Why electrons can't radiate in their atoms' orbits?

It's an old-new question (I found only one similar question with unsatisfactory (for me) answer: Where did Schrödinger solve the radiating problem of Bohr's model?)

It's strange for me how all books simply pass by such an important question, and mentioning strange and mathematically unsupported reasons such as:

• orbits are stationary (while as I know this is just idealization, there is no stationary orbits in reality even for Hydrogen)

• electrons are actually not localized due to uncertainty principle, thus they have no acceleration (while obviously in a non-spherically symmetric orbits a kind of "charge acceleration distribution" always exist)

• vacuum fluctuations play a major role (according to QED).

I'm not interested in how Bohr or Schroedinger explained it, I want to see a rigorous proof with QM, QED or maybe even the standard model as whole. I would like to see how this question was closed.

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This is a duplicate of the question you linked to. Voting to close. –  Ben Crowell Jul 6 '13 at 22:31
On stackexchange, you don't respond to an unsatisfactory answer by asking the same question again. Appropriate responses would be to downvote the answer, make a comment on the answer explaining why you think it's wrong, and offer a bounty on the question to try to attract better answers. –  Ben Crowell Jul 6 '13 at 22:53
@Ben Perhaps the answers was satisfactory for the other question, but not for what TMS wants to ask. That would be a case in which it is appropriate to ask a new question. However (TMS), the new question - this one - should explain explicitly how this question goes beyond the previous question. –  David Z Jul 6 '13 at 23:44
Your question makes no sense. You criticize books for their making completely valid and essential observations and statements while your added statements are all incorrect. The orbits in a QM atom are stationary. The lowest energy eigenstate can't radiate because there's no way to take energy from it - no lower-energy state. Electrons are not quite localized due to the uncertainty principle. It is not true that the acceleration always implies radiation - it only does if there's a lower-energy state. The classical formulae linking radiation to acceleration are just approximations. –  Luboš Motl Jul 7 '13 at 6:41

This question can be answered in the simple framework of non-relativistic quantum mechanics. The electron's electromagnetic charge's density and current — which are the source of the classical electromagnetic field — are given by the electron's probability density and current distributions $$\rho (t,x)=\psi^*(t,x)\,\psi(t,x)\,$$ $$j(t,x)\propto \psi^*(t,x)\,\nabla\psi(t,x)-\psi(t,x)\,\nabla\psi^*(t,x)\,.$$ As in a stationary state $\psi(t,x)=e^{-i\omega\, t}\,\phi(x)$, neither the density nor the current depend on time and therefore they don't emit electromagnetic energy, according to Maxwell equations with $\rho$ and $j$ as sources.

However, when one takes into account the quantum nature of the electromagnetic field, the probability of radiating a photon (quantum of the electromagnetic field) by an atom in a stationary state is different from zero due to the phenomenon of spontaneous emission.

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Please note that I have mentioned in the question that stationary state is just an Idealization, in reality the nuclei "potential" is actually time dependent and such an explanation is fundamentally not precise. –  TMS Jul 6 '13 at 22:44
What do you mean by a time dependent potential? What nuclei? In what framework or approximation? Any link or reference? @TMS –  drake Jul 6 '13 at 22:57
@TMS "fundamentally not precise"?? Unless you both believe in and have a true Theory of Absolutely Everything, this is not an objection to any argument. A good deal of physics - perhaps even the majority of physics - is knowing what to include and what to leave out in any analysis. If your theory has as many bits of information as the class of systems you are trying to study, then you have duplicated nature but you have not done science. –  Chris White Jul 6 '13 at 23:11
This is a nice answer and clearly within the scope of the original question physics.stackexchange.com/q/68381 . I would suggest copying it there and then deleting it from this duplicate question. –  Ben Crowell Jul 6 '13 at 23:39
The electron is not a classical particle in an orbit but a quantum mechanical "entity" characterized by a probability distribution given by the square of the appropriate wave function which shows the space covered as an "orbital" schematically. One does not solve the hydrogen atom using quantum field theory, as one does not use a surgical knife to cut potatoes. –  anna v Jul 7 '13 at 10:55