# Doesn't the uncertainty principle mean all particles with identical energy are indistinguishable and hence have an amplitude for exchange?

I wonder if someone could tell me where my logic is going wrong here?

1. If two particles both have definite energy, then they have indefinite position.
2. As their positions could literally be anywhere in the universe, we cannot tell them apart.
3. If two particles are indistinguishable, then we have to consider the probability of interference due to the particles exchanging places (as in Feynmann Vol 3 Ch 4)
4. Does this mean no two electrons in the entire universe can share the same energy due to the Pauli exclusion principle?
5. Does this break locality?
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1) Uncertainty principle is momentum and position OR energy and lifetime, not energy and position.

2) If we confine the two particles in a infinite square well then they can only be in the well. Their wavefunctions go to zero at the boundaries.

3) True

4) False. Two particles can have the same energy. But thy have to be in two different states. For example, two electrons could have 1ev of energy, but one be in a spin up state and one be in a spin down state.

5) No... Maybe somebody else can explain this one

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To correct (1) I could reword the question to say "momentum". Surely your point in (4) doesn't matter as there are more than 2 electrons in the universe so the same problem arises? Or is it the case that no two electrons can ever share the same momentum? – Sideshow Bob Mar 5 '14 at 0:17
I was simply rebuking your statement, by counterexample, that no two electrons can have the same energy because of pauli exclusion principle. Two electrons can have the same momentum, PEC is only referring to the state of the particle not to the energy or momentum of particle. – jerk_dadt Mar 5 '14 at 0:27
I follow you so far. If two electrons have the same (definite) momentum (and spin), then provided they have enough momentum, their wavefunctions will each be a standing wave that propagates through all space. As they are Fermi particles these wavefunctions would interfere and cancel. Therefore it seems to me they cannot both have the same momentum. Does that make sense? – Sideshow Bob Mar 5 '14 at 12:16