Disclaimer: I lay no claim to the originality of the basic ideas expounded here. Its distinctness lies more in its scope and in the detail of its exposition.
Particles in a 3D Symmetric Potential
In a three dimensional spherically symmetric potential 
the time-independent Schrodinger equation for stationary states of energy E is
This Schrodinger equation in polar coordinates is expressed by
We multiply this equation by 
We then, substitute this wavefunction,
We have
Note that the left side is only a function of r, meaning we can shift the partial derivative to a total derivative,
On the other hand, the left side is a function of $\theta$ and $\varphi$ only,
Note also that they must be equal to a constant because of the different variable dependence. They must be equal to the same constant.
Remember this equation:
Thus, the angular part can be written as,
From the previous section, we've encountered this operator and found that its eigenvalue spectrum are
Now, let's consider the radial part of the equation,
We can substitute our value of 
into this equation,
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