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.

 

[Entropia]

I would first commend you for taking on such a complex problem in the field of quantum mechanics. The solution to the Hydrogen wavefunction in the ground state is a fundamental concept in understanding the behavior of atoms and molecules.

In order to solve this differential equation using Euler's method, it is important to establish the appropriate boundary conditions. These conditions are necessary to ensure that the solution is unique and physically meaningful.

In this case, the boundary conditions are u_nl(0) = 0 and du_nl(0)/dr = 1. However, as you have correctly pointed out, there is a problem with the second derivative at r = 0. The term l(l+1)/r^2 goes to infinity at this point, making it impossible to determine the initial condition for the second derivative.

To resolve this issue, we can use the fact that the wavefunction must be finite and continuous at all points. This means that the second derivative must also be finite at r = 0. Therefore, we can set the initial condition for the second derivative to be any finite value, such as 0.

Alternatively, we can also use a different method of solving the differential equation, such as the Shooting Method, which does not require an initial value for the second derivative.

I hope this helps you in your research and understanding of the Hydrogen atom wavefunction. Keep up the good work!