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2.58: The Wigner Distribution for the 3s State of the 1D Hydrogen Atom

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    This tutorial presents three pictures of the 3s 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 3s wave function is:

    \[ \begin{matrix} \Psi (x) = \frac{2}{243} \sqrt{3} \text{x} \left( 27-18x + 2x^2 \right) \text{exp} \left( - \frac{x}{3} \right) & \int_0^{ \infty} \Psi (x)^2 dx = 1 \end{matrix} \nonumber \]

    Screen Shot 2019-06-22 at 9.44.04 PM.png

    The 3s energy is -0.065 Eh.

    \[ \Phi (p) = \frac{1}{ \sqrt{2 \pi}} \int_0^{ \infty} \text{exp(-i p x)} \Psi (x) \text{dx} \rightarrow \left( -2^{ \frac{1}{2}} \right) 3^{ \frac{1}{2}} \frac{9p^2 + 6ip - 1}{(3ip + 1)^4 \pi^{ \frac{1}{2}}} \nonumber \]

    Screen Shot 2019-06-22 at 9.44.10 PM.png

    The Wigner function (phase‐space representation) for the hydrogen atom 3s state is generated using the momentum wave function.

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

    The Wigner distribution is displayed graphically.

    \[ \begin{matrix} N = 100 & i = 0 .. N & x_i = \frac{30i}{N} & j = 0 .. N & p_j = -2 + \frac{4j}{N} & \text{Wigner}_{i,~j} = \text{W} \left( x_i,~ p_j \right) \end{matrix} \nonumber \]

    Screen Shot 2019-06-22 at 9.44.17 PM.png

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

    This page titled 2.58: The Wigner Distribution for the 3s State of the 1D Hydrogen Atom is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Frank Rioux via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.