Abstract
Schrödinger’s equation took the world of physics by storm. There was nothing exotic about it. It was easy to manipulate: every mathematician and physicist had cut her teeth in college on differential equations. Physicists, young and old, classical and quantum mechanical, could squeeze the S-equation and make it disgorge wave functions. It was applied in its various forms to a multitude of experimental problems; the results were astonishingly good. True, most problems were so intricate that the equation could not be solved in closed form.’ Typically, a problem permitted only numerical solutions of differing degrees of approximation; however, this is a common situation in numerical analysis. As aids to computation became more sophisticated, these approximations improved. But the S-equation worked.
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Endnotes
[Ulam 1976:165]. The quotation is repeated in Am. J. Phys. 62(5) (1994) 469.
Heisenberg (1901–1976), Nobel laureate, 1932. A towering figure in the development of quantum mechanics, Heisenberg played an equivocal role in the politics of his native Germany. See the appendix on him.
[Heisenberg 1958:177].
This pertains to the subject of Fourier analysis. You might review the appendix on this subject.
Width is an imprecise term. It could be made precise: for example, we might use the standard deviation of the packet (whether of a Gaussian shape or not), or perhaps the spread of the packet as measured between points halfway down from the peak amplitude. We will avoid a precise definition and so finesse the intricacies of the algebra. In practice, any reasonable definition of the spread of a wave packet will do. The numbers you arrive at will differ somewhat, depending on the precise definition, but the qualitative conclusions are not affected.
Physicists who want to think of a matter wave as an EM wave come a cropper on this point. EM waves in a vacuum are nondispersive; their speed is the constant c independent of wavelength. There is no medium in which matter waves do not disperse.
We have noted that the definition of width is not precise. Depending on the definition, the r.h.s. of Eq. 1 might read, instead of fi, say 1.811, or 11/2, or, generally, CA where C is some positive number. No reasonable definition of width will give a value of C very different from 1. The precise value of C is rarely important.
See, for example, [Penrose 1994:Part II]. I believe Penrose chose the symbols U and R because the first phase hinges on the mathematical property of being unitary (an esoteric feature which we will not stop to explain), while the second phase has a random element.
John von Neumann showed that no inconsistency arises if the crucial event—the event which causes the collapse—is taken as any one of the long chain of causative events. Far from it being the case that no event causes the collapse, any of a large set of events seems able to. Unfortunately, we cannot identify a single event which is common to all cases of collapse. There is a troublesome liberality of choice here.
Endnotes
Essays, Of Marriage and Single Life.
[Matthew 27:42].
Goudsmit, after the war, commented freely and often on the matter of Heisenberg’s role in the German atomic energy program. So did Heisenberg. The testimony of neither can be accepted uncritically. Thomas Powers, who wrote [1993] an excellent history of the whole matter, inclines to accept Heisenberg’s explanation.
Herman Melville, Moby Dick.
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Grometstein, A.A. (1999). Collapsing the Wave. In: The Roots of Things. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-4877-5_15
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