Reference: The Foundations of Mathematics
QM seems to be 5% experimental knowledge and 95% interpretation. We even call it Copenhagen interpretation.
It is not difficult to start out from the real experimental knowledge and then explore outwards from there keeping in mind that mathematics is just a tool. If we put too much faith in mathematics without checking against reality now and then, then it can be akin to being brainwashed by mathematics.
EXPERIMENTAL OBSERVATION: We cannot idealize a quantum object as a particle that has a precise location.
It is more like a wave packet (snake) of finite bounds. So, it doesen’t make sense to compute the probability of some “particle” as to its location and call it a wave function of the particle. That would be like computing the probability of “part of a snake” at a certain location. Why not compute the location of the whole snake (wave packet)?
The situation is that we cannot fit the whole “snake” in a precise point location as was done for a particle. We have to think in terms of location of the whole wave packet, and the probability of that is always going to be 1. However that location is not going to be a point.
So problem gets to be the fixed idea of thinking in terms of precise “point” locations only.
That is what mathematical integration from minus infinity to plus infinity is doing. I believe that integration no longer works at quantum levels. We need new mathematical tools for better understanding.
Maybe it is time to look at Georg Cantor’s ideas related to infinity and infinite sets, because it seems that the composition of wave packet belongs to a different infinite set then the infinite set in which it is moving.
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Comments
Newton came up with the idea of infinitesimal (dx), continuity and convergence to bring more exactitude to Classical Mechanical concepts. One could now deal with infinity of values in the real number domain much more easily and realistically.
In Classical mechanics a particle was being treated as having an infinitesimal dimension. The concept of locality in space also encouraged the concept of infinitesimal space,
However, when the atomic particles started to get investigated more closely, Einstein recognized the phenomenon of discreteness at a fundamental level. This became the subject of a new mechanics called Quantum Mechanics. We now had “particles” that demonstrated wave-characteristics. The infinitesimal dimension lost its certainty.
This reminded the shock that was felt when irrational numbers were discovered. With that in mind one could now look at the infinitesimal dx having a domain of its own at a new level from minus infinity to plus infinity. Such new levels were investigated by Georg Cantor in his theory of sets.
So this investigation of wave-particle duality seems to be forcing us to look into infinitesimals within infinitesimals.
This game has suddenly become more interesting.
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Doesn’t the wave-particle duality imply that there is a disturbance spread over space in quantum sense, which appears localized as a point in a classical sense?
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The particle seems to have its own structure at a different level. It is like a wave packet with an appearance as a point ‘dx’, but within it there is a disturbance with a certain frequency. The particle ‘dx’ can now be part of a larger disturbance.
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So, there seems to be a disturbance within a disturbance, much like a message through a carrier wave.
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