First, I should point out that this question was raised by a particle physics Professor whose lessons I attended last year. I don't recall exactly how the question was phrased so if anyone would like to suggest an edit feel free.
So, here are some statements that to the best of my knowledge are correct:
- In QM and the Standard Model of particle physics all fundamental particles of the same species are completely equivalent and indistinguishable.
- The wavefunction for every real particle exists throughout the entire Universe. (I'm not sure how to phrase this statement in terms of QFT but I'm pretty sure there is still a non-zero probability of finding any given particle popping up at the edge of the universe.)
- The ground state of the electron in a hydrogen atom has a unique, universal energy eigenvalue.
My question is therefore if (despite their distinct positions) all electrons in hydrogen atoms have the same energy, and their wavefunctions extend across the entire Universe, how does this not violate the Pauli Exclusion Principle?
I remember my Professor saying the resolution to this was that the electrons must therefore have minutely differing energies; either agreed between them from the first instances of the big bang or resulting from the interfering 'tales' of their wavefunction. Clearly the first option doesn't explain hydrogen atoms formed more recently so the second would seem more promising. I should note that by "minute difference" I think he said something like the order of 10^-20 eV so well out of the realms of potential measurement.
Like I said I may not have remembered and phrased this question correctly. Please don't crucify me for any 'school-boy' mistakes but constructive criticism in the comments would be appreciated to help me phrase it better.