Before solving the equation we are going to see how the solutions behave at large (and also large , since these variable are proportional!). For very large, whatever the value of , , and thus we have to solve
(7.11) |
This has two type of solutions, one proportional to and one to . We reject the first one as being not normalisable.
Question: Check that these are the solutions. Why doesn’t it matter that they don’t exactly solve the equations?
Substitute . We find
(7.12) |
so we can obtain a differential equation for in the form
(7.13) |
This equation will be solved by a substitution and infinite series (Taylor series!), and showing that it will have to terminates somewhere, i.e., is a polynomial!