# Exact Fourier transform of Hydrogen atom wavefunction in 3D position space

From wikipedia we know that the bound state wavefunction $\Psi_{n\ell m}(\boldsymbol{r})$ for the hidrogen atom in position space $(\mathbb{R}^3)$ with spherical coordinates $\boldsymbol{r}=\{r=|\boldsymbol{r}|,\theta,\phi\}$ is given by:

$$\Psi_{n\ell m}(\boldsymbol{r})=\Psi_{n\ell m}(r,\theta,\phi) =R_{n\ell}(r)Y_{\ell m}(\theta,\phi),\tag{1}\\$$

$$R_{n\ell}(r)=2a_n^2N_{n\ell}\,\, (2a_nr)^\ell e^{-a_n r}L^{2\ell+1}_{n-\ell-1}(2a_nr),\tag{2}\\$$

where $a_n=n^{-1},N_{n\ell}=\left(\frac{(n-l-1)!}{(n+l)!}\right)^{1/2}$. $L^{\alpha}_{n}(r)$ denotes an associated Laguerre polynomial of degree $n$. And $n=1,2,3,\cdots;\quad\ell=0,1,2,\cdots,n-1;\quad m=-\ell,\cdots,\ell.\\$

The wavefunction $\Psi_{n\ell m}(r,\theta,\phi)$ satisfies the orthonormal condition: $$\int_0^{2\pi}\int_0^\pi Y_{\ell m}(\theta,\phi)Y^{*}_{\ell' m'}(\theta,\phi)\sin\theta\mathrm{d}\theta\mathrm{d}\phi=\delta_{\ell\ell'}\delta_{mm'}\tag{3}\\$$

$$\int_0^\infty R_{n\ell}(r)R_{n'\ell}(r)r^2\mathrm{d}r=\delta_{nn'}\tag{4}\\$$

We verified (3) and (4) using Mathematica 10.0 for several values of $n,\ell,m$.

From wikipedia we also know that the Fourier transform of it, or the wavefunction in momemtum space $(\mathbb{R}^3)$ with spherical coordinates $\boldsymbol{p}=\{p=|\boldsymbol{p}|,\theta_p,\phi_p\}$ is given by:

$$(2\pi)^{-3/2}\int_{\mathbb{R}^3}e^{-i\boldsymbol{p}\cdot \boldsymbol{r}}\Psi_{n\ell m}(\boldsymbol{r})\mathrm{d}\boldsymbol{r} =\Phi_{n\ell m}(p,\theta_p,\phi_p)=S_{n\ell}(p)Y_{\ell m}(\theta_p,\phi_p)\tag{5}\\$$

$$S_{n\ell}(p)={\color{red}{(-i)^\ell}}\,\,\ell !\,\,a_n^2N_{n\ell}\,\,\frac{2^{2\ell+3} }{\sqrt{2\pi}}\frac{(a_np)^\ell}{(p^2+a_n^{2})^{\ell+2}} C^{\ell+1}_{n-\ell-1}\left(\frac{p^2-a_n^{2}}{p^2+a_n^{2}}\right)\tag{6}\\$$

where $C^{\alpha}_{n}(\kappa)$ denotes a Gegenbauer polynomial of order $n$ and parameter $\alpha$.

The wavefunction $\Phi_{n\ell m}(p,\theta_p,\phi_p)$ also satisfies the orthonormal condition:

$$\int_0^\infty S_{n\ell}(p)S^{*}_{n'\ell}(p)p^2\mathrm{d}p=\delta_{nn'}\tag{7}\\$$ We verified (7) using Mathematica 10.0 for several values of $n,\ell$.

We remark that the phase factor ${\color{red}{(-i)^\ell}}$ is absent in wikipedia. Hage-Hassan obtained a phase factor of ${\color{red}{i^\ell}}$ for 3-Dimensional Hydrogen atom wavefunction in Momentum space. But Hage-Hassan obtained a phase factor of ${\color{red}{-i^\ell}}$ for N-Dimensional Hydrogen atom wavefunction in Momentum space. We are quite confused and want to figure out which one is right.

