Hydrogen Atom Wavefunction Boundary conditions

[Chris333]

Hi, I have been given a differential equation to use in order to solve for the Hydrogen wavefunction in the ground state using Euler's method.

d^2u_nl/dr^2 -(l(l+1)/r^2)*u_nl + 2k*(E_nl-V(r))*u_nl = 0

V(r) = -a/r where a = 1/137.04

I have been given initial conditions u_nl(0) = 0 an du_nl(0)/dr = 1
However I need an initial value for the second derivative in order to proceed.

At r = 0, l(l+1/r^2 goes to infinity as does V(r) so I'm not sure what initial condition to use. Any help would be much appreciated.

 

[soarce]

You already have two initial conditions for u_{nl} and there is no need for a third one.
You will find the right wavefunction by adjusting the value of the energy, which in this case is called the shooting parameter. Varying the value of the energy you will find the wavefunction decay at zero at infinity.