What is the relationship between spherical harmonics and the schrodinger equation? spherical harmonics (below image)  
 
Schrödinger equation(below image)

What is the relationship between spherical harmonics and the schrodinger equation?
 A: Quite the opposite: Spherical harmonics are found to be eigenfunctions for the angular momentum operator $L^2$, which all comes from the Schrödinger equation (SE). If you want to see how to get from SE to the spherical harmonics, I suggest you pick up an introductory text on Quantum Mechanics - it is far too big an endeavor to undertake here. 

Edit to answer OPs edit:
SE (in the eigenbasis) is $$\hat{H}\psi=E\psi,$$
where $\hat{H}$ is the Hamiltonian. The Hamiltonian is an operator (which is not a number, but more like a matrix... that is the reason for the "hat") that basically specifies what system you're looking at. A lot of problems in QM goes like "write down $\hat{H}$ for the system at hand, then solve SE." 
Now, $\hat{H}$ is special because the eigenvalues associated with it are the allowed energy states for the system, which determines how it behaves. As you may know, the energy of a system can be split up into the kinetic energy $\hat{T}$ and the potential energy $\hat{V}$. We can therefore write $\hat{H}=\hat{T}+\hat{V}$. Here's the connection with the spherical harmonics: Whenever $\hat{V}$ is spherically symmetric (and SE is separable), the eigenfunctions $\psi$ of $\hat{H}$ will have a polar part. This polar part is precisely the spherical harmonics. This is for instance the case in the famous example of the hydrogen atom.
A: The Schrödinger equation, and in particular the Schrödinger equation with $1/r$ electric potential function, is not something that you can "derive" from mathematics.  Mathematically it would be perfectly consistent for  photons to be massive (which would result in a Yukawa potential rather than $1/r$).
