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\)
Section 22.3 Schrödinger’s Equation in Spherical Coordinates
Schrödinger’s equation is
\begin{equation}
\Hop\Psi = i\hbar\frac{\partial\Psi}{\partial t}\tag{22.3.1}
\end{equation}
For one-dimensional waves, the Hamiltonian is
\begin{equation}
\Hop= -\frac{\hbar^2}{2\mu}\frac{\partial^2}{\partial x^2} + U(x)\tag{22.3.2}
\end{equation}
In a central potential the role of the second derivative with respect to \(x\) is played by the Laplacian operator \(\Lap\) and the potential energy is a function only on the separation variable \(U=U(r)\text{,}\) making the Hamiltonian:
\begin{equation}
\Hop= -\frac{\hbar^2}{2\mu}\Lap + U(r)\tag{22.3.3}
\end{equation}
Because of the parameter \(r\text{,}\) this problem is clearly asking for the use of spherical coordinates, centered at the origin of the central force.
In rectangular coordinates, we know that the Laplacian \(\Lap\) is given by:
\begin{equation*}
\Lap= \frac{\partial^2}{\partial x^2} +
\frac{\partial^2}{\partial z^2}
+ \frac{\partial^2}{\partial z^2}
\end{equation*}
What is the Laplacian in spherical coordinates? Since \(\Lap\defeq\grad\cdot\grad\text{,}\) combine the spherical coordinate definitions of gradient and divergence
\begin{align}
\grad V \amp=
\frac{\partial V}{\partial r}\,\rhat
+ \frac{1}{r}\frac{\partial V}{\partial \theta}\,\that
+ \frac{1}{r\sin\theta}\frac{\partial V}{\partial \phi}\,\phat\tag{22.3.4}\\
\grad\cdot{{\vec{v}}} \amp=
\frac{1}{r^2} \frac{\partial}{\partial r}\left({r^2}v_{r}\right)
+ \frac{1}{r\sin\theta}\frac{\partial}{\partial\theta}
\left({\sin\theta}\,v_{\theta}\right)
+ \frac{1}{r\sin\theta}\frac{\partial v_\phi}{\partial\phi}\tag{22.3.5}
\end{align}
to obtain:
\begin{align}
\Lap
\amp = \frac{1}{r^2}\frac{\partial}{\partial r}
\left(r^2\frac{\partial}{\partial r} \right)\tag{22.3.6}\\
\amp \qquad
+ \frac{1}{r^2}\left[\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}
\left( \sin\theta\frac{\partial}{\partial\theta} \right)
+ \frac{1}{\sin^2\theta}\frac{\partial^2}{\partial\phi^2}
\right]\tag{22.3.7}
\end{align}
To focus our attention on an important physical quantity, we will give the combination of angular derivatives which appears in
(22.3.7) a new name:
\begin{equation}
\Lop
\defeq -\hbar^2 \left[
\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}
\left( \sin\theta\frac{\partial}{\partial\theta} \right)
+ \frac{1}{\sin^2\theta}\frac{\partial^2}{\partial\phi^2}
\right]\tag{22.3.8}
\end{equation}
Notice the conventional factor of
\(-\hbar^2\text{.}\) \(\hbar\) is a constant, with dimensions of angular momentum. Its value is
\(1.05459\times10^{-27}~\text{ergsec}\) or
\(6.58217\times10^{-16}~\text{eV-sec}\text{.}\) The operator
\(\Lop\) represents the square of the angular momentum of the system. With this definition,
(22.3.7) divides into radial and angular pieces
\begin{equation}
\Lap
= \frac{1}{r^2}\frac{\partial}{\partial r}
\left({r^2}\frac{\partial}{\partial r}\right)
- \frac{1}{\hbar^2r^2} \Lop^2\tag{22.3.9}
\end{equation}
