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I find this very strange that an electron can 'teleport' from one energy level to another. So what evidence suggests this?

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    $\begingroup$ "I find this very strange" is the most common reaction to true quantum phenomena. $\endgroup$ – ACuriousMind Jan 25 '15 at 14:14
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Electron can't teleport from one energy level to another. Rather, when you shine light of the frequency, corresponding to the given transition between levels, on the atom in initial energy state, the probability of finding this atom in the other energy state increases with time. This probability can be computed via Fermi's golden rule.

The idea of "teleporting" from one level to another is likely what you've learned from Bohr model. But this model was an early attempt at fixing the problems of Rutherford model, and is not fully satisfactory. Modern description should be via Schrödinger's equation. In this picture electron no longer has definite trajectories, and its state is instead described by a complex-valued function, called wavefunction, defined in the whole space.

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A moderate amount, or very little, depending on your perspective.

The quantum leap is an outdated theoretical construct. The maths of modern QM do not require instantaneous jumps to be performed by electrons shifting between energy levels, or in any other situation.

We now understand that all things including the electron are dynamically evolving smoothly according to the Schrödinger equation.

It is merely when two wavefronts are brought into close proximity, for example during measurements, that the probability of the system being found in a state wherein the energy level is changed may become highly likely. Then the level will or will probably change, and a photon may be emitted or absorbed or not. It may appear to suddenly change, but this is only because all we can see is whether a photon is emitted. Everything else is effectively invisible to us.

This phenomena is sometimes labelled as a 'quantum jump' or the 'collapse of the wave function.'

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There is evidence of particles having both wave behaviour (by diffraction e.g.) and particle behaviour.

The only way to understand wave behaviour is to accept that it points towards a new understanding of a particle as not a point-particle but more a propability of where it is right now. If the wave function overlaps several energy levels then you can call it "jumping" to or just "being" in another energy level suddenly, because the probability has changed.

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