THE RADIAL EQUATION

The equation that follows depends on coordinate r and potential function V(r). To solve this equation, we need to know the form of the potential function. In this section, we will not solve this equation. Instead, we will closely look at its form.

.........(1)

This equation can be further simplified by substituting the following function:

..........(2)

and the equation becomes:

..........(3)

This equation is identical to a 1-dimensional Schrodinger equation except for the second term. The second term is called effective potential:

........(4)

The second term in the previous equation is due to angular momentum, and it acts as a repulsive potential. This repulsive potential is similar to centrifugal force in that it is responsible for keeping the particle from having angular momentum away from the origin. For particles with l = 0, (zero angular momentum), this potential becomes zero. With all this understanding, we are prepared to tackle the problem of hydrogen atom in the next section.