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bhobba

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Suppose two systems interact, and the result is several possible outcomes. We imagine that, at least conceptually, these outcomes can be displayed as a number on a digital readout. Such is an observation in QM. You may think all I need to know is the number. But I will be a bit more general than this and allow different outcomes to have the same number. To model this, we write the number from the digital readout of the ith outcome in position i of a vector. We arrange all the possible outcomes as a square matrix with the numbers on the diagonal. Those who know some linear algebra recognise this as a linear operator in diagonal matrix form. To be as general as possible, this is logically equivalent to a hermitian matrix in an assumed complex vector space where the eigenvalues are the possible outcomes. Why complex? That is a profound mystery of QM - it needs a complex vector space. Those that have calculated eigenvalues and eigenvectors of operators know they often have complex eigenvectors - so from an applied math viewpoint, it is only natural. But just because something is natural mathematically does not mean nature must oblige.

So we have the first Axiom of Quantum Mechanics:

To every observation, there exists a hermitian operator from a complex vector space such that its eigenvalues are the possible outcomes of the observation. This is called the Observable of the observation.

But nothing is mystical or strange about it, just a common sense way to model observations. The only actual assumption is it is from a complex vector space.

Believe it or not, that is all we need to develop Quantum Mechanics. This is because of Gleason's Theorem:

https://www.arxiv-vanity.com/papers/quant-ph/9909073/

This leads to the second axiom of QM.

The expected value of the outcome of any observable O, E(O), is E(O) = trace (OS), where S is a positive matrix of unit trace, called the state of a system.

These are the two axioms of QM from Ballantine.

Thanks

Bill