[time-nuts] Modern Rb atomic reference vs classic Cs

[Attila Kinali]

On Tue, 17 Mar 2020 22:16:38 +0000
Greg Maxwell <gmaxwell at gmail.com> wrote:

> Given that it's possible measure frequency differences very precisely
> (and without a multi billion dollar particle accelerator), it might be
> interesting to search for some of these.  However, the frequency
> ratios might not be the most natural ratios to search for. E.g. it
> might be useful to first back out various effects which we know how to
> account for precisely and see if there is a simple expression for the
> residual.

No, it would not. If there would have been a simple solution, then
the formulas would be simple which means we would be able to calculate
them. At the same time, if you try to find a pattern you will find one
if you just look hard enough. Same as when you turn your TV set (yeah,
an old CRT one) to an empty channel and look at the noise.. at some
point it will start moving around and patterns will arise, where no
pattern resides.

Let me quickly explain what makes this so hard:

We are looking at a set of differential equations that describe the
movement, or rather the energetic states of the electrons around
the atoms. Now, if you have a simple equation, you can find a solution
... maybe. Differential equations have the nasty behavior, that even the
smallest complication, like adding a constant, can make the equation
unsolvable. If you add multiple simple differential equations together
you might also get something solvable...until it isn't anymore. The
famous three body problem ( https://en.wikipedia.org/wiki/Three-body_problem )
is the most famous example where just adding "more of the same" makes the
differential equations unsolvable even at a level where it is still almost
trivial. Now, for a Rubidium atom, you have 38 bodies: 37 electroncs and the
nucleus. This is so far beyond the 3 body problem that there is no hope
that this will ever lead to a closed formula. Keep in mind that all 38
elements interact with each other and cannot be really seperated.
And looking at it as an many body problem is still just a classical
description and ignores all the quantum mechanic dynamics, that have to
be described with more differential equations. This makes the
problem so hard, that even numerical solutions are very very time
consuming, to the point that it is not feasible for anything but
very simple atoms.  

Hence, the most common way is to simplify the inner electrons and
the nucleus as a single object, with only a single electron orbiting.
This way, the atom can be treated as a Hydrogen atom, for which we can
find solutions. Of course, how accurate these solutions are depends on
how well we can approximate the effect of the nucleus and the inner
electrons.

For a more thorough explanation of the problem with differential
equations, see 3blue1brown's videos on this:
https://www.youtube.com/playlist?list=PLZHQObOWTQDNPOjrT6KVlfJuKtYTftqH6

Unfortunately, I don't know an easily accessable resource for calulating
the energy levels of an atom.


			Attila Kinali

-- 
<JaberWorky>	The bad part of Zurich is where the degenerates
                throw DARK chocolate at you.