Are the protons and neutrons in the nucleus arranged in any particular way? I was wondering this: suppose you have two oxygen atoms. They will both have 8 protons and 8 neutrons in the nucleus (at least if they are the most common isotope). Now, will all those particles be arranged in the same way in both atoms? If they are, why would that be, and if not, does that affect the element's properties in any way?
But then I also thought that maybe the uncertainty principle doesn't let us even ask this question. Maybe you can't tell the particles' positions so accurately, so all you can say is that you have 8 protons and 8 neutrons all together in a small space.
So, which one is it? Can we even tell where all the particles are, and if we can, does it matter exactly how they are arranged?
 A: 
So, which one is it? Can we even tell where all the particles are, and if we can, does it matter exactly how they are arranged?

We can tell/model the distribution of the particles in the nucleus, as dmckee states, and it does matter how they are arranged as we go up the periodic table of elements. The internal arrangement/distribution give rise to the dipole and quadrupole moments of the nuclei .
Thus in aggregate the distribution of protons in the nuclei do characterize them.
A: Well smaller elements the  protons and neutrons are not arranged in sphere but more like disk in the nucleus. Once the element get larger the protons and neutrons are arranged in sphere like structure because of proton repulsive forces and the limited amount of neutrons to block the repulsive force. Most scientist believe that strong forces holding the protons and neutrons together are more like clips like structure  rather than magnetic forces or electrical force. This hold the key to there arrangement in larger isotope elements. The strong forces maybe like the way trains connect with each other. With pressure they clip on each other The protons and neutrons are clipped together. One thing for sure there is all ways more neutrons than protons . The reason has to be that the neutron shield the repulsive forces. Im working on this problem when i have time. There arrangement would determine if they are more positive on one end. They say that the larger the element the faster the speed of the electron on the outer shell. A neutron being neutral could come and go from a nucleus of atom . Unaffected by either charge or gravity. So one has to believe that when a proton turns into a neutron they must still be clipped onto each other like trains clip onto each other.  Just food for thought.
A: Javier, try looking at What is an intuitive picture of the motion of nucleons? to start. 
The short answer is that it is as reasonable to say that they are identical as it is to say that the configuration of electrons in multiple atoms of a single element are identical. That is, there is a set of position (or momentum1) distributions to which they conform.

1 The position and momentum distributions turn out to be linked to each other by Fourier transformations, so information required to specify one is the same as that required to specify the other. Nuclear physicist mostly concern themselves with the momentum distributions.
