Who will win -- Pauli or Einstein?

[Kaushik_KS]

I met with a little conflict between Pauli and Einstein? Can you please help. Its a thought experiment.
Consider a single crystal which is 1km long. During its formation, due to Pauli’s exclusion principle, no two electron will have same quantum state. Now consider two electron, one with E and other with E+δE. Practically they are next to each other in quantum states, so their energy varies this minimum value. But they are located, spatially, 1km apart at the extremes. One with higher energy will always try to get down but as the state is already occupied, it won’t be able to get there. My questions are –
Q1. As they are at infinity to each other, i.e. can’t feel each other’s field, what physically resist this transition from E+δE to E?
Q2. Suppose we provide energy to one with energy E. This would be resisted too as the higher states are also occupied. Then what happens to that one which was actually in the state before? Where will it go and what compels it to move?
Q3. One with E+δE will drop down to E. Is this instantaneous? How the message of vacancy in lower state being conveyed to one that occupies it? Does the reaction time depend upon separation between the electrons? If these messages are conveyed instantaneously the relativity is at stake.

Who will win Pauli or Einstein?

 

[Kaushik_KS]

For answer: Pauli Exclusion only applied to electrons within the same poly-electron atom.
Explanation: Fermi Dirac statistics and the concept of Fermi energy level for electron sea model originates from Pauli Exclusion principle. Had there been no such principle, all electrons would have occupied the lowest energy level in metals not the lowest POSSIBLE energy level, hence no Fermi Dirac. MB stats would have worked.

 

[mikeyork]

Your questions are much too vague. You need to take into account the electrons' position (or momenta) and spin states and the likely presence of many other electrons.

If you are asking what it is that prevents two electrons from occupying the same quantum state, then the answer is that two-particle states in a Hilbert space are formed from the direct product of the Hilbert spaces of each particle but since that direct product is order-dependent and nature doesn't care about what order we use, the result is a quantum superselection rule. The same logic that applies to electrons applies to all identical fermions and bosons and the superselection rule is that L+S must be even in the CM frame of the two particles if they have the same energy. (L and S are the total orbital and spin quantum numbers of the composite system.)

When two electrons form such a superselected composite state they behave like a boson and any other electrons effectively ignore them as distinct electrons and are not superselected by the bosonic pair. Look up "Cooper pairs" if you want to understand what happens to electrons in their lowest energy state in your meter long crystal.

 

[PeterDonis]

During its formation, due to Pauli’s exclusion principle, no two electron will have same quantum state.

That is true; but you are not actually using this assumption. You are using the much stronger (and false) assumption that no two electrons will have the same energy. In fact there will be multiple different quantum states with the same energy.

they are located, spatially, 1km apart at the extremes

Which means it is perfectly possible for them to both have the same energy--because they have different positions.

Pauli Exclusion only applied to electrons within the same poly-electron atom.

This is not correct. The Pauli Exclusion principle applies to all fermions, everywhere.

 

[Kaushik_KS]

Its not that those two energy levels are of different state but the reverse. Consider two different states at E and E+dE.

 

[PeterDonis]

Consider two different states at E and E+dE.

Your argument assumes that the electron at energy E + dE cannot drop to energy E. That assumption amounts to claiming that no two electrons can have the same energy, even if they are 1 km apart. That assumption is false. The electron at E + dE can drop to energy E, so your argument breaks down.

 

[Khashishi]

In a crystal state, you have a total energy for the crystal, but it's not possible to state that individual electrons have energy E and E+$\delta$E. Energy is a global property for the state.

Position eigenstates for the electrons are not energy eigenstates of the crystal. If you are able to state that two electrons are on different sides of the crystal, it probably doesn't have a definite value of energy. I think a crystal this large would not likely be in an energy or position eigenstate because of constant interactions with the environment.

 

[ZapperZ]

I met with a little conflict between Pauli and Einstein? Can you please help. Its a thought experiment.
Consider a single crystal which is 1km long. During its formation, due to Pauli’s exclusion principle, no two electron will have same quantum state. Now consider two electron, one with E and other with E+δE. Practically they are next to each other in quantum states, so their energy varies this minimum value. But they are located, spatially, 1km apart at the extremes. One with higher energy will always try to get down but as the state is already occupied, it won’t be able to get there. My questions are –
Q1. As they are at infinity to each other, i.e. can’t feel each other’s field, what physically resist this transition from E+δE to E?
Q2. Suppose we provide energy to one with energy E. This would be resisted too as the higher states are also occupied. Then what happens to that one which was actually in the state before? Where will it go and what compels it to move?
Q3. One with E+δE will drop down to E. Is this instantaneous? How the message of vacancy in lower state being conveyed to one that occupies it? Does the reaction time depend upon separation between the electrons? If these messages are conveyed instantaneously the relativity is at stake.

Who will win Pauli or Einstein?

Pauli exclusion principle is part of a more general symmetry principle, that the overall wave function must be antisymmetric. The spatial part in this scenario is already antisymmetric. I do not see the problem here.

Unfortunately, the person who will lose in this one is YOU.

Zz.