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, , .
The coordinates r,
and
are related to the standard ones
by
x
=
rcossin
y
=
rsinsin
/TD>
=
rcos
(11.6)
where
0 < r < ,
0 <
<
and
0 <
< 2. In these new coordinates
we have