It has only been within the last few years that I learned the atomic model I grew up with (the Bohr model) was wrong, and that I should instead be thinking about electron orbitals as a cloud of probabilities instead. This led to a question about how clouds of probability can have velocity and this led to a decent explanation that I THINK I understood a little, but what I got from that answer was that the probability cloud does not change over time, the probabilities are fixed. If I am wrong about that, please correct me.

Then, today I watched a Sixty Symbols video on the Nobel Prizes awarded for attosecond lasers. At the 1:24 mark, Ed Copeland explains the breakthrough thusly:

"And, what the people have done is they've developed the technology, and built the experiments which allow you to actually probe the movements of electrons in an atom.

Later at the 3:04 mark, he says:

In twenty attoseconds, the electron could have gone from one side of an atom to another side of an atom as it orbits. Electrons orbit atoms, and these can probe that. So, they can actually sort of see where it's maximized here, and 20 attoseconds later, see where it's over here.

Then, he goes on to explain (3:04 that the "typical" atomic radius is an angstrom or 10^-10 meters. (For those interested in a comparison. The radius of a hydrogen atom in its ground state is 0.53 angstroms; phosphorus, sulfur, and chlorine, are ; palladium is 1.4Å; uranium, neptunium, plutonium, and americium are 1.75Å; and cesium is the largest at 2.6 Å.) Copeland explains that the reason why attosecond lasers are significant is that the speed of light is 10^8 meters/second, he generalizes, and that light travels approximately 1 angstrom in 1 attosecond, and that this allows us to see the electron at different parts in its orbit.

Later (15:19), Copeland clarifies that the uncertainty principle still applies, meaning that we are still unable to exactly pinpoint both the velocity and the speed. Still, from my understanding, what this video is getting at is that attosecond lasers allow us to measure within the electron cloud orbitals to then use a time factor that allows us to further constrain the location of the electron within the already determined probability clouds for a more accurate measure of where the electron is? So, my question is is my understanding of this video correct? Do attosecond lasers allow us to further constrain the location of the electron within its orbitals, based on time?

***As an added bonus, at 14:31 in the video, Copeland says that the Nobel committee had been "quite quick" in giving the prize to this field in that the technology is still in its "early days", and that the only kind of physics it has been applied to is "academic", and while he does sort of explain what he means by "academic", he leaves it at that; he does not explain what breakthrough or application he was hoping for that would deservedly (or just maybe not quite as quickly) win the Nobel prize. I was wondering if anyone knew more. What I think he is meaning is that the lasers developed so far are still off from measuring an electron at different places in its orbital.

What I mean is that, from what I have found, the fastest attosecond laser built so far (according to Phys.org and Guinness World Records) is 43 attoseconds. When taking into account that cesium's atomic radii is 2.6Å, meaning its circumference is just over 8Å, this means the best laser is still unable to probe the largest electron orbit by a factor of more than 5.

Also I believe I found something complicating the electron orbital measuring problem in that (according to Phys.org) the most accurate time measuring capabilities still have an uncertainty of 12 attoseconds. So, do you think that these two setbacks (the laser being off by a factor of more than 5 and the uncertainty of time measurement), combined with the fact that there appears to be a few advances in the field per decade is why he thought that the Nobel prizes were a little premature? Obviously, no one can read minds, but it could be possible that other physicists felt the same way. Also possible, if someone is a student or knows Copeland in some other way, they could ask.

Lastly, I wanted to make sure I got the speed factor the laser was off by is correct. If either of these additions are too much, off-topic, etc., tell me and I will delete them. But, when I first looked at this problem with the distance light travels in an attosecond, I concluded you would need a laser ~ < 1/3. My reasoning was that the speed of light is faster than 10^8 m/s by a factor of almost 3; meaning, to probe an electron orbital of 1Å, you would actually need a sub-attosecond laser (<1/3Å).

Then, it hit me that the orbital of an atom with an atomic radius of 1Å would be 3.14Å So, the factor of 3 Copeland generalized away was made up for by the factor 3 for pi, right? This also means the factor of 5 was approximately correct? But, orbitals are not perfect circles, correct? So, I imagine this makes the orbitals even longer? But, I have no idea how much longer. If there is something I am missing, something about the fuzziness of the quantum world, could you please explain that?

I appreciate any help, thanks.

  • $\begingroup$ If the electron is in a stationary state (as 1s, or 2p for example), the wave function doesn't change with time. But if it is in a temporary superposition state (1s + 2p for example) it does. it is well discussed in physics.stackexchange.com/a/293413/195949 $\endgroup$ Commented Oct 11, 2023 at 21:51

1 Answer 1


In general, yes, it changes with time. The states that don't change with time are energy eigenstates. The universe would be a boring place if that were all that existed, but it isn't: most states are not energy eigenstates and behave more interestingly under time evolution. Among them there are even orbitals that look rather classical: an electron localized to a region significantly smaller than its distance from the nucleus, and moving along a conic-section trajectory.

I know nothing about attosecond lasers, but Emilio Pisanty wrote a long Q&A recently from which I gather that they can be used to look at the time evolution of orbital states that aren't energy eigenstates. A weighted sum (superposition) of two energy eigenstates will oscillate with a period of $h/ΔE$ (where $h$ is Planck's constant). For typical atomic energy levels, that works out to somewhere in the tens to hundreds of attoseconds. So with a fast enough strobe light you should be able to photograph the oscillation in the manner of Eadweard Muybridge—or at least that's how I choose to imagine it. (These are not the quasi-classical states that I mentioned in the previous paragraph, which are more complicated combinations of many energy eigenstates, and are actually easier to observe.)

The states that don't change with time, but nevertheless move, you can imagine as similar to a current in a wire, or a spinning featureless sphere, since those systems also have the property that there is motion despite a snapshot at any moment looking the same. In the energy eigenstates, the electron's position is maximally indeterminate, but, wherever it may be, it's likely heading elsewhere. The chance that it leaves a point P in a time interval is balanced by the chance that it arrives at P from elsewhere in the same time interval, so the chance of finding it at P doesn't change.


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