The Radial Equation

Students should be able to:

1. (RadialSeries) In this problem students solve the differential equation for the Laguerre Polynomials. The question is worded so that students see this as a practice problem on series solution methods rather than specifically the Laguerre equation which they can look up. By timing the assignment of this homework problem appropriately, it is possible to arrange to have the students complete this problem just before beginning the radial solution to the hydrogen atom; thus saving time in class solving the Laguerre equation.

   1. By hand, find the recurrence relation for a power series solution $H(\rho)$ of the equation:

      $$\rho {d^2 H\over d\rho^2} +(2\ell+2-\rho){dH\over d\rho} +(\lambda-\ell-1) H=0$$

      where $\ell$ is a known positive integer, and $\lambda$ is an unknown constant.

   2. Suppose that you want a solution to (a) which is a polynomial of degree 4. Assume that $\ell=2$. What does that tell you about the unknown constant $\lambda$?

   3. Find the polynomial of degree 4 solution to the differential equation in part (a) assuming $\ell=2$. Assume anything you need to about $\lambda$.