I Heisenberg's matrix mechanics

1. Aug 13, 2016

Sophrosyne

I have been trying to read about Heisenberg's matrix mechanics on my own, and I am getting hopelessly lost. I understand it has something to do with anharmonic oscillators. I am no physicist, so please take it easy with the explanations.
Also, I read somewhere that these, along with Max Born's formulation of them into Matrix form, which were the inspiration for Dirac's notation later. Is there a relationship between the two?

2. Aug 13, 2016

Mentz114

3. Aug 13, 2016

bhobba

Of course.

And its easy in the Dirac notation, but without the math forget it.

The fundamental thing is given an orthonormal basis |bi> Σ|bi><bi| = 1.

Now one of the foundational axioms of QM is given any observable you can find a Hermitian operator O whose eigenvalues yi are the possible outcomes of the observation. Now associated with any eigenvalue yi is an eigenvector |bi> and it turns out they form an orthonormal basis (there are a few subtleties - but that is pretty much it). Just as an aside its really the only axiom - but that is a whole new thread where the beautiful Gleason's theorem is discussed.

So here is what happens. You simply insert Σ|bi><bi| = 1 twice - O = Σ|bi><bi| O Σ|bj><bj| = ΣΣ|bi><bi|O|bj><bj| = ΣΣ<bi|O|bj>|bi><bj|. <bi|O|bj> is called the matrix representation of O and it turns out for eigenvectors is diagonal.

If the above is gobbly-gook then I am sorry - there is no out - you must learn the math:
http://quantum.phys.cmu.edu/CQT/chaps/cqt03.pdf

Thanks
Bill