48Physics Formulary by ir. J.C.A. Wevers10.10SpinFor the spin operators are defined by their commutation relations:[Sx, Sy] =i¯hSz. Because the spin operatorsdo not act in the physical space(x, y, z)the uniqueness of the wavefunction is not a criterium here: also halfodd-integer values are allowed for the spin. Because[L, S] = 0spin and angular momentum operators do nothave a common set of eigenfunctions. The spin operators are given byS=12¯hσ, withσx=0110,σy=0-ii0,σz=100-1The eigenstates ofSzare calledspinors:χ=α+χ++α-χ-, whereχ+= (1,0)represents the state withspin up (Sz=12¯h) andχ-= (0,1)represents the state with spin down (Sz=-12¯h). Then the probabilityto find spin up after a measurement is given by|α+|2and the chance to find spin down is given by|α-|2. Ofcourse holds|α+|2+|α-|2= 1.The electron will have an intrinsic magnetic dipole momentMdue to its spin, given byM=-egSS/2m,withgS= 2(1 +α/2π+· · ·)the gyromagnetic ratio. In the presence of an external magnetic field this gives
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