Skip to main content
Engineering LibreTexts

1.12: Expectation values of position

  • Page ID
    49375
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    Given that P(x) is the probability density of the electron at position x, we can determine the average, or expectation value of x from

    \[ \langle x\rangle =\frac{\int^{+\infty}_{-\infty} xP(x)dx}{\int^{+\infty}_{-\infty} P(x)dx} \nonumber \]

    Of course if the wavefunction is normalized then the denominator is 1.

    We could also write this in terms of the wavefunction

    \[ \langle x\rangle =\frac{\int^{+\infty}_{-\infty} x|\psi(x)|^{2} dx}{\int^{+\infty}_{-\infty} |\psi(x)|^{2}dx} \nonumber \]

    Where once again if the wavefunction is normalized then the denominator is 1.

    Since \(|\psi(x)|^{2} = \psi(x)^{*}\psi(x)\),

    \[ \langle x\rangle =\frac{\int^{+\infty}_{-\infty} \psi(x)^{*}x\psi(x) dx}{\int^{+\infty}_{-\infty}\psi(x)^{*}\psi(x) dx} \nonumber \]


    This page titled 1.12: Expectation values of position is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Marc Baldo (MIT OpenCourseWare) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.