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In non-relativistic quantum mechanics, the spectrum of a particle in a 1D closed box is $E_{n} = \frac{n^{2}\pi^{2}\hbar^{2}}{2mL^{2}}$, in order to get this result one has to consider boundary conditions at both ends of the box. Those boundary conditions globally force a certain quantization of the wavelength.

Given that one treats space and time on equal footing in relativity, can one consider the same experiment but "rotated" along time? For example a particle created at one time and destroyed at another. If the same reasoning applies, would different boundary conditions in time globally force a certain quantization of the wave period?

EDIT: I know that of course I am dealing with non-relativistic QM, so I shouldn't expect these cases to make any sense, my question is more about what happens to these two cases when one starts reasoning in terms of a relativistic extension of QM, can one re-interpret these scenarios in one of those frameworks in a way that makes the issue go away?

EDIT: Would somebody please delete the question? what the hell are you trying to do with all these edits anyway?

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    $\begingroup$ How can this be compatible with relativity? Non-relativistic quantum mechanics isn’t supposed to be compatible with relativity. $\endgroup$ Commented Dec 2, 2024 at 17:52
  • $\begingroup$ That's fair, but i'm still confused about what happens in the second case, once somebody seemingly takes relativity into account $\endgroup$ Commented Dec 2, 2024 at 17:54
  • $\begingroup$ You are wading into muddy territory by taking the results of a time-independent problem (the potential is time independent for a particle in a box) and then trying to staple that result onto a time-dependent system in an ad hoc way. $\endgroup$ Commented Dec 2, 2024 at 17:57
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    $\begingroup$ Then why don't you consider what happens when you apply the "PW formalism" to this problem? $\endgroup$ Commented Dec 2, 2024 at 18:14
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    $\begingroup$ For the first scenario, the system starts in a very low energy state, if the wavelength of the particle is comparable to the size of a box that light takes a significant amount of time to cross. Because of the energy-time uncertainty principle $\Delta E \Delta t \gtrsim \hbar$, to measure the energy at all (and separate it from zero/no particle present) you're going to need to wait a very long time. Shaking the wall will change the energy of the system, but it'll take time to observe this. For your second scenario, I don't understand the "rotation"; you should try to write some equations. $\endgroup$ Commented Dec 2, 2024 at 19:59

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In non-relativistic quantum mechanics…

…how can this be compatible with relativity, given that the information about the fact that the potential at the other end has changed should travel at the speed of light?

Given that one treats space and time on equal footing in relativity…

Nonrelativistic quantum mechanics is not compatible with relativity.

You’re using incompatible theories together. Nonrelativistic quantum mechanics of course can be made to produce results incompatible with relativity; that’s why it’s called non-relativistic. Quantum field theory is relativistic and does deal with some quantum relativity effects, but that’s not the framework you’re using, so you don’t get those relativity-compatible results. (QFT in curved spacetime would be even more accurate, if a bit more involved mathematically).

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