# Unstable energy levels

Well, reading about "Raman Effect" I saw that when the electron absorb some energy, with frequency $\omega_{abs}$, that is different from $\omega_{n} - \omega_{n-1} \neq \omega_{abs1}$, it go to an "virtual energy level" that is unstable, but nevertheless we could measure its lifetime. And more, the electron could stay in this unstable level enough time to absorb another electron with frequency $\omega_{abs2}$, such that $\omega_{n} - \omega_{n-1} = \omega_{abs1} + \omega_{abs2}$, and than finally go to an stable energy level, that is foreseen by the Quantum Mechanics.

My question is: Is this right? An electron could stay a little time in a unstable energy level?

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In reality, there are always different lattices excitation modes (phonon) coupled with electrons. Electron states that coupled with phonons could be unstable, is it included in your case? For the formula you showed, its excite a electron by two photons like the case of up-conversion which is mainly studied in solar cell related fields. I am not sure if that is related. –  Z.Sun Mar 18 '11 at 16:11
@Z.Sun I know that the Raman effect include phonons, but was not my intention with the question. However, thank you! –  Rodrigo Thomas Mar 18 '11 at 22:34

All excited energy levels are unstable to this or that extent. So, yes, it can. If below a given energy level $E_n$ there are many lower possible levels $E_{n'} < E_n$, there are usually several channels of transition, each with its own probability and energy difference so we can observe emission of different frequencies.

On the other hand, absorbing two different photons, although possible in an intensive EM wave, normally happens rarely. I think it is one-photon absorption (excitation) and many-photon deactivation of atoms and molecules that normally happen (cascade decay).

If you mean existence of unstable energy levels between two neighboring levels, then no, they do not exist.

EDIT: I remember deriving the atom ionization probability by an intensive EMW of low frequency, following the work by Keldysh. Atom can be excited and even ionized with a strong constant or low-frequency EMF and it is a simultaneous multi-photon process, when a single $\hbar \omega$ is smaller than the energy $E_n - E_{n'}$ required for a given transition.

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Is this right? An electron could stay a little time in a unstable energy level?

Short answer is -- yes, this is a correct standard description of the process.

Let me elaborate. In quantum mechanics all observables, including energy, are described with hermitian operators. Eigenvectors of these operators are the states with definite value of the observable. Energy levels are the eigenvectors of the operator of energy -- the Hamiltonian.

"Virtual level" in the current context is not a state with definite energy. So it's not an energy level in a strict sense. Check this on Wikipedia -- the part on "quantum state".

I must also add that all that discussion is more about an interpretation than the actual derivation. The math is straightforward and independent on how you call things in your equations. Your question is similar to the question "do virtual particles exist". And answer depends on what you mean under "exist", so you are getting into philosophy instead of physics.

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I don't want too fall into philosophical speculation. My question arise from this: If the gap between two energy levels its 2eV, an electron could absorbe two photons, in distincts instants of time, of 1eV each? –  Rodrigo Thomas Mar 18 '11 at 22:43
I'm formally asking @Vladimir Kalitvianski too! –  Rodrigo Thomas Mar 18 '11 at 22:44
Two-photon (and multi-photon) absorption is possible but much more improbable. One needs a highly intensive EM filed to have a multi-photon absorption. –  Vladimir Kalitvianski Mar 19 '11 at 9:33
Alternatively two-photon processes are possible under aid of a nonlinear medium. But all this is not Raman-Effect! Rodrigo should make clear, what his question is. Shooting at dangling targets is not what I like. –  Georg Mar 23 '11 at 16:49

Firstly, Raman scattering, photoelectric effect, Compton scattering, etc., all produce change in energy levels, due to absorption/release of photon energy $h\Delta\nu$, typically in a molecule, in an atom, in an electron, etc., respectively. Secondly, the virtual energy level is the (sometimes somewhat artificial) adding of the photons energy to the energy level of the system that is hit by the incoming photon. It may (but not always) only be a mathematical abstraction if the system releases an outgoing photon as well. Thirdly, in quantum mechanics we are typically allowed to ask about probabilities, half times, etc, of various energy levels of the system. It does not make sense to ask how much time did the system spend with specific energy $E<0$, with no uncertainty in $E$, if $E$ does not correspond to an actual energy level of the system.

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