Wavefunction and shroedinger equation

[renegade05]

Homework Statement

Homework Equations

The Attempt at a Solution

First, I got the wavefunction to look like the one in the question. I think the wavefunction should be n=1 not n=0. So Y(theta,psi) = A constant, that is where the C comes from. But how can I plug this into the shrodinger equation? How can I answer this question?

 

[ShayanJ]

In the Schrodinger equation, there is a differentiation operator ($-\frac{\hbar^2}{2m} \nabla^2$ )and a multiplication-by-a-function($-\frac{kZe^2}{r}$) operator. You just should apply those operators to $\phi_{0,0,0}(\vec r)$ and add the results and check whether you get a constant times $\phi_{0,0,0}(\vec r)$.
For $-\frac{\hbar^2}{2m} \nabla^2$, you should first take the gradient of $\phi_{0,0,0}(\vec r)$ which gives you a vector field which you should get the divergence of. Then multiply by $-\frac{\hbar^2}{2m}$.

 

[Orodruin]

Alternatively to the two step approach in first taking the gradient and then the divergence of the gradient, you could apply the Laplace operator in spherical coordinates directly:
$$
\nabla^2 = \frac{1}{r^2}\frac{\partial}{\partial r} r^2 \frac{\partial}{\partial r} = \frac{\partial^2}{\partial r^2} + \frac{2}{r} \frac{\partial}{\partial r} ,
$$
where I have removed the angular part since your wave function does not depend on the angles.