How is an electron in an atom always spinning without losing any energy?

You are not the first person to ask this question. The first person to ask this question was this guy:

His name was Niels Bohr.  In 1911, the popular model of the atom — due to the experiments of Ernest Rutherford , was the “planetary model” of negatively-charged electrons orbiting a positively-charged nucleus. That’s still the popular model of the atom, as shown by this popular schematic.

But Bohr pointed out that a negative charge in circular orbit would release electromagnetic radiation, losing energy and eventually impacting the nucleus. This didn’t happen, so Bohr proposed that there were fixed, stable orbits about the nucleus where the electron did not shed energy: these are what are known as the “electron’s orbitals”. When an electron absorbed energy, it would jump to a higher orbital (the original “quantum leap”); when it lost energy, it would fall to a lower orbital, but it could never fall lower than the lowest orbital (the “ground state”). A full understanding only came with the subsequent work of Fermi , who explained that there was a fixed number of electrons in any state, and Linus Pauling, who built on Fermi’s work to develop our modern understanding of the orbitals of the electron, and explained all of chemistry on that basis.

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