1.30: From Coordinate Space to Momentum Space and Back
- Page ID
- 143937
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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.
