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Recently I had a conversation with someone about quantum mechanics, I was asking if it meant everything was quantised. If space and our ability to move through it and the positions matter could take within it were continuous or quantised. If, when I sweep my arm through the air, is it a continuous motion or am I just clicking through predetermined positions like a knob on a blender?

For a reply I got several things including...

"Strictly speaking, even motion is space becomes quantised if the atom or molecule is inside a container, although for a container with macroscopic dimensions the allowed energy (and momentum) values are so close together as to be indistinguishable from a continuum in practice. Look up "particle in a box" for more about this."

I tried to ask what that meant but the person in question wouldn’t tell me. I read the description on Wikipedia and looked at some articles about it and as best I can tell, the particle in the box means that if the edges of the container get close enough to the particle then it can only adopt a finite number of quantised positions. Is that right? Is it just a metaphor? Has this behaviour been observed? When they say a container, do they mean a specific container or anything like the walls of a house? Particles exist in any of a continuous number of places? Does the particle in the box mean that a particle can only exist in a finite number of predetermined positions in space? And if it doesn’t mean that then what does it mean?

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    $\begingroup$ Not quite a duplicate but you might find the answers here helpful. $\endgroup$ – jacob1729 Sep 22 at 0:42
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    $\begingroup$ We don't know if spacetime is quantized (we probably need a quantum gravity theory to answer that), but if it is, it happens somewhere around the Planck scale, far far smaller than the proton. Please see physics.stackexchange.com/q/384913/123208 and the linked & related questions. $\endgroup$ – PM 2Ring Sep 22 at 0:47
  • $\begingroup$ Thanks but I really wanted an answer to what The Particle in a box is all about. $\endgroup$ – Tailspin Sep 22 at 8:58
  • $\begingroup$ I definitely read that with a small enough box, it's positions become finite. $\endgroup$ – Tailspin Sep 22 at 21:20
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1) The solution of the 'particle in a box' problem does not result in a finite number of quantized positions. The particle can be anywhere inside the box, which is most certainly an infinite number of positions. The energy is the thing what is quantized, and that is not the same as the position.

2) The problem should be taken as a model for a solvable quantum system, which is actually surprisingly rare to find. It is not supposed to model the whole spacetime as it is. The closest you will get to an application of the model is that it can sometimes be used to explain molecular spectra of conjugated olefine molecules, where the electron is confined in the molecule. Again, this is energy, not position. Other than this niche application, it is just a model, do not take away too much conclusions from it.

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In QM space is continuous, not quantized. If a particle is in an impenetrable box there are positions where the particle is more likely to be found and positions where the particle cannot be found. But this is a consequence of the fact that the particle interacts with the box, it has nothing to do with a granular structure of space itself.

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