \infty} \Psi (x)^2 dx = 1 \end{matrix} \nonumber \]

The ground state energy is -0.5 E_h.

\[ \frac{-1}{2} \frac{d^2}{x^2} \Psi (x) - \frac{1}{x} \Psi (x) = E \Psi (x) \text{ solve, E} \rightarrow \frac{-1}{2} \nonumber \]

The momentum wave function is generated by the following Fourier transform of the coordinate space wave function.

\[ \Phi (p) = \frac{1}{ \sqrt{2 \pi}} \int_0^{ \infty} \text{exp(-i p x)} \Psi (x) dx \rightarrow \frac{2^{ \frac{1}{2}}}{ \text{(i p + 1)} \pi^{ \frac{1}{2}}} \nonumber \]

The Wigner function for the hydrogen atom ground state is generated using the momentum wave function.

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

The Wigner distribution is displayed graphically.

\[ \begin{matrix} N = 60 & i = 0 .. N & x_i = \frac{6i}{N} & j = 0 .. N & p_j = -5 + \frac{10j}{N} & \text{Wigner}_{i,~j} = \text{W} \left( x_i,~ p_j \right) \end{matrix} \nonumber \]

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