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phys documents (dragged) 54

# phys documents (dragged) 54 - 48 Physics Formulary by ir...

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48 Physics Formulary by ir. J.C.A. Wevers 10.10 Spin For the spin operators are defined by their commutation relations: [ S x , S y ] = i ¯ hS z . Because the spin operators do not act in the physical space ( x, y, z ) the uniqueness of the wavefunction is not a criterium here: also half odd-integer values are allowed for the spin. Because [ L, S ] = 0 spin and angular momentum operators do not have a common set of eigenfunctions. The spin operators are given by S = 1 2 ¯ h σ , with σ x = 0 1 1 0 , σ y = 0 - i i 0 , σ z = 1 0 0 - 1 The eigenstates of S z are called spinors : χ = α + χ + + α - χ - , where χ + = (1 , 0) represents the state with spin up ( S z = 1 2 ¯ h ) and χ - = (0 , 1) represents the state with spin down ( S z = - 1 2 ¯ h ). Then the probability to find spin up after a measurement is given by | α + | 2 and the chance to find spin down is given by | α - | 2 . Of course holds | α + | 2 + | α - | 2 = 1 . The electron will have an intrinsic magnetic dipole moment M due to its spin, given by M = - eg S S/ 2 m , with g S = 2(1 + α / 2 π + · · · ) the gyromagnetic ratio. In the presence of an external magnetic field this gives
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