10.6 Other coordinate systems 307
This result can be rearranged to give the equation
and this is essentially the Schrödinger equation of the hydrogen atom, with fthe 2 p
xorbital. The result of Example 10.10 is the same equation for the 1 sorbital.
0 Exercises 28–33
EXAMPLE 10.12Show that the functionf(r) 1 = 112 rsatisfies the Laplace equation
in three dimensions.
The Laplace equation is
∇
2f 1 = 10
As in Example 10.10, the function does not depend on the angles θand φ. Therefore,
Nowdf 2 dr 1 = 1 − 12 r
2, d
2f 2 dr
21 = 122 r
3. Therefore
We note that this demonstration is not valid when r 1 = 10 , and this singular point
requires special treatment in physical applications.
This example shows that the gravitational and Coulomb potential functions satisfy
the Laplace equation (see Example 9.20 for the two-dimensional case).
0 Exercises 34–37
10.6 Other coordinate systems
In addition to the cartesian and spherical polar coordinates, other systems of
coordinates have been found useful for the description of physical systems. The most
important of these are of the type called orthogonal curvilinear coordinates.
2In the cartesian system (Figure 10.1), the position of a point P(x, 1 y, 1 z)is defined
by the intersection of three mutually perpendicular surfaces (planes); x 1 = 1 constant
df
dr
r
df
dr
rr
r
2233222
- =−= 00 ()≠
∇= +
2222
f
df
dr
r
df
dr
−∇−
=−
1
2
11
8
2r
ff
2Curvilinear coordinates were introduced by Lamé in his Sur les coordonnées curvilignes et leurs diverses
applications. Gabriel Lamé (1795–1870), engineer and professor at the École Polytechnique, made contributions
to the theory of the elasticity of solids and was involved in the construction of the first railways in France.