# 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