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\)
Section 17.1 States for the Hydrogen atom
The energy eigenstates for the hydrogen atom are
\begin{equation}
\psi_{n\ell m}(r,\theta,\phi)
= R_{n\ell}(r)Y_\ell^m(\theta,\phi)\tag{17.1.1}
\end{equation}
where the radial functions
\(R_{n\ell}(r)\) were found in
(16.3.13) and the angular functions are spherical harmonics
\(Y_\ell^m(\theta,\phi)\) found in
Section 15.4 .
The pure state solutions to Schrödinger’s equation are
\begin{equation}
\Psi_{n\ell m}(r,\theta,\phi,t)
= R_{n\ell}(r)Y_\ell^m(\theta,\phi) e^{-iE_nt/h}\tag{17.1.2}
\end{equation}
where
\(E_n\) are the energy eigenvalues given in
(16.3.13) .
As is always the case for quantum systems, it is possible for the atom to exist in mixed states that are not pure eigenstates. These states can be described by appropriate linear combinations of the eigenfunctions given in
(17.1.2) :
\begin{equation}
\Psi(r,\theta,\phi,t)
= \sum_{n=1}^\infty \sum_{\ell=0}^{n-1} \sum_{m=-\ell}^\ell c_{n\ell m}
R_{n\ell}(r)Y_\ell^m(\theta,\phi) e^{-iE_nt/h}\tag{17.1.3}
\end{equation}
Given any arbitrary (but mathematically well-behaved) function \(\Psi(r,\theta,\phi,0)\) that represents the state of the atom at \(t=0\text{,}\) we can find the expansion coefficents \(c_{n\ell m}\) by expanding the initial condition \(\Psi(r,\theta,\phi,0)\) in terms of the energy eigenstates
\begin{equation}
\Psi(r,\theta,\phi,0)
= \sum_{n=1}^\infty \sum_{\ell=0}^{n-1} \sum_{m=-\ell}^\ell
c_{n\ell m} R_{n\ell}(r)Y_\ell^m(\theta,\phi)\tag{17.1.4}
\end{equation}
where
\begin{equation}
c_{n\ell m}
= \int_0^\infty r^2 dr \int_0^\pi \sin\theta\,d\theta
\int_0^{2\pi} d\phi\;
(R_{n\ell}^*(r)Y_\ell^{m}{}^*(\theta,\phi))\tag{17.1.5}
\end{equation}