We can expand the plane wave factor as: $$e^{-i\boldsymbol{p}\cdot \boldsymbol{r}} =4\pi \sum_{\ell'=0}^{\infty}\sum_{m'=-\ell'}^{\ell'}(-i)^{\ell'}j_{\ell'}(pr)Y^{*}_{\ell',m'}(\theta,\phi)Y_{\ell',m'}(\theta_p,\phi_p),\tag{8}\\$$ where $j_{\ell}(r)=\sqrt{\frac{\pi}{2r}}J_{\ell+1/2}(r)$ and $J_{\nu}(r)$ is the Bessel function of first kind. Substituting of (1) and (8) into the left hand side of (5) and integrating over the range of $\theta,\phi$ leads to

$$4\pi (-i)^{\ell}\frac{a_n^2N_{n\ell}}{(2\pi)^{3/2}}\int_{0}^\infty j_{\ell}(pr)(2a_nr)^\ell e^{-a_nr}L^{2\ell+1}_{n-\ell-1}(2a_nr)r^2\mathrm{d}r\\ ={\color{red}{(-i)^\ell}}\,\,\ell !\,\,a_n^2N_{n\ell}\frac{2^{2\ell+3}}{\sqrt{2\pi}}\frac{(a_np)^{\ell}}{(p^2+a_n^{2})^{\ell+2}} C^{\ell+1}_{n-\ell-1}\left(\frac{p^2-a_n^{2}}{p^2+a_n^{2}}\right),\tag{9}\\$$

From (9) we see that the factor $(-i)^\ell$ is needed to balance the phase factor $(-i)^{\ell}$ from the plane wave expansion.

A little bit simplification leads to $$\int_{0}^\infty j_{\ell}(pr)(2a_nr)^\ell e^{-a_nr}L^{2\ell+1}_{n-\ell-1}(2a_nr)r^2\mathrm{d}r\\ =\ell !{2^{2\ell+{\color{red}{2}}}}\frac{(a_np)^{\ell}}{(p^2+a_n^{2})^{\ell+2}} C^{\ell+1}_{n-\ell-1}\left(\frac{p^2-a_n^{2}}{p^2+a_n^{2}}\right),\tag{10}\\$$

If the exponent $\color{red}{2}$ in (10) is replaced by $\color{red}{1}$, then we are able to symbolically verify (10) for several values of $n,\ell$.

So we are seeking for the closed-form formulas for exact Fourier transform of Hydrogen atom wavefunction in 3D position space.

• How do you sub (1) into (8) when (1) depends on $R_{n\ell}(r)$ whereas (8) is an expression for $e^{-ipr}$? – ZeroTheHero Aug 22 '17 at 17:59
• @ZeroTheHero. I did not sub (1) into (8). I sub (1) and (8) into the left hand side of (5). Thanks for checking! – mike Aug 22 '17 at 19:53
• I see... my bad... :( so you're chasing a factor of $2$. As an aside I think your (8) is missing a factor of $1/(pr)$ on the right hand side. I wonder if this is enough to make things work for you (unlikely but one never knows). You can compare your (8) with Eq.(81.16) of Practical Quantum Mechanics by Siegfried Flugge. – ZeroTheHero Aug 22 '17 at 20:03
• @ZeroTheHero. Thanks a lot for checking (8) for me. The definition of $j_\ell(r)$ in Practical Quantum Mechanics by Siegfried Flugge is $j_\ell(r)=\sqrt{\frac{\pi r}{2}}J_{\ell+1/2}(r)$. The definition I used here (just added) is $j_\ell(r)=\sqrt{\frac{\pi }{2r}}J_{\ell+1/2}(r)$. So Eq.(81.16) is identical to (8) here. They both contain $\sqrt{\frac{\pi }{2r}}J_{\ell+1/2}(r)$. Best- – mike Aug 23 '17 at 8:15