As a sort of introductory concept to quantum mechanics, I've heard many explain that there's a small but nonzero probability of unlikely events happening: your hand quantum tunnels through the desk, all the air molecules bunch up to one corner of the room.

While some of these silly unlikely examples have been worked out mathematically, I think it would be a helpful/interesting experience to try to attempt to get an approximate answer for this question:

What is an approximate probability of a person's wavefunction collapsing to the moon?

Some details that may or may not be important to this approximation:

  • The person must still be "alive" upon collapse (so upon appearing on the moon, all of their atoms/molecules cannot be scrambled or spread out on the moon.).
  • For consistency in answers, let's assume
    • the person is "average" (so average weight, height, etc.).
    • any small details (other than special quantum preparation) that make the probability smaller is allowed. For example, you can choose that the person is on the surface of the earth closest to the moon, and (if you feel like making the effort) you can even pick the distance the moon is closest to the Earth in its elliptical orbit.
  • No special preparation is allowed. While this question might sound similar to asking how traditional quantum teleportation might eventually teleport large molecules or humans - this is not what I'm asking for. In this question, I am asking for an approximate raw probability of due to random quantum fluctuations, a person's wavefunction collapsing to the moon (similar to the example used at the end of this question).
  • Try to use benchmark examples first. Here are some example problems that might be helpful in building a complete answer:
    • The probability of an electron in space, in a groundstate wavefunction of a harmonic oscillator ($|\psi_0 \rangle \propto C e^{-x^2}$), collapsing to a distance away of $4\times10^8 \text{m}$.
    • The probability of a chain of harmonic oscillators (each in their respective ground states) collapsing to a distance away of $4\times10^8 \text{m}$. (Intuitively, you would think that with a chain of 3 particles, all three particles collapsing to the same location would have a smaller probability than the individual probabilities cubed.)
    • The probability of this chain collapsing the same distance, but now in a gravitational potential well. (And then with an atmosphere, if desired.)
    • Some way of dealing with the wavefunction of the nucleus of an atom. Could this be approximated by this chain of harmonic oscillators? Would other calculations need to be made before having even a crude approximation?
  • $\begingroup$ What is meant by a "person's wavefunction" ? $\endgroup$ – N. Steinle Dec 10 '18 at 19:18
  • $\begingroup$ IMO, if I were to attempt this, I'd just treat the entire person as one big 70kg particle. If we tried to account for all of the atoms in someone's body with harmonic oscillators... well... that would be a mess. $\endgroup$ – Hanting Zhang Dec 10 '18 at 19:21
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    $\begingroup$ The odds of you being the next lunar astronaut are only of the order of 1 in 8 billion. The odds of you appearing on the moon any other way seem a $\text{lot}^{\text{lot}}$ less. $\endgroup$ – StephenG Dec 10 '18 at 19:23
  • $\begingroup$ @HantingZhang A mess indeed, unless one greatly simplified it to only a few springs (1 for body, 1 for each leg, 1 for each arm, 1 for the head). StephenG, do you mean rather, a $lot^{lot}$ more? $\endgroup$ – N. Steinle Dec 10 '18 at 19:23
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    $\begingroup$ @StephenG I'm aware the number is very small. Feel free to use uparrow notation if necessary. My first attempt gave 1/e^10^30^30 (I'll post my answer and thoughts when I have time), but I suspect the answer is much smaller. $\endgroup$ – Steven Sagona Dec 10 '18 at 19:41

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