###

Solution of Hydrogen Radial Equation *****

The differential equation we wish to solve is.

First we change to a **dimensionless variable**
,

giving the differential equation

where the constant

Next we look at the equation for **large**
.

This can be solved by
, so we explicitly include this.

We should also pick of the **small**
behavior.

Assuming
, we get

So either
or
.
The second is not well normalizable.
We **write as a sum**.

The differential equation for
is

We plug the sum into the differential equation.

Now we **shift the sum** so that each term contains
.

The **coefficient of each power of must be zero**, so we can derive the **recursion relation**
for the constants
.

This is then the power series for

unless it somehow terminates.
We can **terminate the series** if for some value of
,

The number of nodes in
will be
.
We will call
the principal quantum number, since the energy will depend only on
.
Plugging in for
we get the **energy eigenvalues**.

The **solutions** are

The recursion relation is

We can rewrite
, substituting the energy eigenvalue.

Jim Branson
2013-04-22