# Does particle indistinguishability and quantised enery levels (in bound states) violate the Pauli Exclusion Principle?

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.

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The Pauli principle says that two fermions can't be found in the exactly same dynamical state. A dynamical state is specified by 3 coordinates and spin, or by n,l,m,s in the Hydrogen atom, or (for an electron in 3 spatial dimensions) a set of 4 quantum numbers, 3 of which can take many values. The previous sentence also says that the Pauli principle doesn't say that two electrons can't have the same (just) spin; and it surely doesn't say that two electrons can't have the same constant properties such as mass: all of them have the same mass, charge, and the potential to behave in any context. – Luboš Motl Feb 9 '11 at 6:02
@Lubosh - Thanks for the comment but this, to me, seems like the clearest attempt to actually answer the question; hence the +1. I agree with what you have said and therefore think I must have remembered the question incorrectly. Perhaps, instead, we could consider all the Universe's fermions in an instant just after the Big Bang. If the available space was sufficiently small (say less than the size of one Hydrogen atom) would this not require all the fermions to adopt minutely differing energies, so as not to violate the PEP? – qftme Feb 11 '11 at 14:40

Electrons are indistinguishable and the Pauli Exclusion Principle applies to them. Theoretical solution for the 1s wavefunction of an electron in hydrogen atom $\Psi_{1s}=Ne^{-r/a}$ is calculated neglecting all particles in the world except one electron and one proton. If another electron exists in the same quantum state, both would avoid each other and the wavefunction would be distorted leading to a change in energy of a total system.

I emphasize it because - remember - electrons are indistinguishable and interaction of two hydrogen atoms cannot lead to electron A having lower energy than electron B. This does not contradict PEP — the principle applies to quantum states, not potential energies. Both hydrogen atoms will exhibit the same shift of electron energy levels. It may help to visualize the system as almost dissociated hydrogen molecule.

Note that the wavefunction decays much faster than the dipole electrostatic potential of an atom, so for distances larger than $a$ (Bohr radius, $0.53 A$) the effect of wavefunction overlap is negligible compared to electrostatic interactions with other atoms.

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@gigacyan. Wrt your first paragraph, if indeed the wavefunctions of electrons bound in distinct Hydrogen atoms do indeed overlap - however negligable the effect, I believe it is to this that I was refering. In all honesty I'm now not so sure that the PEP even applies to electrons bound in distinct atoms?? Perhaps it is this part of the question that I remembered incorrectly.. – qftme Feb 11 '11 at 14:47
@qftme: PEP does apply: electrons are indistinguishable so you cannot say that electron a belongs to atom 1. What I mean to say is that the effect of the wavefunction overlap on the electron energy is much smaller than effect of electrostatic interaction of two atoms. – gigacyan Feb 11 '11 at 15:23
@gigacyan: So just to clarify, are you saying that all the electrons bound in Hydrogen atoms do indeed have slightly different energies, beit due to electromagnetic interaction or wavefunction overlap? Given the effect isn't measurable I'm not too concerned as to whether the prodominant factor is one of the other, just that A: the PEP still applies and B: the energies must therefore be different. – qftme Feb 11 '11 at 16:02
@qftme: A,B:yes. Analytic solution treats proton and electron as if they were the only two particles in the world. Any other particle will influence the solution albeit usually in a negligible way. – gigacyan Feb 11 '11 at 19:06
@gigacyan: Having just finished another read-through of this page I wondered whether you would edit your answer to include the essence of our discussion in its comments. After which, I would be happy to accept it :) – qftme Jun 16 '11 at 20:40

First, the wave states for the electrons of atom A are different from the wave states for the electrons of atom B; the two waves have their maxima at two different regions. So there's no violation of the HUP for them (actually in combining them into a single multi-particle wave function).

But what you're really supposed to do is to compute an overall wave function for all the electrons. If you have 8 electrons, this means having 8 copies of the universe (x,y,z,t) coordinates. And the wave function is antisymmetric under the exchange of two electrons.

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The Pauli exclusion principle is a direct consequence of indistinguishability, so there is no way to indistinguishable particle violate this rule.

When you have two or more indistinguishable particles, the total wavefunction is an antisymmetric combination of single-particles wavefunctions. The overlap of these wavefunctions will decrease the total energy in an amount known as exchange energy.

The exchange energy is important for the electronic structure calculations of atoms, molecules and solids, but is not so important when there are a large distance between these electrons.

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