Initially we shall just restrict ourselves to those cases where the wave function is independent of and , i.e.,
(11.8) |
In that case the Schrödinger equation becomes (why?)
(11.9) |
One often simplifies life even further by substituting , and multiplying the equation by at the same time,
(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 , the allowed values of range over the surface of a sphere of radius . The area of such a sphere is . Thus the integration over can be reduced to
(11.11) |
Especially, the normalisation condition translates to
(11.12) |