Skip to main content
Chemistry LibreTexts

1.73: Time‐dependent Wigner Function for Harmonic Oscillator Transitions

  1. Last updated
  2. Save as PDF
  • Page ID
    156463

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

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

    \( \newcommand{\dsum}{\displaystyle\sum\limits} \)

    \( \newcommand{\dint}{\displaystyle\int\limits} \)

    \( \newcommand{\dlim}{\displaystyle\lim\limits} \)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

    \(\newcommand{\longvect}{\overrightarrow}\)

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

    \(\newcommand{\ket}[1]{\left| #1 \right>}\)
    \(\newcommand{\bra}[1]{\left< #1 \right|}\)
    \(\newcommand{\braket}[2]{\left< #1 \vphantom{#2} \right| \left. #2 \vphantom{#1} \right>}\)
    \(\newcommand{\braopket}[3]{\left< #1 \vphantom{#2}\vphantom{#3} \right| #2 \vphantom{#1}\vphantom{#3} \left| #3 \vphantom{#1}\vphantom{#2} \right>}\)
    \(\newcommand{\qmvec}[1]{\mathbf{\vec{#1}}}\)
    \(\newcommand{\op}[1]{\hat{\mathbf{#1}}}\)
    \(\newcommand{\expect}[1]{\langle #1 \rangle}\)
    \(\newcommand{\dfn}[1]{\emph{\textbf{#1}}}\)

    \(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)

    Initial state:

    \[
    \mathrm{m} :=0 \qquad \mathrm{E}_{\mathrm{m}} :=\mathrm{m}+\frac{1}{2}
    \nonumber \]

    Final state:

    \[
    \mathrm{n} :=1 \qquad \mathrm{E}_{\mathrm{n}} :=\mathrm{n}+\frac{1}{2} \qquad t : = \text{FRAME}
    \nonumber \]

    Define Wigner distribution function for a linear superposition of the initial and final harmonic oscillator state.

    \[
    W(x,p) : = \frac{1}{\pi^{\frac{3}{2}}}\int_{-\infty}^{\infty}\left[\frac{1}{\sqrt{2^{\mathrm{n}} \cdot \mathrm{n} ! \sqrt{\pi}}} \cdot \operatorname{Her}\left(\mathrm{n}, x+\frac{s}{2}\right) \cdot \exp \left[-\frac{\left(\mathrm{x}+\frac{\mathrm{s}}{2}\right)^{2}}{2}\right] \cdot \exp \left(\mathrm{i} \cdot \mathrm{E}_{\mathrm{n}} \cdot \mathrm{t}\right)+\frac{1}{\sqrt{2^{\mathrm{m}} \cdot \mathrm{m} ! \cdot \sqrt{\pi}}} \cdot \operatorname{Her}\left(\mathrm{m}, \mathrm{x}+\frac{\mathrm{s}}{2}\right) \cdot \exp \left[-\frac{\left(\mathrm{x}+\frac{\mathrm{s}}{2}\right)^{2}}{2}\right] \cdot \exp \left(\mathrm{i} \cdot \mathrm{E}_{\mathrm{m}} \cdot \mathrm{t}\right)\right] \\ \cdot \exp (\mathrm{i} \cdot \mathrm{s} \cdot \mathrm{p}) \cdot \left[\frac{1}{\sqrt{2^{\mathrm{n}} \cdot n ! \cdot \sqrt{\pi}}} \cdot \operatorname{Her}\left(\mathrm{n}, \mathrm{x}-\frac{\mathrm{s}}{2}\right) \cdot \exp \left[-\frac{\left(\mathrm{x}-\frac{\mathrm{s}}{2}\right)^{2}}{2}\right] \cdot \exp \left(-\mathrm{i} \cdot \mathrm{E}_{\mathrm{n}} \cdot \mathrm{t}\right)\\+\frac{1}{\sqrt{2^{\mathrm{m}} \cdot \mathrm{m} ! \cdot \sqrt{\pi}}} \cdot \operatorname{Her}\left(\mathrm{m}, \mathrm{x}-\frac{\mathrm{s}}{2}\right)\cdot\exp\left[-\frac{\left(\mathrm{x}-\frac{\mathrm{s}}{2}\right)^{2}}{2}\right]\cdot \exp\left(\mathrm{-i}\cdot\mathrm{E}_{\mathrm{m}}\cdot\mathrm{t}\right)\right]\mathrm{ds}
    \nonumber \]

    Display Wigner distribution:

    \[
    \mathrm{N}=60 \qquad \mathrm{i} :=0 \ldots \mathrm{N} \qquad \mathrm{x}_{\mathrm{i}} :=-2.5+\frac{5 \cdot \mathrm{i}}{\mathrm{N}} \\ \mathrm{j} :=0 \ldots \mathrm{N} \qquad \mathrm{P}_{\mathrm{j}} :=-2.5+\frac{5 \mathrm{j}}{\mathrm{N}} \qquad \text{Wigner}_{i,j}: = W(x_{i},p_{j})
    \nonumber \]

    clipboard_e1614b0d7f94d556da2ee3535dc873b4f.png

    This page titled 1.73: Time‐dependent Wigner Function for Harmonic Oscillator Transitions was last modified on Wed, 11 Jan 2023 07:23:08 GMT and 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.