By 1920s it had become clear that the Bohr model was not sufficient to explain the quantum behavior of atoms and molecules beyond hydrogen. Yet, the Bohr model had been a stunning success for hydrogen itself. The quantization of angular momentum had to be part of the solution. Actually, it was quite clear what was needed since the wave equation could be solved for simple cases. But, the mathematics behind the hydrogen atom was quite involved. It is actually a remarkable fact that Schrodinger, who first solved the wave equation for hydrogen, did not need to invoke much truly new mathematics. The series solutions of the differential equations had been solved previously as tbe LeGendre and LaGuerre polynomials. This is not to belittle the task of putting these solutions together with separation of variables to solve the entire problem. First, Schrodinger had to consider the separation of the atomic translational motion from the motion of the electron relaive to the proton in the center-of-mass reference frame. Then he had to consider the mathematical form of "del-squared", the second derivative operator in the spherical polar coordinates.
The method of separation of variables is a well established technique in the solution of differential equations. The first stage involved the separation of r from the angular part. The separation of the angular part into the theta and phi equation came next.
The angular part of the Schrodinger equation describes a standing wave on a sphere. This is also known as a spherical harmonic. The problem of a spherical harmonic had been solved by LeGendre using a series method. Therefore, once the operator could be written only in terms of a theta and a phi equation, the solution was known. The solution of the phi equation is trivial and we have already seen it in the particle-on-a-circle model. This solution could then be combined the LeGendre polynomials to give the spherical harmonics provided. These steps defined the separation constant, which in turn defined the radial equation.
The solutions to the radial equation are the LaGuerre polynomials. These present radial nodes that complement the nodes in the spherical harmonics. The total wave function must have a fixed number of nodes for each solution. The n = 1 solution has zero nodes. The n = 2, 3 .. solutions have n = 1 nodes, which may be either radial or angular or both.
