Next: 11.4 The hydrogen atom
Up: 11. 3D Schrödinger equation
Previous: 11.2 Spherical coordinates
Initially we shall just restrict ourselves to those cases where
the wave function is independent of
and
, i.e.,
In that case the Schrödinger equation becomes (why?)
-    r2 R(r)
+ V(r)R(r) = ER(r). |
(11.9) |
One often simplifies life even further by substituting
u(r)/r = R(r),
and multiplying the equation by r at the same time,
-  u(r) + V(r)u(r) = Eu(r). |
(11.10) |
Of course we shall need to normalise solutions of this type. Even though
the solution are independent of
and
, we shall have to
integrate over these variables. Here a geometric picture comes in handy.
For each value of r, the allowed values of x range over the surface
of a sphere of radius r. The area of such a sphere is 4
r2. Thus
the integration over
r,
,
can be reduced to
f (r) dxdydz = f (r) 4 r2dr. |
(11.11) |
Especially, the normalisation condition translates to
| R(r)|2 4 r2dr = | u(r)|2 4 dr = 1 |
(11.12) |
Next: 11.4 The hydrogen atom
Up: 11. 3D Schrödinger equation
Previous: 11.2 Spherical coordinates