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.

See also:

- https://physics.stackexchange.com/questions/233330/why-do-electrons-jump-between-orbitals
- https://physics.stackexchange.com/questions/117417/quantum-mechanics-scattering-theory/522220#522220
- https://physics.stackexchange.com/questions/430268/stimulated-emission-how-can-giving-energy-to-electrons-make-them-decay-to-a-low/430288