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1.22: Relationship Between the Coordinate and Momentum Representations

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    143928

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    A quon has position \(x_{1} :| x_{1} \rangle\)

    Coordinate space \(\Leftrightarrow\) Fourier Transform \(\Leftrightarrow\) Momentum space

    \[
    \langle x | x_{1}\rangle=\delta\left(x-x_{1}\right)= \xrightleftharpoons[\int\langle x | p\rangle\langle p | x_{1}\rangle d p]{\int\langle p | x\rangle\langle x | x_{1}\rangle d x} \langle p | x_{1}\rangle=\exp \left(-\frac{i p x_{1}}{\hbar}\right)
    \nonumber \]

    clipboard_ea35cae035f07fff91a64e87fac9a5b68.png

    A quon has momentum \(p_{1} :| p_{1} \rangle\)

    Coordinate space \(\Leftrightarrow\) Fourier Transform \(\Leftrightarrow\) Momentum space

    \[
    \langle x | p_{1}\rangle=\exp \left(\frac{i p_{1} x}{\hbar}\right) \xrightleftharpoons[\int\langle x | p\rangle\langle p | p_{1}\rangle d p]{\int\langle p | x\rangle\langle x | p_{1}\rangle d x} \langle p | p_{1}\rangle=\delta\left(p-p_{1}\right)
    \nonumber \]

    clipboard_ed5858b6a155b0ce64d8e73f349f84359.png

    Please note the important role that the coordinate and momentum completeness relations play in these transformations.

    \[
    \int | x \rangle\langle x|d x=1 \quad \text { and } \quad \int| p\rangle\langle p|d p=1
    \nonumber \]

    This page titled 1.22: Relationship Between the Coordinate and Momentum Representations 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.