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 ]
σ_y = [ 0 -i ]
[ i 0 ]
σ_z = [ 1 0 ]
[ 0 -1 ]

Eigenfunctions of Sz

The eigenfunctions of S_z are:

S_z α = (ħ/2) α, S_z β = -(ħ/2) β

where the eigenstates are:

α = [ 1 ]
[ 0 ]
β = [ 0 ]
[ 1 ]

Eigenstate for Sx

If a beam of spin-1/2 particles is prepared with spin along the x-direction, the corresponding eigenstate of S_x is:

|+x⟩ = (1/√2) ( α + β )

Similarly, if the spin was along -x, the eigenstate would be:

|-x⟩ = (1/√2) ( α – β )