# Pauli exclusion principle

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## Main Question or Discussion Point

What is the explanation for the case that fermion's can't occupy the same quantum state? Is there some logic behind this or is it just observation?

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DrClaude
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The short answer is that the Pauli principle states that a valid state for identical fermions must be anti-symmetric with respect to the exchange of two particles. You can't construct an anti-symmetric state where two fermions are in the same single-particle state.

As to why the Pauli principle applies, you have to look up the spin-statistics theorem. Ultimately, the answer is "because Nature is that way."

Ultimately, the reason is that nature doesn't care which particle a potential observer thinks is which. A similar rule applies to bosons. The rules are usually expressed as symmetry or anti-symmetry of the wavefunction depending on spin. An alternative expression of the rule (and one that relates directly to observables) is that L+S must be even in the CM frame (equal and opposite momentum) for any pair of identical particles, where L is the net orbital angular momentum and S is the net spin angular momentum. In a frame where the particles have equal and parallel momentum, then the rule becomes that S must be even and, in the case of electrons, this means S=0 so they must have opposite spin. In the case of atomic electrons, where it becomes the Pauli rule, it is stated as "no two electrons can occupy the same state" because if all other quantum numbers are identical then the spins must be opposite so they sum to 0 (with the corollary that if the spins are the same, then the states must differ in some other way).

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DrClaude
Mentor
UAn alternative expression of the rule (and one that relates directly to observables) is that L+S must be even in the CM frame (equal and opposite momentum) for any pair of identical particles, where L is the net orbital angular momentum and S is the net spin angular momentum.
Do you have any reference for this statement? Does it mean that having two electrons, one with ##l=0,m_l=0,m_s=1/2## and the other ##l=2,m_l=2,m_s=1/2## (so ##L=2##, ##S=1##) is not allowed?

Do you have any reference for this statement? Does it mean that having two electrons, one with ##l=0,m_l=0,m_s=1/2## and the other ##l=2,m_l=2,m_s=1/2## (so ##L=2##, ##S=1##) is not allowed?
Good question! The L+S rule comes from SU(2) couplings given the usual anti-symmetry. Off the top of my head I think the answer to your question is that you can't get those two specific orbital states in the CM frame because of the spatial symmetry relating the angular co-ordinates (their momenta point in opposite directions). So, for instance you could have l1 = l2 = 1 and then the L=1 state would require s1 = s2. From memory I think the original rule comes from the classic Jacob & Wick* paper on helicity states, but you can also find it in my spin-statistics papers (which I haven't linked because they are not accepted mainstream for other reasons). PM me if you want further information.

*M. Jacob and G. Wick, “On the general theory of collisions for particles with spin,” Annals Phys. 7 (1959) 404–428. (I can't locate my copy at this moment but, IIRC, they give the partial wave analysis of identical particle scattering in the CM frame.)

Later edit: Also check out Rose "Elementary Theory of Angular Momentum", Chapter XII, section 38 ("Identical Particles in L-S coupling").

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