Skip to main content
Chemistry LibreTexts

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

  • Page ID
    158590
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    \( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)

    ( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)

    \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

    \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)

    \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

    \( \newcommand{\Span}{\mathrm{span}}\)

    \( \newcommand{\id}{\mathrm{id}}\)

    \( \newcommand{\Span}{\mathrm{span}}\)

    \( \newcommand{\kernel}{\mathrm{null}\,}\)

    \( \newcommand{\range}{\mathrm{range}\,}\)

    \( \newcommand{\RealPart}{\mathrm{Re}}\)

    \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

    \( \newcommand{\Argument}{\mathrm{Arg}}\)

    \( \newcommand{\norm}[1]{\| #1 \|}\)

    \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

    \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    \( \newcommand{\vectorA}[1]{\vec{#1}}      % arrow\)

    \( \newcommand{\vectorAt}[1]{\vec{\text{#1}}}      % arrow\)

    \( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vectorC}[1]{\textbf{#1}} \)

    \( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)

    \( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)

    \( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)

    \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    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:

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

    Screen Shot 2019-06-20 at 9.59.08 PM.png

    The ground state energy is -0.5 Eh.

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

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

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

    Screen Shot 2019-06-20 at 10.02.44 PM.png

    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.

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

    Screen Shot 2019-06-22 at 9.16.12 PM.png


    This page titled 2.55: The Wigner Distribution for the 1s 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.