Do electrons spontaneously jump from high orbitals to lower ones emitting photons?
Explaining this was was one of the key initial achievements of the Dirac equation.
Yes, but this is not predicted by the Schrödinger equation, you need to go to the Dirac equation.
A critical application of this phenomena is laser.
TODO understand better, mentioned e.g. at Subtle is the Lord by Abraham Pais (1982) page 20, and is something that Einstein worked on.
Photon hits excited electron, makes that electron go down, and generates a new identical photon in the process, with the exact same:This is the basis of lasers.

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