# How fast does an electron jump between orbitals?

I'm wondering what speed electrons jump from level to level. I've been told only that they emit light when doing so and need energy to be inputed in order to occupy orbitals closer to the nucleus.

I will explain the reasoning for asking this question after I understand the logic behind the answer.

• Toasty, do you mean "speed" like "the speed the electron moves from place to place"? Or do you mean "speed" like "this reaction occurs rapidly, it's fast"? – Maury Markowitz Apr 5 '18 at 14:18
• The lifetime of non-stationary states (if I can be allowed to introduce so poorly defined a concept) can be understood by studying multi-photon excitation processes, and at a conceptual level you could understand the evolution time from one state to another as being a few of these. Though I would be the first to admit that this argument has rather more hand-waving that rigor. – dmckee Apr 5 '18 at 14:26
• @MauryMarkowitz He meant the former, since it originally said 'velocity' and I edited it to speed for consistency. – knzhou Apr 5 '18 at 14:26

First let's consider a different situation. Light waves have polarization. If you imagine a light wave coming out of this screen, its electric field can be polarized vertically, horizontally, diagonally, etc., and this is also true for each individual photon.

If I pass a photon through a vertical polarizing filter, I only ever get two results: either the whole photon gets through or nothing gets through at all. So I'll only get two possible results from the measurement: vertical (gets through) or horizontal (gets blocked).

There exist materials that can rotate polarization. So you might ask, when I put a horizontally polarized photon through such a material, what is the moment when it turns from horizontal to vertical? There has to be an instantaneous jump, because it can only be horizontal or vertical, right? But that's not right at all. The polarization just smoothly rotates, through a superposition of horizontal and vertical, as we can see using diagonal polarizing filters. Just because a particular measuring device can only see two options doesn't mean only two options exist.

The same goes for your question. Now it doesn't really make sense to talk about the 'speed' of a jump because the electrons don't even have definite positions; you're just having one delocalized cloud turn into another. But the orbitals do have definite angular momentum, so you could ask how fast the angular momentum jumps. Same answer as for polarization; it just interpolates through a superposition, even though a measurement at any intermediate point will always give an integer angular momentum.

Perhaps something closer to what you want would be an electron in a double well. Starting in one well, the electron can tunnel to the other. The process is governed by the Schrodinger equation and is perfectly continuous in time. I have a feeling you're looking for a way to travel faster than light and you can in this model, but only because we're doing nonrelativistic quantum mechanics. In a relativistic theory everything would properly obey causality.

• So if I posed my question slightly differently then: how fast does the angular momentum jump, would this be 3x10^8 m/s in a vacuum? – Toast45.toasty Apr 5 '18 at 13:19
• @Toast45.toasty No, it just doesn't jump at all, it changes continuously. For example, halfway through the process we would have an equal superposition, of say, zero angular momentum and $\hbar$ angular momentum. The expected value would be $\hbar/2$. – knzhou Apr 5 '18 at 13:21
• @Jasper The point of those paragraphs is that it's obvious you can rotate polarization continuously, and this is exactly the same as how electrons move between orbitals continuously. – knzhou Apr 5 '18 at 13:59
• @MauryMarkowitz That just depends on the system, though. It's like asking how long it takes for an apple to fall from a tree. It just depends on the tree, the local gravity, etc. – knzhou Apr 5 '18 at 14:16
• @MauryMarkowitz I am pretty sure the OP wanted to know if the "speed" was faster than light or equal to the speed of light, and the answer is simply no. – knzhou Apr 5 '18 at 14:16

If you look at the spectral lines emitted by transiting electrons from one energy level to another, you will see that the lines have a width . This width in principle should be intrinsic and calculable if all the possible potentials that would influence it can be included in the solution of the quantum mechanical state.

Experimentally the energy width can be transformed to a time interval using the Heisneberg Uncertainty of

$ΔΕΔt> h/2π$

So an order of magnitude for the time taken for the transition can be estimated.

• Isn't that $\Delta t$ the lifetime of the excited state, not the length of time it takes an individual atom to transition from the upper state to the lower state? I think the OP is looking for the latter, but the question is open to interpretation. – garyp Apr 5 '18 at 15:30
• @garyp I consider it as a decay time? a muon decays and transitions to decay products, an electron transitions to a lower energy level. – anna v Apr 5 '18 at 16:00
• The way I think of decay time: consider an ensemble of identically prepared particles (atoms, muons, ...). The decay time is the time that the population of the ensemble reduces to a fraction (say, $1/e$) of the original size. This is equivalent to the average lifetime of a particle. I think this is a different question from "What is the duration of the transition itself?". I'm not sure that that question even makes sense in quantum mechanics. – garyp Apr 5 '18 at 16:52
• @garyp well, the HUP gives an envelope in time, the probability in time of finding the electron at the lower level. Lifetimes have to do with the wid,th for example in the e+e- plots pdg.lbl.gov/2014/hadronic-xsections/… . see also this i2u2.org/elab/cms/posters/… – anna v Apr 5 '18 at 17:05

I dont know abaout the speed, but I would say they need to release energy to occupy closer orbitals and that energy is usually the light, it can be released as thermal energy