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Probability and Schrodinger's Mechanics

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The presentation and interpretation of (non-relativistic) quantum mechanics is a very well-worked area of study; there have to be very good reasons for adding to the literature on this subject.

My reasons are (obviously) that I am far from satisfied with much of the published work and find difficulties with some points, in particular:

• Any abstract formalism is much less rich than the structure from which it has been abstracted; a fact that even Wittgenstein had to come to terms with in the latter part his adult life. Language is richer than (cannot be reduced to) a representation of logic, Schrodinger's mechanics is richer than (cannot be reduced to) a representation of Hilbert space. Just as language contains more structures than logic so Schrodinger's mechanics contains more structures than those of Hilbert space.

• The use of probability in quantum theory is arbitrary, eccentric, out of step with modern probability theory and is the source of the majority of "paradoxes" in the interpretation of quantum theory. While these paradoxes are the bread and butter of some of the more popular expositions of quantum theory, I cannot say that I am fond of paradoxes in physical theories.

• Although positivism is discredited as a philosophy of science it has left a huge clutter of verbal and conceptual debris strewn across the field of quantum theory. Positivism in its most aggressive form (instrumentalism) makes the mistake of confusing the meaning of a concept with the way in which numerical values of the variables involved in that concept may be determined. This attitude has, of course, added to the confusion about probabilities; "defining" them in terms of the ways in which they might be measured thus reinforcing the view that probabilities are frequency ratios and the everyday opinion that probabilities are applicable to individual events.

• More mundanely, the prescriptions for generating Schrodinger's mechanics from classical mechanics in the vast majority of texts are wrong; they simply do not work. We are saved from total chaos by the fact that the form of the Schr6dinger equation is known and used independently of these formal prescriptions by working scientists. I am at a loss to explain why this central point is ignored in text after text, both on the applications of quantum theory and on its interpretation.

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