Physics 310

Solving the Radial Part of the Hydrogen Atom Wave Function

1. The three-dimensional Schroedinger equation can be written in spherical coordinates as
   

   $${-\hbar^2 \over 2\mu} \left \{ {1\over r^2} {\partial \ \o... ...al^2 \ \over \partial \phi^2} \right \}\psi + V(r)\psi = E\psi$$ (1)

   
   
   where $\mu$ is the reduced mass. Assuming the solution is separable so $\psi_E = R(r) \Theta(\theta) \Phi(\phi)$ and using the eigenvalues of the hydrogen atom show that Equation 1 can rewritten as
   

   $$\frac{\partial}{\partial r} \left ( r^2 \frac{\partial R}{\p... ... \left[ E - V - \frac{\hbar^2}{2\mu r^2}l (l+1) \right ] R = 0$$ (2)

   
   

2. Let
   

   $$R(r) = \frac{U(r)}{r} \qquad W = -E \quad {\rm and} \quad V = -\frac{e^2}{r}$$ (3)

   
   
   and show Equation 2 can be rewritten as
   

   $$-\frac{\partial^2 U}{\partial r^2} + \left [ \frac{l(l+1)}... ...{\hbar^2}\frac{e^2}{r} + \frac{2\mu W}{\hbar^2} \right ] U = 0$$ (4)

   
   

3. Now let
   

   $$\rho = \sqrt{\frac{2\mu W}{\hbar^2}}~r \quad \lambda = \sqrt... ...u}{W \hbar^2}} \quad {\rm and} \quad U(r) = e^{-\rho} v(\rho)$$ (5)

   
   
   and rewrite Equation 4 as
   

   $$\frac{\partial^2 v}{\partial \rho^2} - 2\frac{\partial v}{\p... ...frac{\lambda e^2}{\rho} - \frac{l(l+1)}{\rho^2} \right ] v = 0$$ (6)

   
   

4. Here we apply the method of Frobenius to generate a solution to Equation 6. Assume the following form for $v(\rho)$.
   

   $$v(\rho) = \rho^{l+1} \sum_{k=0}^\infty a_k \rho^k$$ (7)

   
   
   and generate the recurrence relationship
   

   $$a_{k+1} = \frac{2(l+k+1) - \lambda e^2}{[2(l+1) + k](k+1)} a_k$$ (8)