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A Little Explanation!

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Posted by Richard D. Stafford, Ph.D. on July 11, 2001 03:14:10 UTC

Harv,

I am glad to hear that you are not yet finished quizzing me. I was beginning to get the impression that you had decided not to pursue the matter and I am not finished by a long shot!

With regard to the definitions underlying Quantum Mechanics. You must recognize that the original equations of Quantum Mechanics have their source in relations developed in what is called Hamiltonian Mechanics. At one time there was great effort expended in Geometrical interpretation of physical laws. Many very difficult problems were solved by transforming the geometry of the problem into a geometry where the forces vanished (a purely mathematical procedure). This whole idea evolved into the field called Hamiltonian Mechanics. I do not know if this field is even taught to the average graduate student any more. One of the significant facts proved when the field was strong was the fact that there existed no three dimensional geometry which would make gravity a consequence of the geometry. A very disturbing fact considering the circumstance (that the force due to gravity is proportional to mass, a sure sign of a pseudo force caused by geometric effects). -- Just a minor aside!

The square of the wave function was interpreted to be the probability density of seeing the event (that is, the event the wave function was describing). Under this interpretation, the Momentum was associated with the “expectation value” of a particular differential operator (the association comes from analysis developed under Hamiltonian Mechanics). Momentum was originally defined by Newton as the mass times its velocity (a quantity apparently conserved in collisions).

Now, if the probability density of seeing this event was given by a function of the coordinates of the event, then certainly the position of the event was uncertain (if we are discussing probability, we are discussing uncertainty)! Having defined (or associated) the momentum to be a very specific differential operator, one could show mathematically that the momentum of the object was also uncertain. A consequence of the mathematical relationship between the two operators (x, the position of the event and the differential operator, which was its momentum) was that the product of the uncertainty was constant!! This is purely a consequence of the structure of the differential operator.

I know what I have said will sound over technical to you, but I think if you read it carefully, I have said nothing which you cannot understand.

>>> In any case, where in the mathematical heavens did this constant come from?

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