Skip to main content
Engineering LibreTexts

1.18: The Uncertainty Principle

  1. Last updated
  2. Save as PDF
  • Page ID
    50119

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

    Now that we see that k is simply related to momentum, and ω is simply related to energy, we can revisit the uncertainty relation of Equation (1.10.13)

    \[ |\sigma_{x}||\sigma_{k}| \geq \frac{1}{2} \nonumber \]

    which after multiplication by ħ becomes

    \[ \Delta p \Delta x \geq \frac{\hbar}{2} \nonumber \]

    This is the celebrated Heisenberg uncertainty relation. It states that we can never know both position and momentum exactly.

    For example, we have seen from our Fourier transform pairs that to know position exactly means that in k-space the wavefunction is \(\Psi(k)=\text{exp}[-ikx_{0}]\). Since \(|\Psi(k)|^{2}=1\) all values of k, and hence all values of momentum are equiprobable. Thus, momentum is perfectly undefined if position is perfectly defined.

    This page titled 1.18: The Uncertainty Principle 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.