Kepler's laws at atomic level

Question

If Coulomb's inverse square Law of forces operates between two electrostatic charges and at an atomic level as well (as in Newton's gravitational Law), has it been verified that Kepler's Laws which are their basis also hold good at that level? We are comparing the universe to elementary atomic models like for Hydrogen, Helium.

Answer 1

Newton's gravitation law has been tested at levels down to about a tenth of millimeter. It's a weak enough force that at atomic levels it is not been able to be measured at this point.

We've measured down pretty well to about a tenth of a millimeter, but much lower it becomes more difficult. They've tried down to a micron or so ($10^{-6}$ meters, 0.001 mm), with negative results pretty definitive to a few tenths of microns (ie, around 0.1 mm).

See some references below. No deviations have been found.

The main reason people have been trying to look for deviations is not to see how it compares with atomic or electrical forces, but whether is goes like $1/R^2$ or it varies from it, and if that could provide: 1) any reasons for some modification of the law of gravitation, at small but still larger than quantum scales (note that a quantum theory of gravity that has enough evidence to be acceptable still does not exist). 2) any indication that there might be more than the 3 dimensions to space. If there are variations in the inverse square law it could/would be evidence for microscopic spatial extra dimensions around those sizes. Extra spatial dimensions are required for String Theory (and could be observable at the micron or smaller scales) or some other unproven physics quantum gravity theories.

There may be more recent results, I didn't find them in a quick search, but there's not been much of anything about any anomalies found at the small distances.

References:

1) Trying for sub micron: 2010: http://www.livescience.com/8789-gravity-small-scales-remains-mystery.html

2) Also around 100 to 1 micron: 2004: http://www.slac.stanford.edu/econf/C040802/papers/MOT004.PDF

3) Results down to 56 micron: 2007: plus results back to 1997: a ppt in a pdf, MB's (note, it also discusses the Pioneer anomaly, at solar system astronomical ranges, but I think that's been explained already):
http://moriond.in2p3.fr/J07/trans/wednesday/reynaud.pdf From there: 'At 95% confidence, a Yukawa interaction with gravitational strength must have a range <56μm'

Answer 2

Kepler's Second Law is one way of expressing the conservation of angular momentum of a system operating under a Central Force. That holds up in quantum In a straight forward way. By the Eherenfest Theorem and its many variations, not only is angular momentum conserved, there's even an analog to Newton's Second Law that applies to the relevant classical kinematic statements, e.g. $<dp/dt>=<-\nabla V>$. More generally, Any observable that commutes with the Hamiltonian is conserved.

Kepler's third law implies $<r^3>=c<\tau^2>$. A variation of the Ehrenfest Theorem applies here.

I'm not sure if Kepler I holds up though.

That would require $<1/r>=c_1+c_2<\cos{\theta}>+c_3<\sin \theta>$

Classically or otherwise $\cos{\theta}=\sin \theta =0$.

So we have only that $<1/r>=1/(a_0 n^2)$ with no reference to the angles.