Based on what we know about atomic structure all atoms have an electron density function which describes the uncertainty in the position of an electron. Thus theoretically 2 atoms of the same element can be different in terms of positions of their respective elections.
why is it then that no matter where they are found 2 atoms of the same element tend to have the exactly same physical and chemical properties ?
Best Answer
All bound states (groups that are not decaying away, they're more "orbiting" in some way) of nuclei and electrons may be written as a (linear combination of) energy eigenstates – eigenstates of the Hamiltonian $H$.
The Hamiltonian (energy operator) is always the same – it captures all the information about the laws of Nature and there are the same. The possible eigenstates and eigenvalues of this operator are "discrete", i.e. sharply separated from each other, so their set is countable. We may uniquely identify these solutions – eigenstates – by labels, e.g. by integers.
If two atoms have the same label, they are exactly the same because they are exactly the same eigenstates of the same Hamiltonian which defines the laws of physics and those are also the same everywhere.
Incidentally, two atoms are also perfectly identical in a stronger sense: if we exchange them, the wave function remains exactly the same (up to the sign flip, if they are fermions). It has to be so because the atoms' dynamics is described by "field theory" where the fields are either bosonic or fermionic, and two or more atoms are achieved by the action of two or more "creation operators" and those either commute or anticommute with each other which guarantees the symmetry or antisymmetry of the wave function under the permutations of the atoms.