- #1
Dazzabaijan
- 9
- 0
I'm having trouble with trying to find the expansion coefficients of a superposition of a Gaussian wave packet.
First I'm decomposing a Gaussian wave packet
$$\psi(\textbf{r},0) = \frac{1}{(2\pi)^{3/4}\sigma^{3/2}}\text{exp}\left[ -\frac{(\textbf{r} - \textbf{r}_0)^2}{4\sigma^2} + i\textbf{k}_0 \cdot \textbf{r}\right]$$
into a linear superposition of eigenfunctions of the hydrogen atom, where the initial position and momentum is chosen as ##\textbf{r}_0## and ##\textbf{p}_0 = \hbar \textbf{k}_0##, and also a spatial width ##\sigma## such that the initial wave packet ##\psi(\textbf{r},0)## can to a sufficient approximation be superimposed by bound-state eigenfunctions ##\varphi_\textit{nlm}(\textbf{r})##,
$$\psi(\textbf{r},0) = \sum_{n=1}^{\infty}\sum_{l=0}^{n-1}\sum_{m=-l}^{l} b_{\textit{nlm}}\varphi_\textit{nlm}(\textbf{r})$$
Having solved the Schrödinger Equation of a hydrogen atom with a Hamiltonian containing the Coulomb potential, the eigenfunctions were obtained to be a product of the Radial Wave solution and Spherical Harmonics
$$\varphi_\textit{nlm}(\textbf{r}) = R_{nl}(r) Y_{lm}(\theta, \phi)$$
where
$$R_{nl}(r) = \frac{1}{a^{3/2}}\frac{2}{n^2}\sqrt{\frac{(n-l-1)!}{(n+1)!}}\left(\frac{2r}{na}\right)^l e^{-\frac{r}{na}} L^{2l+1}_{n-l-1}\left(\frac{2r}{na}\right)$$
where ##L^{2l+1}_{n-l-1}\left(\frac{2r}{na}\right)## is the Laguerre Polynomial which has the form of
$$L^{k}_{p}\left(x\right) = \sum_{k}^{p} (-1)^s \binom {p+k}{p-s} \frac{x^s}{s!}$$
and
$$Y_{lm}(\theta, \phi) = (-1)^m \sqrt{\frac{2l+1}{4\pi}\frac{(l-m)!}{(l+m)!}} P^{m}_{l} (\cos\theta) e^{im\phi}$$
with the Legendre Polynomial contained within the Associated Legendre Functions
$$P^{m}_{l} (x) = (1 - x^2)^{\frac{m}{2}} \frac{d^m}{dx^m}(P_{l}(x))$$
$$P_{l}(x) = \frac{1}{2^l l! }\frac{d^l}{dx^l}[(x^2 - 1)^l]$$
What I ultimately would like to find is the closed form expansion coefficients ##b_{nlm}## of the eigenfunction superposition, now from my knowledge of finding the expansion coefficients of a single-indexed superposition of a wavefunction, the same method can be applied on this superposition which involves 3 different indices, such that
$$
\begin{equation*}
\begin{split}
\langle\varphi_\textit{nlm}(\textbf{r})|\psi(\textbf{r},0)\rangle &= \langle\varphi_\textit{nlm}(\textbf{r})|\sum_{n=1}^{\infty}\sum_{l=0}^{n-1}\sum_{m=-l}^{l} b_{\textit{nlm}}\varphi_\textit{nlm}(\textbf{r})\rangle\\
&= b_{\textit{nlm}}\sum_{n=1}^{\infty}\sum_{l=0}^{n-1}\sum_{m=-l}^{l}\langle\varphi_\textit{n'l'm'}(\textbf{r})| \varphi_\textit{nlm}(\textbf{r})\rangle\\
&= b_{\textit{nlm}}\sum_{n=1}^{\infty}\sum_{l=0}^{n-1}\sum_{m=-l}^{l}\delta_{n'n}\delta_{l'l}\delta_{m'm}\\
&= b_{nlm}
