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# 1.30: From Coordinate Space to Momentum Space and Back


The 2s state of the one-dimensional hydrogen atom is used to illustrate transformations back and forth between the coordinate and momentum representations.

$\Psi_{2}(x) :=\frac{1}{\sqrt{8}} \cdot x \cdot(2-x) \cdot \exp \left(-\frac{x}{2}\right) \nonumber$

The 2s state is Fourier transformed into momentum space (using atomic units) and the magnitude of the momentum wave function is displayed.

$\langle p | \Psi_{2}\rangle=\int_{0}^{\infty}\langle p | x\rangle\langle x | \Psi_{2}\rangle d x \quad \text { where } \quad\langle p | x\rangle=\frac{1}{\sqrt{2 \pi}} \exp \left(\frac{-i p x}{\hbar}\right) \nonumber$

$\Psi_{2}(\mathrm{p}) :=\frac{1}{\sqrt{2 \cdot \pi}} \int_{0}^{\infty} \exp (-\mathrm{i} \cdot \mathrm{p} \cdot \mathrm{x}) \cdot \Psi_{2}(\mathrm{x}) \mathrm{d} x \text { simplify } \rightarrow \frac{2}{\pi^{\frac{1}{2}}} \cdot \frac{2 \cdot \mathrm{i} \cdot \mathrm{p}-1}{(2 \cdot \mathrm{i} \cdot \mathrm{p}+1)^{3}} \nonumber$

The return to coordinate space is carried out in the numeric mode, integrating over the range of momentum values shown above ($$\pm$$10 is effectively $$\pm \infty$$).

$\langle x | \Psi_{2}\rangle=\int_{-\infty}^{\infty}\langle x | p\rangle\langle p | \Psi_{2}\rangle d p \quad \text { where } \quad\langle x | p\rangle=\frac{1}{\sqrt{2 \pi}} \exp \left(\frac{i p x}{\hbar}\right) \nonumber$

$\Psi_{2}(x) :=\int_{-10}^{10} \frac{1}{\sqrt{2 \cdot \pi}} \cdot \exp (i \cdot p \cdot x) \cdot \Psi_{2}(p) d p \nonumber$

The graphical display below shows that we have successfully returned to coordinate space.

This page titled 1.30: From Coordinate Space to Momentum Space and Back 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.

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