Just what we needed: a somewhat forced tail [sic] of metaparadox designed to clarify wavefunction collapse.
Imagine if you will, a world with an infinite number of scientist clones named Schrödinger. Two of them, lets call them Schrödinger A and Schrödinger B, watch a box in a room in a laboratory. The box has a cat and a Zyklon B gas canister rigged to an avalanche photodiode sensitive to cosmic rays. If a cosmic ray comes with enough energy and the right trajectory, hey presto, the cat is dead. If not, the cat is still alive. For the moment, the curtains are drawn on the box and both scientists are looking rather contemplative as they observe the system.
Schrödinger A believes that his cat is both alive and dead, as he has not collapsed the wavefunction by observing the system directly.
Schrödinger B believes that his cat is either alive or dead, but not in reality both at once despite his not having observed the cat to be either alive or dead with his own eyes. His idea is that the wavefunction is already collapsed, one way or the other, trough interaction with the material in the box though of course he has no proof of that or knowledge of whether the cat is alive or dead.
Schrödinger A holds in his head the uncollapsed wavefunction, while Schrödinger B holds in his head the concept of a collapsed wavefunction, just not yet observably clear as to which way it has collapsed.
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A third and fourth Schrödinger clone observe the room from a neighboring room via closed circult cameras. Schrödinger C believes that both Schrödinger A and Schrödinger B are correct at once. C’s wavefunction includes a superposition of A and B’s world, namely that the cat is both alive and dead, but also that the cat is either dead or alive but not both, at the same time. Schrödinger D is of the opposite opinion, that either Schrödinger A or Schrödinger B is correct, but not both at once, as at some level the wavefunction representing the (original first) two Schrödingers has collapsed due to natural evolution of the system even though it is unclear which scientist is correct.
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A third room holds another pair of Schrödingers. Schrödinger E is of the opinion that Schrödinger C and D are both right simultaneously in their description of A and B. Schrödinger E’s wavefunction is a superposition of the two possibilities represented by the wavefunctions of C and D.
Schrödinger F is of the opposite opinion, that either Schrödinger C or Schrödinger D is correct but not both at once, though he isn’t sure yet which one of the two is correct.
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At this point, you are probably expecting that we are going to describe a Hilbert Laboratory, with an infinite number of rooms, and Schrödingers going on forever with each ones wavefunctions describing a superposition or non-superposition of the previous Schrödingers.
This infinite sequence of superposed and non superposed wavefunctions exists at this point as a possibility, a potential observable which might emerge from collapse of the wavefunction representing this post.
Sure that would be cool, but…. well if you were expecting this post to go that direction, forgive me for collapsing your wavefunctions abruptly, but we aren’t going there.
Instead that we are privy to some insider information, and have in fact we are looking at nothing more than a single pair of Schrödingers watching themselves on the closed circult television with a delay so they don’t know it is themselves!!
At this point in the reading of the story, an infinite amount of wavefunctions have just collapsed, and yet the two Schrödingers we watch still hold that infinite heirarchy of wavefunctions in the uncollapsed state in their heads.
So what can we conclude from this gedankenexperiment? Wavefunctions are on the one hand a sophisticated tool to describe quantum mechanics, and on the other hand a very feeble human concept prone to collapse. We look around and learn something about our environment and they collapse like Hilbert’s dominoes with a great big whooshing sound.