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I am wondering why black body emission spectrum is continous. Assuming only quantum effects that is electrones falling to less energetic orbitals it should be discrete. Is the continous part emerging classically from oscilating charges?

@lionelbrits your answer is too high level for me. Let me rephrase my question. Let consider black body appproximation like a star. I understand why the ideal black body radiates continous spectrum (charged oscilators on springs). But I cannot connect this picture to the quantum view of atoms and how they emit radiation - that is through emitting photons with energy equal to the difference between two energy levels.

How this (discrete emission):

enter image description here

Can lead to that(continous spectrum)

enter image description here

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marked as duplicate by John Rennie, tpg2114, Frederic Brünner, Qmechanic Nov 30 '13 at 15:22

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Only electrons confined to a bounded phase space have a discrete spectrum. The electrons you are imagining are somehow already "in orbit", and, in that case, I would agree with you, but an electron that is coming in on an otherwise hyperbolic trajectory (asymptotically speaking) that happens to pass too close will probably be captured, so there is a continuum of energies to pick from. Furthermore, the black hole spectrum is due to viewing the vacuum from an accelerating reference frame. There is nothing here that suggests a discrete spectrum.

Finally, if we foliate space outside the black hole, then near the event horizon the volume keeps getting bigger, so I'm not sure the phase space is bounded in any case. That is, if you solve the wave equation in $t,r,\Omega$ coordinates outside the event horizon, applying appropriate boundary conditions (no reflection), I'm not sure you will end up with solutions that have a lower bound between level spacing. I'll invite someone else to comment on this, as I'm not really sure.

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