The differential equation we wish to solve is.
Solution of Hydrogen Radial Equation *
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
, we get
The second is not well normalizable.
We write as a sum.
The differential equation for
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
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.