2.60: The Wigner Distribution for the 4s State of the 1D Hydrogen Atom

This tutorial presents three pictures of the 4s 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 position 4s wave function is:

\[ \begin{matrix} \Psi (x) = \frac{x}{4} \left( 1 - \frac{3}{4} x + \frac{1}{8} x^2 - \frac{1}{192} x^3 \right) \text{exp} \left( \frac{-x}{4} \right) & \int_0^{ \infty} \Psi (x)^2 \text{dx} = 1 \end{matrix} \nonumber \]

The 4s energy is -0.03125 E_h.

\[ \frac{-1}{2} \frac{d^2}{dx^2} \Psi (x) - \frac{1}{x} \Psi (x) = \text{E} \Psi (x) \text{solve, E} \rightarrow \frac{-1}{32} = -0.03125 \nonumber \]

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

\[ \Phi (p) = \frac{1}{ \sqrt{2 \pi}} \int_0^{ \infty} \text{exp(-i p x)} \Psi (x) dx \rightarrow (-2) 2^{ \frac{1}{2}} \frac{64 \text{i p}^3 - 48 \text{p}^2 - 12 \text{i p} + 1}{(4 \text{i p} + 1)^5 \pi^{ \frac{1}{2}}} \nonumber \]

The Wigner function (phase‐space representation) for the hydrogen atom 4s 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.

\[ \begin{matrix} N = 100 & i = 0 .. N & x_i = \frac{50i}{N} & j = 0 .. N & p_j = -2 + \frac{4j}{N} & \text{Wigner}_{i,~j} = \text{W} \left( x_i,~ p_j \right) \end{matrix} \nonumber \]

Just as for the 2s and 3s states, the Wigner distribution for the 4s state takes on negative values.

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