What does "orbital velocity" mean in a quantum setting?

[sshai45]

Hi.

I once heard somewhere that atoms of heavy elements like gold get unusual properties such as gold's yellow color due to "relativistic contraction" of the orbitals due to their high "orbital velocity". But what on Earth does orbital velocity mean in quantum mechanics? These are quantum thingies, they are not little planets orbiting a sun! There isn't a classical velocity, but a wave function. What is nearing the speed of light? Is it the expected value of velocity? Isn't that zero? Does it have to do something with the probability distribution of velocity having a fatter tail closer to "c"? Is whatever I heard totally bollox? If so, how do you understand the orbital contraction when nothing is "orbiting"? I want to point out I am not expert with quantum mechanics, hence these questions, and am also not sure what to put down for the level tag.

 

[houlahound]

I think you have to distinguish between electron probability density and motion of individual electrons.

 

[PeterDonis]

I once heard somewhere

If it wasn't a textbook or peer-reviewed paper, pay no attention to what it says. You should not be trying to learn science from pop science sources.

A reasonably accurate description of what is actually going on with atoms like gold is that the differences in energy between the energy levels that the electrons can occupy (more precisely, the energy levels that are involved in the transitions that contribute to the observed color of the metal) are smaller than they would be expected to be if relativistic effects were not taken into account. "Relativistic contraction of orbitals" is a very heuristic pop science way of describing this. The result is that the photon frequencies associated with the energy level transitions are in the visible range, as opposed to the ultraviolet range for an atom like silver where the relativistic effects are negligible. This means that metallic gold, unlike metallic silver, absorbs certain visible light frequencies instead of reflecting them, giving it a discernible color.

 

[secur]

@sshai45, you're right: the color of gold is explained by the high "velocity" of the inner electrons. In such heavy atoms the "velocity" is so high that relativity must be taken into account. You're also right that the concept of velocity needs unpacking in this context. QM considers the electron orbital to be a stationary "cloud" or wavefunction. That electron orbit has an expected ("average") radius, and an angular momentum. From these two things, plus electron mass, it's easy to calculate a "pseudo-velocity" which in this case is near c. We sometimes talk about the velocity even though there isn't an actual moving electron, like a planet orbiting the sun. It's just a convenient picture. The truth is in the math - which involves angular momentum, not "velocity".

 

[PeterDonis]

the color of gold is explained by the high "velocity" of the inner electrons.

Actually, "inner" is somewhat inaccurate, since the main energy level transition involved is the 5d - 6s transition, and the 6s is not an "inner" electron in the usual sense of that term (which refers to lower energy levels, none of which contain electrons that contribute to gold's color, since those levels are all filled). The 6s electron does have more of its wavefunction close to the nucleus, which is why it is much more affected by the relativistic correction than the 5d (or other non-s orbitals).

The truth is in the math - which involves angular momentum, not "velocity".

And energy; it's the energy level differences that determine the light frequencies involved.