Due to the position-momentum uncertainty, it is impossible to measure the position of a microscopic particle exactly.
This means that it is impossible to measure the probability density, i.e. the square of the absolute value of the wave function, pointwisely, i.e. at any individual point, and by extension, the values of the wave function itself at any individual point of the physical space are irrelevant from the perspective of physics, they are, so to speak, “non-physical”.
Then, wouldn’t be more consistent, from the perspective of physics, to impose any mathematical condition on small regions of physical space instead of points?
Of course, if the wave function is to be continuous, then a condition imposed on a neighborhood of some point is translated to a condition on the point itself, but this is a consequence that follows from a mathematical property, it is not a physical requirement.
Besides, since the values of the wave function are not physically measurable, its continuity is not physically measurable either.
The conditions on the wave function are imposed in phase space, not spacetime-that's a big source of confusion. They represent conditions on the state of the system.
However the wave function is just a way of describing the probability density, which, has the particular property-quite common among probability distributions, that are used in applications, that it can be reconstructed by its moments. The uncertainty principle is a relation between the second moments of pairs of canonically conjugate variables, such as position and momentum. It does not imply that the probability density of the position of the particle can't be known exactly. Indeed it is possible to compute the fluctuations of the probability density itself in quantum mechanics and find that they vanish. A more precise statement can be found here: Article Normal Typicality and von Neumann's Quantum Ergodic Theorem
In my opinion ,we can imagine an observation that both position and momentum can be observed simultaneously with in a quanta of time. We can apply mathematical conditions for that physical point. But when we observe it, signal velocity plays a role and the position and momentum will be separated(we can not observe it simultaneously). means, position of the point has been changed. At this moment the previously applied condition itself will tell us the uncertainty. So nothing wrong in application of mathematics to describe such system That mathematics is valid for describing probability distribution of those observables. Physics has limitation to describe the system where as mathematics can describe it substantially.
Spiros Konstantogiannis
Dear Spiros,
You ask: "Then, wouldn’t be more consistent, from the perspective of physics, to impose any mathematical condition on small regions of physical space instead of points?"
This is precisely what de Broglie proposed as QM was in process of being defined in the 1920's.
Actually, the Schrödinger wave function was meant to define a "resonance volume" within which the electron would be in axial resonance mode.
One of the possible trajectories of the infinite set of the Feynman's path integral can even be established as completely electromagnetism compliant, despite his own opinion to the contrary:
Ref: Michaud, A. (2018). The Hydrogen Atom Fundamental Resonance States. Journal of Modern Physics, 9, 1052-1110. doi: 10.4236/jmp.2018.95067
https://www.scirp.org/pdf/JMP_2018042716061246.pdf
Best Regards, André
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