\end{split}
\end{equation*}
$$
My attempt:
To avoid doing the scalar product directly and having to compute the whole integral, I was thinking of using the decomposition of stationary harmonic plane wave into partial waves, as such
$$e^{\textbf{k}\cdot\textbf{r}} = e^{ikz} = e^{ikr\cos\theta} = \sum^{l=0}_{\infty}(2l+1)i^l j_l(kr) P_l(\cos\theta)$$
where ##i## is the imaginary number, ##\textbf{k}## and ##\textbf{r}## are the wave and position vectors, ##j_l## are the spherical Bessel functions, ##P_l## are Legendre Polynomials. But according to Wikipedia (https://en.wikipedia.org/wiki/Plane_wave_expansion), it can also be rewritten as
$$e^{\textbf{k}\cdot\textbf{r}} = 4\pi \sum_{l=0}^{\infty}\sum_{m=-l}^{l} i^l j_l(kr)Y^m_l(\hat{\textbf{k}})Y^{m*}_l(\hat{\textbf{r}})$$
In theory, this should now mean that I can rewrite the scalar product in terms of these sum of spherical waves, so that
$$
\begin{equation*}
\begin{split}
b_{nlm} &= \langle\varphi_\textit{nlm}(\textbf{r})|\psi(\textbf{r},0)\rangle\\
&= \langle R_{nl}(r) Y_{lm}(\theta, \phi)|\frac{1}{(2\pi)^{3/4}\sigma^{3/2}}\text{exp}\left[ -\frac{(\textbf{r} - \textbf{r}_0)^2}{4\sigma^2}\right]\cdot 4\pi\sum_{l=0}^{\infty}\sum_{m=-l}^{l} i^l j_l(kr)Y^m_l(\hat{\textbf{k}})Y^{m*}_l(\hat{\textbf{r}})\rangle\\
\end{split}
\end{equation*}
$$
and so this is where I'm at. I'm not sure how to go from here. I hope I've clarified my question well.
First I'm decomposing a Gaussian wave packet
$$\psi(\textbf{r},0) = \frac{1}{(2\pi)^{3/4}\sigma^{3/2}}\text{exp}\left[ -\frac{(\textbf{r} - \textbf{r}_0)^2}{4\sigma^2} + i\textbf{k}_0 \cdot \textbf{r}\right]$$
into a linear superposition of eigenfunctions of the hydrogen atom, where the initial position and momentum is chosen as ##\textbf{r}_0## and ##\textbf{p}_0 = \hbar \textbf{k}_0##, and also a spatial width ##\sigma## such that the initial wave packet ##\psi(\textbf{r},0)## can to a sufficient approximation be superimposed by bound-state eigenfunctions ##\varphi_\textit{nlm}(\textbf{r})##,
$$\psi(\textbf{r},0) = \sum_{n=1}^{\infty}\sum_{l=0}^{n-1}\sum_{m=-l}^{l} b_{\textit{nlm}}\varphi_\textit{nlm}(\textbf{r})$$
Having solved the Schrödinger Equation of a hydrogen atom with a Hamiltonian containing the Coulomb potential, the eigenfunctions were obtained to be a product of the Radial Wave solution and Spherical Harmonics
$$\varphi_\textit{nlm}(\textbf{r}) = R_{nl}(r) Y_{lm}(\theta, \phi)$$
where
$$R_{nl}(r) = \frac{1}{a^{3/2}}\frac{2}{n^2}\sqrt{\frac{(n-l-1)!}{(n+1)!}}\left(\frac{2r}{na}\right)^l e^{-\frac{r}{na}} L^{2l+1}_{n-l-1}\left(\frac{2r}{na}\right)$$
where ##L^{2l+1}_{n-l-1}\left(\frac{2r}{na}\right)## is the Laguerre Polynomial which has the form of
