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In class, we were first introduced to classical mechanics, and how Bohr used the hydrogen electron to explain the atomic emission spectrum of that element. And it made sense. But then classical mechanics fails to explain other things, so we jump to quantum mechanics. And we learn about Newton's ideas of particles of light, and how Einstein said that light quanta are called photons, and how Louis de Broglie suggested that particles of matter can also behave like waves.

But I'm frustrated because, while the book does explain the history of quantum mechanics and the Heisenberg uncertainty principle, it doesn't explain how quantum mechanics has anything to do with atomic emission spectra. So my question is, how does quantum mechanics explain the atomic emission spectrum of elements better than classical mechanics does?

And how much of classical mechanics fails to account for what quantum mechanics does? So is the idea that an electron moving from one high energy level to a lower energy level in an atom releases energy/light is only true for hydrogen, and false for all the other elements? How can this be true?

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Bohr atom is already quantum mechanics, only a special (restricted) kind of it. In classical mechanics there would be no discrete energy levels in atom, and indeed no atom itself, for an electron would rapidly lose its energy through radiation and fall to the nucleus. So if you ask how does classical mechanics fail to explain the spectra of elements, the answer is: completely.

As we move on to the atoms with more than one electron, Bohr model reaches its limit and ceases to be helpful, so we need the "real" quantum mechanics with $\psi$ function and Schrödinger equation. It contains many counter-intuitive implications, like the electron being delocalized and staying everywhere at once, but in return for that it grants us the ability to calculate the spectra of all elements (and many, many more) with great precision.

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First of all, classical mechanics does not explain the emission spectrum at all, since there is no energy quantization out there. Bohr introduced energy quantization in his model in an ad hoc way by requiring the angular momentum to be an integer multiple of a fixed unit $\hbar = h/2\pi$ (which in turn resulted in energy also being quantized). So, Bohr model is not a classical mechanical model, since there is quantization of angular momentum and energy. But it is also not a quantum mechanical one since quantization of angular momentum is not justified, but rather asserted.

I'm saying all this just to correct OP: it was not a fight of classical vs. quantum mechanics; classical mechanics was already knockouted and quantum mechanics didn't yet fully emerged. It was a battle between semiclassical-semiquantum theories and reality which today could be viewed as well as a fight between semiclassical-semiquantum theories and the real quantum theory invented later.

Anyway, there are in general many possible electron transitions for each atom (or molecule) that contribute to its emission spectrum and each transition has a specific energy difference and probability. The Bohr semiclassical model had shortcomings with respect to both these characteristics of electron transitions:

  1. Multi-electron atoms do not have energy levels as predicted by the Bohr model, and consequently, the model predicts wrong positions of the spectral lines. In fact, it fails already for helium atom.
  2. There is no general way of calculating probabilities of transitions, thus, it is not possible to provide relative intensities of spectral lines.
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