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.
Answer
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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