Does this concept apply? Is there a method of measuring  the "distance" (I have a spacetime distance in mind, probably unrealistically) of "individual" observations in any way other than statistically? 

For example I assume radioactive decay used in timing devices   is measured statistically for that purpose. Does  it make any sense to attempt to consider two "neighbouring" emissions?

19 minutes ago, geordief said:

For example I assume radioactive decay used in timing devices   is measured statistically for that purpose. 

Caesium-133 used in atomic clocks is not radioactive isotope of Caesium. Radioactive is Caesium-137, one of products of fission of Uranium-235.

 

Of course distance applies.

If you have some electrostatic interaction, the interaction energy (Hamiltonian operator) will have a term that varies as 1/r

What you don't have is the notion that a particle will be at a specific point, so the useful solution you are getting might involve integrating over all space (depends on what you are calculating) The result (e.g. the energy levels of the hydrogen atom) might not specifically have any dependence on distance, but you could also take the wave function and solve for the most probable value of r for the ground state, and you would get the Bohr radius.

 

29 minutes ago, geordief said:

 For example I assume radioactive decay used in timing devices   is measured statistically for that purpose. Does  it make any sense to attempt to consider two "neighbouring" emissions?

For radioactive decay timing, I'm not sure — it's possible that there is some correlation, if one decay could induce another one, but then we'd notice the half-life varying with concentration of the material, and I am not aware of anyone noticing this.

For excitation, this is definitely the case, and is how lasers work — an emitted photon from one atom induces another excited atom to release a photon (in phase), and so on.