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Let say there is an electron in an infinite potential well. Since it's trapped in the well, I assume (which I'm not sure of whether it's valid to assume such thing) that it will bouce off the wall and change its direction of propagation. Now I know, from classical electromagnetism, when a charged particle accelerates, it produces electromagnetic wave. So in this particular situation, will the electron emmit photon and will they perpetually interact each other? And I'm not really confident about my understanding about that, that the energy of this system will remain unchanged unless some observer outside of the system mesures the energy. If that changes everything, then, is solving quantum (Schroedinger) equation only meaningful when we assume that the system that we are theoretically dealing with is not disturbed by observer?

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    $\begingroup$ Since an infinite potential well is a purely theoretical construct, a clear answer to the question would require either (1) specifying the model that should be used to answer the question, or (2) replacing the infinite potential well with something more realistic so that the question can be answered empirically. If option 1 is intended, then the wording of the question suggests that the simplest compatible model is something like non-relativistic quantum electrodynamics; many textbook potential-well problems use single-particle quantum mechanics instead, which doesn't have any photons at all. $\endgroup$ Jan 17, 2019 at 3:11
  • $\begingroup$ @DanYand Thanks for your reply. I don't have much knowledge in quantum physics, I just started studying basic quantum mechanics and while thinking about the matter the inquiry came up in my mind. So I can't give you any such model as I don't even know what models are out there to begin with. But I guess a theory that kind of deals with this matter is quantum electrodynamics. $\endgroup$
    – user575201
    Jan 17, 2019 at 3:18
  • $\begingroup$ isn't this the question, w.r.t. to atoms, that kick started quantum mechanics? $\endgroup$
    – JEB
    Jan 17, 2019 at 4:24
  • $\begingroup$ @user575201 Understood. It's a good question, and my comment wasn't meant to be demanding. It was only meant to point out that you might get different answers depending on how the answerer interprets the question. Some answers might have a lack of EM radiation built into the model itself, whereas other answers might try to deduce whether or not that assumption is valid. It's a good question, the kind of question that could be asked again and again as more and more insight is accumulated. $\endgroup$ Jan 17, 2019 at 14:56

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In the example that you give, you are thinking electron as a particle. However in QM its important to realize that electron is not actually just a particle but its also a wave, which position and time is represented by a wavefunction.

In the classical atom theory, since the motion of electron is circular, electron is accelerating. In this case electron should have emit photons, lose enegy and fall into the proton. But this is not the case. because electrons acts like a wave which is theoratically described by De Broglie wavelength. It says that,

$$\lambda=\frac {h} {p}$$

So only particular orbits are stable, where the De Broglie wavelength of the electron does not cancel out. From here we can write the stable orbit condition as

$$2\pi r=n \lambda$$ for $n=1,2,3,...$ and r is the radius of orbital.

Now lets talk about the infinite potential box. Essentially its the same basic idea. We should think electron as a wave, but not like a particle. And in this case we can see that only some of the wavelength's can exist in the box without canceling each other ( like electron in an atom case). You can think electron as a standing wave which is composition of the two waves that goes opposite directions in the box.

From here we can see that only certain wavelengths are allowed. And this wavelength's can be described with respect to the size of the box,$L$

$$\lambda=2L/n$$

In this case electrons can be stable in the box without losing energy.

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