Spin States are not a function of space
Spin Angular Momentum
Definition
For a spin-1/2 particle, spin operators are represented by the Pauli matrices:
S_x = (ħ/2) σ_x, S_y = (ħ/2) σ_y, S_z = (ħ/2) σ_z
where
σ_x = [ 0 1 ]
[ 1 0 ]
[ 1 0 ]
σ_y = [ 0 -i ]
[ i 0 ]
[ i 0 ]
σ_z = [ 1 0 ]
[ 0 -1 ]
[ 0 -1 ]
Eigenfunctions of Sz
The eigenfunctions of are:
S_z α = (ħ/2) α, S_z β = -(ħ/2) β
where the eigenstates are:
α = [ 1 ]
[ 0 ]
[ 0 ]
β = [ 0 ]
[ 1 ]
[ 1 ]
Eigenstate for Sx
If a beam of spin-1/2 particles is prepared with spin along the x-direction, the corresponding eigenstate of is:
|+x⟩ = (1/√2) ( α + β )
Similarly, if the spin was along -x, the eigenstate would be:
|-x⟩ = (1/√2) ( α – β )
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