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11.2: Spherical Coordinates

Last updated

Nov 4, 2024
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- 11.1: The momentum operator as a vector
- 11.3: Solutions independent of angular variables

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The solution to Schrödinger's equation in three dimensions is quite complicated in general. Fortunately, nature lends us a hand, since most physical systems are "rotationally invariant", i.e., $V(x)$ depends on the size of $x$ , but not its direction! In that case it helps to introduce spherical coordinates, as denoted in Fig. $\PageIndex{1}$ .

Figure $\PageIndex{1}$ : The spherical coordinates $r, \theta, \varphi$ .

The coordinates $r, \theta$ and $\phi$ are related to the standard ones by

$\begin{aligned}x & =r \cos \varphi \sin \theta \\ y & =r \sin \varphi \sin \theta \\ & =r \cos \theta\end{aligned}$

where $0<r<\infty, 0<\theta<\pi$ and $0<\phi<2 \pi$ . In these new coordinates we have

$\Delta f(r, \theta, \varphi)=\frac{1}{r^2} \frac{\partial}{\partial r}\left(r^2 \frac{\partial}{\partial r} f(r, \theta, \varphi)\right)-\frac{1}{r^2}\left[\frac{1}{\sin \theta} \frac{\partial}{\partial \theta}\left(\sin \theta \frac{\partial}{\partial \theta} f(r, \theta, \varphi)\right)+\frac{\partial^2}{\partial \varphi^2} f(r, \theta, \varphi)\right] .$

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