Is science sometimes in danger of getting tunnel vision? Recently published ebook author, Ian Miller, looks at other possible theories arising from data that we think we understand. Can looking problems in a different light give scientists a different perspective?

Comprehensible Quantum Mechanics?

In the May edition of “Chemistry World” there was an item regarding “leaps of faith” in quantum mechanics, and this item quoted a paper published in Proc. Nat. Acad. Sci. showing how the Schrodinger equation can be arrived at from the classical Hamilton Jacobi equation. What puzzled me was, why was this published? After all, in the chapter “Classical Mechanics” in “Fundamental Formulas of Physics” (Dover, 1962), essentially the same thing was published, and no claim to originality was made. The book was a summary of well-known physics, so this was presumably well-established by then.
 
So, why is quantum mechanics so weird? One possibility is that it is not at all weird, and requires no great leaps of faith at all. The only problem is that we do not understand it, which in turn might mean nothing more than there is more to sort out. One problem was that we were deeply committed to Newtonian mechanics, so anything non-Newtonian was, perforce, weird. Within its set of assumptions, Newtonian mechanics are, in my view, completely correct, but as I noted in my ebook, Elements of Theory 1, there are two statement implied by Newtonian mechanics that are not correct. The first is, force acts instantaneously at a distance. By far Einstein’s greatest contribution to science was to propose that that was wrong, and force is mediated at a velocity. Further, the statement that when you see something, you cannot say, “It is there,” but rather, “It was there when the photons set off.” The second erroneous assumption is inherent to Newton’s first law. Newton’s first law is often regarded as a bit redundant, because it is essentially the second law with a zero applied force. However, there is one further part to Newton’s first law, and that is that motion is continuous. In more detail, what the physicists call action is continuous. In my opinion, that is wrong, and it is where the problems in comprehension lie. Instead, I regard action as discrete, and specifically, in units of Planck’s quantum of action. That, as far as I can tell, is the only required difference between classical and quantum mechanics. The derivation of the Schrodinger equation immediately follows from the Hamilton-Jacobi equation if the quantum of action defines a period of the wave. The Uncertainty Principle and the Exclusion Principle also follow.
 
I think another problem in understanding what is going on follows from an obsession with another part of Hamiltonian mechanics, namely the canonical equations. You will often see that these partial differential equations enable us to represent momentum, p, and positional coordinate, q, as equivalent, from which we can make phase space diagrams, etc. However, action is an integral of motion, and if the discreteness of action is the fundamental essence of quantum mechanics, then some care has to be taken with conclusions based on partial differentials. An example I gave in the ebook is this. ∫pdq has a simple meaning: a particle travelling along a coordinate with uniform momentum. Now, consider ∫qdp; a particle at constant position with a continual change of momentum? Strictly speaking, both integrals give you action, except one is ridiculous. As for the first, ∫pdq, consider that if you integrate over a period you should get a wavelength. If so, = h, the quantum of action, and we have the de Broglie equation. Action can also be represented as ∫Edt, where E is the energy. If τ is the periodic time, then it follows again that = h, from which, bearing in mind frequency is 1/τ, then E = hν, as required. This does not require "leaps of faith", and is reasonably straightforward, but how many chemists get shown things like that in their courses on quantum mechanics? Oh no! What tends to happen is that massive equations get put up, or obscure formalism using the "Sledge Hammer" approach: "Trust me, I know what I am doing."
 
Feynman said that nobody understands quantum mechanics. What I think he meant was that nobody as yet completely understands quantum mechanics, but I think you can get a lot closer to it if you take the trouble to get a few things in order. Ask what is really fundamental, and watch what follows.
Posted by Ian Miller on May 27, 2013 12:58 AM Europe/London

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