1.22: Relationship Between the Coordinate and Momentum Representations

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

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

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

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

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

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

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

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

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