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11.2 Spherical coordinates

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. 11.1.

  
Figure 11.1: The spherical coordinates r, $ \theta$, $ \varphi$.
\includegraphics[width=6.0cm]{Figures/spherical.eps}

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

x = rcos$\displaystyle \varphi$sin$\displaystyle \theta$  
y = rsin$\displaystyle \varphi$sin$\displaystyle \theta$  
/TD> = rcos$\displaystyle \theta$ (11.6)

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

$\displaystyle \Delta$f (r,$\displaystyle \theta$,$\displaystyle \varphi$) = $\displaystyle {\frac{1}{r^2}}$$\displaystyle {\frac{\partial}{\partial r}}$$\displaystyle \left(\vphantom{r^2 \frac{\partial}{\partial r}f(r,\theta,\varphi)}\right.$r2$\displaystyle {\frac{\partial}{\partial r}}$f (r,$\displaystyle \theta$,$\displaystyle \varphi$)$\displaystyle \left.\vphantom{r^2 \frac{\partial}{\partial r}f(r,\theta,\varphi)}\right)$ - $\displaystyle {\frac{1}{r^2}}$$\displaystyle \left[\vphantom{\frac{1}{\sin \theta} \frac{\partial}{\partial \t...
...arphi)\right)+\frac{\partial^2}{\partial \varphi^2}f(r,\theta,\varphi) }\right.$$\displaystyle {\frac{1}{\sin \theta}}$$\displaystyle {\frac{\partial}{\partial \theta}}$$\displaystyle \left(\vphantom{\sin\theta \frac{\partial}{\partial \theta}f(r,\theta,\varphi)}\right.$sin$\displaystyle \theta$$\displaystyle {\frac{\partial}{\partial \theta}}$f (r,$\displaystyle \theta$,$\displaystyle \varphi$)$\displaystyle \left.\vphantom{\sin\theta \frac{\partial}{\partial \theta}f(r,\theta,\varphi)}\right)$ + $\displaystyle {\frac{\partial^2}{\partial \varphi^2}}$f (r,$\displaystyle \theta$,$\displaystyle \varphi$)$\displaystyle \left.\vphantom{\frac{1}{\sin \theta} \frac{\partial}{\partial \t...
...arphi)\right)+\frac{\partial^2}{\partial \varphi^2}f(r,\theta,\varphi) }\right]$. (11.7)


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© 1998 Niels Walet, UMIST
Email Niels.Walet@umist.ac.uk