I believe that most of the problems which you point to here, are a product of the means of conceptualization. I will try not to be offensive or insulting, because we are all only human after all, but I can say that the concepts employed here are deficient, showing a lack of understanding of the concepts. This leads to odd problems much like the conceptualizations employed by Zeno led to Zeno's paradoxes. We can see with Zeno's paradoxes, that the problem is the result of the means of description, the conceptualizations employed, and how they are employed. Things are described in a way which produces the appearance of paradox. Such is the case in your description above.
So let me start with this idea that the electron (or whatever proposed particle) takes a path, a "trajectory". It is actually impossible that the particle takes a path, and this fact is imposed by the concept of "energy". Energy, in its conceptual formulation is a wave feature. A person might think that a massive object, or a particle, moves from A to B, and brings with it "energy" which is transferred to another object at point B. But the energy according to conventional principles is understood as being transferred from one body to another through wave principles. A moving body has velocity, mass, and momentum according to Newtonian principles, but it does not have "energy". Energy is a feature of the body's relation to something else, and according to conventional principles (Einsteinian), the "something else" is light, or electro-magnetism. Since electro-magnetism is understood and represented by wave principles, the concept of "energy" dictates that energy transmission from one place to another is in the form of a wave. Therefore it makes no sense at all, to say that energy moves from A to B as a particle, because the very concept of "energy" dictates that energy can only be transmitted as a wave. The further discussion as to which trajectory the particle takes therefore, is completely moot, having no significance whatsoever, because whatever it is which transmits from A to B is conceptualized as energy, and energy transmits as a wave, not a particle. There is no particle which moves from A to B, only energy, and by conventional principles energy is transmitted as a wave. The possibility of a particle with a trajectory is excluded by the conceptualization employed.
This idea of the absolute square is important. It is how we get from the non-physical wavefunction to a real thing, even as abstract as probability. Why is the wavefunction non-physical? Because it has real and imaginary components: u = Re{u} + i*Im{u}, and nothing observed in nature has this feature. The absolute square of the wavefunction is real, and is obtained by multiplying the wavefunction by its complex conjugate u* = Re{u} - i*Im{u} (note the minus sign). Remembering that i*i = -1, you can see for yourself this is real. We'll come back to this. — Kenosha Kid
I believe the wave function, as you mention here, is artificial. It has been created in an attempt to establish consistency between the two incompatible representations of space, 1) massive bodies moving through empty space, and 2) energy moving as waves in space.
Notice, what you say later, that what makes the wave function "real" is the capacity for reversal. This is another feature of conventional conceptualization. We understand radiant energy, such as radiant heat, through its absorption, not through an understanding of the process of radiation. So this is a necessary condition of radiant energy, that it is absorbed. The concept of radiant energy is based in the absorption of energy into a body. Sure we can talk about something radiating energy into empty space, into an infinite vacuum or some such thing, but this is not consistent with the concept which is based in objects receiving energy, not in objects emitting energy.
This is a fundamental feature of our means of understanding, which is observation. The observer is always on the receiving end of the radiation, so our understanding of radiation is based in its reception. It doesn't really make sense to talk about observing radio wave emissions because the act of observing is itself a reception. And, though we can observe changes to the body which emits the radiation, this is not a true observation of emission itself. So, to facilitate mathematical calculations we simply assume that emission is an inversion of reception, and voila, the wavefunction is claimed to be real, but really there is a hole in the understanding here.
The back screen is a macroscopic object that cannot be treated precisely with quantum mechanics. — Kenosha Kid
I believe the macroscopic/microscopic division is not an adequate representation of the real divide. The real divide is the division between two incompatible conceptions of "space". One understanding of space is as a medium full of waves, and the other is as an empty vacuum with massive bodies moving around. You can see how the two conceptualizations are incompatible, and where the two conceptualizations meet, radiation is absorbed or emitted from a massive body, and there is confusion due to the incompatibility of the two. That the issue is not macroscopic/microscopic is evident from the fact that macroscopic things can be represented by the wave model. That the wave model representation of macroscopic objects is inaccurate is due to that hole in the understanding.
The back screen is a high-entropy object compared with the electron. That is, at any time, it may occupy one of hundreds of thousands or millions of microstates: particular configurations that are energetically equivalent to one another. At one instant t', a position on the screen r' may not admit an electron because it already has one there. At a subsequent instance t'', it might admit an electron at r'. The screen will explore these microstates in a thermodynamic way. i,e, in the same way that a box of gas will have different but energetically equivalent configurations of gas molecules one instant to the next. — Kenosha Kid
This is a good example of the incompatibility. You are describing the screen as an area of space within which there are subdivisions, some of which will allow for the existence of a particle. But the incoming radiation, to be absorbed into that space is being received as waves within this empty space. So it is necessary to have a transformation principle whereby space (as a medium) with energy moving as waves, is compatible with the conception of empty space with moving particles. Conventional wisdom tells us that the wave formulation is far more advanced, providing a much higher degree of understanding of the reality of the situation, so we ought to dispense this conception of empty space with bodies or particles moving around, and replace it with a consistent wave model. The whole idea described here, that the screen consists of an area with subareas which might or might not provide for the existence of a particle is the wrong approach. The entire area (screen, or macroscopic object) needs to be represented as an interaction of waves to be able to properly understand how the incoming waves of radiant energy will react. However, as I described above, the relationship between emission and reception of radiation is not well understood, and we cannot simply assume an inversion.
Because the electron's birth and death are the true boundary conditions of its wavefunction! — Kenosha Kid
This is the principal misleading, or misguided principle right here. If the radiant energy travels as waves, as necessitated by the concept "energy", the death of the electron is the moment that the energy is emitted. Its birth is when the energy is received. There is no continuity of the particle during transmission. The particle only exists as a part of a massive object. So this is where you need to turn your model around. You cannot represent a photon or electron as being emitted. The photon, as a particle only exists as a part of an object. If energy is emitted by that object, it is emitted as waves, and this constitutes the end of the photon or electron, not its beginning. Conversely, the beginning of the photon or electron, is when radiation is absorbed into an object. We must maintain these principles because the spatial conception which represents energy traveling from here to there, does not allow that energy travels as a particle, it necessitates waves. The particle only exists within the other spatial conception of objects existing in empty space. Remember, "boundary conditions" are applied as deemed required, so if you want your boundary conditions of the electron or photon to be true, you need to represent the true existence of the particle, as allowed for by the conceptions employed. If the conceptions deny the possibility of a particle transmitting energy from one object to another, you cannot employ boundary conditions of the particles, which allow that the particle exists while the energy is being transmitted as waves.
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