$$L^{k}_{p}\left(x\right) = \sum_{k}^{p} (-1)^s \binom {p+k}{p-s} \frac{x^s}{s!}$$
and
$$Y_{lm}(\theta, \phi) = (-1)^m \sqrt{\frac{2l+1}{4\pi}\frac{(l-m)!}{(l+m)!}} P^{m}_{l} (\cos\theta) e^{im\phi}$$
with the Legendre Polynomial contained within the Associated Legendre Functions
$$P^{m}_{l} (x) = (1 - x^2)^{\frac{m}{2}} \frac{d^m}{dx^m}(P_{l}(x))$$
$$P_{l}(x) = \frac{1}{2^l l! }\frac{d^l}{dx^l}[(x^2 - 1)^l]$$
What I ultimately would like to find is the closed form expansion coefficients ##b_{nlm}## of the eigenfunction superposition, now from my knowledge of finding the expansion coefficients of a single-indexed superposition of a wavefunction, the same method can be applied on this superposition which involves 3 different indices, such that
$$
\begin{equation*}
\begin{split}
\langle\varphi_\textit{nlm}(\textbf{r})|\psi(\textbf{r},0)\rangle &= \langle\varphi_\textit{nlm}(\textbf{r})|\sum_{n=1}^{\infty}\sum_{l=0}^{n-1}\sum_{m=-l}^{l} b_{\textit{nlm}}\varphi_\textit{nlm}(\textbf{r})\rangle\\
&= b_{\textit{nlm}}\sum_{n=1}^{\infty}\sum_{l=0}^{n-1}\sum_{m=-l}^{l}\langle\varphi_\textit{n'l'm'}(\textbf{r})| \varphi_\textit{nlm}(\textbf{r})\rangle\\
&= b_{\textit{nlm}}\sum_{n=1}^{\infty}\sum_{l=0}^{n-1}\sum_{m=-l}^{l}\delta_{n'n}\delta_{l'l}\delta_{m'm}\\
&= b_{nlm}
\end{split}
\end{equation*}
$$
My attempt:
To avoid doing the scalar product directly and having to compute the whole integral, I was thinking of using the decomposition of stationary harmonic plane wave into partial waves, as such
$$e^{\textbf{k}\cdot\textbf{r}} = e^{ikz} = e^{ikr\cos\theta} = \sum^{l=0}_{\infty}(2l+1)i^l j_l(kr) P_l(\cos\theta)$$
where ##i## is the imaginary number, ##\textbf{k}## and ##\textbf{r}## are the wave and position vectors, ##j_l## are the spherical Bessel functions, ##P_l## are Legendre Polynomials. But according to Wikipedia (https://en.wikipedia.org/wiki/Plane_wave_expansion), it can also be rewritten as
$$e^{\textbf{k}\cdot\textbf{r}} = 4\pi \sum_{l=0}^{\infty}\sum_{m=-l}^{l} i^l j_l(kr)Y^m_l(\hat{\textbf{k}})Y^{m*}_l(\hat{\textbf{r}})$$
In theory, this should now mean that I can rewrite the scalar product in terms of these sum of spherical waves, so that
$$
\begin{equation*}
\begin{split}
b_{nlm} &= \langle\varphi_\textit{nlm}(\textbf{r})|\psi(\textbf{r},0)\rangle\\
&= \langle R_{nl}(r) Y_{lm}(\theta, \phi)|\frac{1}{(2\pi)^{3/4}\sigma^{3/2}}\text{exp}\left[ -\frac{(\textbf{r} - \textbf{r}_0)^2}{4\sigma^2}\right]\cdot 4\pi\sum_{l=0}^{\infty}\sum_{m=-l}^{l} i^l j_l(kr)Y^m_l(\hat{\textbf{k}})Y^{m*}_l(\hat{\textbf{r}})\rangle\\
\end{split}
\end{equation*}
$$
and so this is where I'm at. I'm not sure how to go from here. I hope I've clarified my question well.
