Bill Hobba Posted February 25 Report Share Posted February 25 Like many people, I have struggled to understand what QM is and what it says. Recently, after many years I have formed the following answer. There is a reality out there, independent of us and amenable to rational analysis by the conscious mind. Some parts of that reality we directly interact with every day. Others, like Electric fields, are necessary for well-understood laws to hold (eg Wigner proved if there were no electric fields, then conservation of momentum would not hold in violation of Noether's Theorem.) This is everyday stuff. But we know things like electrons exist, and we cannot directly interact with them; all we can do is interact with them using other things and find out what happens. They are equally as real as all the other things - it is just to know about them; we must interact with them somehow. At rock bottom, QM is a theory about such interactions. It follows from a straightforward model of interactions and their outcomes. 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 called an observation, but it is an interaction between two systems. 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 we have seen there is nothing mystical or strange about it - it is just a common sense way to model observations. The only actual physical 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 a theorem called Gleason's Theorem, a simple proof of which has recently been found: 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. Believe it or not, this is all that is needed to derive QM. See Ballentine - Quantum Mechanics - A Modern Development. https://www.amazon.com/QUANTUM-MECHANICS-MODERN-DEVELOPMENT-2ND-dp-9814578576/dp/9814578576 Quote Link to comment Share on other sites More sharing options...
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