Heisenberg's Matrix Mechanics: An Understanding Review

[Pollock]

I would like to know more about how Heisenberg developed his matrix mechanics.He wanted to represent the quantum state in terms of observable quantities such as spectral frequencies and intensities,rather than via the more abstract wave function of Schrödinger.But how did he assemble the arrays of numbers that Born eventually recognised as non-commuting matrices ?.Can someone tell me or refer me to an easily understandable review or paper ?.I guess I would find Heisenbergs original papers too difficult.

 

[Simon Bridge]

Can someone tell me or refer me to an easily understandable review or paper ?.I guess I would find Heisenbergs original papers too difficult.

... in that case, nobody can tell you and is unlikely to be able to get you a review with more details than you can find from googling for history books. I doubt anyone actually knows anyway.
The process someone went through, of getting to a new model, representation, or paradigm is usually not very useful to others - creativity is so personal.
However, someone will be able to get you some idea if you would tell us what you hope to gain from the answer.

 

[Dr Transport]

Try try this book,
https://www.amazon.com/dp/0716722437/?tag=pfamazon01-20

I remember it giving a fairly nice account of the history and Heisenberg's reasonings...

 

[bhobba]

I would like to know more about how Heisenberg developed his matrix mechanics.He wanted to represent the quantum state in terms of observable quantities such as spectral frequencies and intensities,rather than via the more abstract wave function of Schrödinger.But how did he assemble the arrays of numbers that Born eventually recognised as non-commuting matrices ?.Can someone tell me or refer me to an easily understandable review or paper ?.I guess I would find Heisenbergs original papers too difficult.

The following may help:
http://www.mathpages.com/home/kmath698/kmath698.htm

Actually Schroedinger developed wave mechanics after matrix mechanics and it was a competitor for a while. The reason Schroedinger hypothesised his equation is also quite interesting:
http://arxiv.org/abs/1204.0653

Even more interesting was how Dirac dramatically extended Heisenbergs ideas by means of analogies to Poisson brackets and his so called q numbers:
http://arxiv.org/pdf/1006.4610.pdf

Eventually Dirac, in late 1926, came up with his transformation theory, which is basically QM as we know it today, that showed all three approaches were really the same. This also has an interesting history, being tied up with that damnable Dirac delta function so loved by applied mathematicians, but sending pure mathematicians insane:
http://www.lajpe.org/may08/09_Carlos_Madrid.pdf

That issue was resolved with the development of Rigged Hilbert Spaces by Gelfland and others, and forms the rigorous mathematical foundation of modern QM as usually practiced by physicists, rather than Von Neumanns approach. Still Von-Neumanns - Mathematical Foundations of QM is an instructive read - I cut my teeth in QM from that book.

Thanks
Bill