### 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