1.30: From Coordinate Space to Momentum Space and Back
- Page ID
- 143937
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.