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It is true that two electrons can't have identical quantum numbers, but spin itself is a quantum number. That is, the state with quantum numbers 111 can hold two electrons: one of spin up and one of spin down. So when you are finding the ground state, for example, find the four lowest energy eigenstates (ignoring spin), and the ground state of eight ...


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The idea behind the notation is that the operator $F$ is supposed to count the number of fermions in an expression, i.e. $$[F,A_n]= n A_n$$ if the operator $A_n$ contains $n$ fermions, what that means. Then $$[f(F),A_n]= f(n) A_n$$ for a sufficiently well-behaved function $f:\mathbb{C}\to \mathbb{C}$. In particular, for $f(x)=(-1)^x $, one has ...


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They shouldn't be thought of as operators i.e. $q$-numbers; instead, they should be thought of as $c$-numbers. They're mutually anticommuting but otherwise they play exactly the same role as $\Delta x^\mu$ for translations or angles $\varphi$ for rotations. They're spinor variables which means that under a Lorentz transformation $\Lambda\in SO(3,1)$, they ...


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Note: All products between $\chi$'s are to be understood as tensor products. Assuming the fermions are both spin 1/2 particles, one recalls that the spin-part of the total wavefunction in the spin 1 triplet is either one or a linear combination of $$ \chi(j=1,m=1) = \chi(1/2,1/2)\chi(1/2,1/2) $$ $$ \chi(j=1,0) = \frac{1}{\sqrt{2}}\big( ...



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