::Baccyak4H, and everybody: 

 ::I am stuck on the math, but in general the path up to Heisenberg's breakthrough is clear, and it is easy to explain how everything worked to explain things that had previously been unexplainable: 
 ::This is just an outline: 
 ::Balmer: λn=Å•3646•(n2/(n2-4) where (n=3,4,....) 
 ::Rydberg and Ritz: 1/λn = RH(1/4 - 1/n2) where (n = 2, 3, 4...) 
 ::E = hf  (Planks's constant) 
 ::f = c/λ (nothing new here, but novices need to see how to get from frequencies to wavelengths and back. 
 ::Eelectron = hc/λ 
 ::Planck-Einstein relation: 
 ::Ephoton = Ei - Ej = hf 
 ::Bohr's scheme of energy states predicts many spectral lines: 
 ::1/λ = RH( 1/nf 2 - 1/ni 2 ) where (nf = 1,2,3... and ni > Nf) 
 ::So on this basis, one could predict the spectral lines for hydrogen, and the energy of the photons associated with each of the lines. The only thing that would not be known on this basis would be the intensities of each of the spectral lines. By studying the best available secondary sources I have realized that the problem for Heisenberg, and the discovery that Heisenberg made to solve it, was how to account for the intensities of the hydrogen bright-line spectrum. Although every step in the math up to this point is almost too simple to believe that it can account for so much, the calculations that he had to perform take him about 16 pages just to sketch out  and people in the field even complain that they can't understand it. If only he had written in a little more detail it would have been clearer. I say this because some people have since puzzled it out.  Right now I'm trying to get a "proof of concept" grasp on the math so that I won't say anything ridiculous on how the matrices are to be written out. What Heisenberg actually wrote and published did not involve a single matrix, but my colleague in the physics department took one look at the equation that a secondary source spotlights as "Heisenberg's law for matrix multiplication," and said that it gave clear directions for how to write a matrix.  I do not think that the novice reader is going to object to the article's not giving a clear treatment regarding how the math was worked out. The important thing to see is that once this one step was taken there was a complete quantum theoretical picture of how hydrogen produces light. It also provided the key to the Heisenberg indeterminacy principle.  
 ::I am less familiar with the developments after that point. The math appears to be more abstract, which may be just the way things are, but some of the discussion like the "h-bar" part clearly do not help novice readers. 
 ::I plan to finish my rewrite on the basis of what I know now, and keep trying to work through the math so that if I learn anything that contradicts what has been put up then I can correct it  assuming somebody else has not already caught it.  Maybe I can finish up the draft over this weekend. As far as I remember, the material up to Heisenberg is accurate. Historical details can be removed.