### 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