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How does de Broglie's hypothesis explain the stability of electron orbits in Bohr's atom? How does making them standing waves mean the electrons don't accelerate any more and don't emit radiation?

I understand how it means that the electrons reside in quantised energy levels, as they can be considered to be standing waves (why standing waves, not another type of wave?), in which case the circumference of orbit is an integer number of wavelengths (and thus Bohr's quantised angular momentum).

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It doesn't. The Bohr model postulates that electrons can only exist in circular orbits which have orbital angular momenta $L = mvr = nh$. It offers no explanation for this fact, but accepting it implies (i) that electrons cannot spiral into the nucleus so atoms are stable, and (ii) that gases absorb and emit specific, discrete wavelengths of light rather than continuous spectra. Furthermore, the measured emission spectra match well with what the model would predict.

Obviously this is unsatisfactory from a theoretical standpoint, but that's why the full machinery of quantum mechanics was developed. It turns out that Bohr's model predicts the correct energies by a happy coincidence of dimensional analysis, but other than that, it is not the correct way to view an atom. The Bohr model is a useful tool and an important step in the development of quantum mechanics, but that's about as far as it goes.

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