Brief Tutorial for 
Measurement of Quantum-mechanical Spin-1/2 Systems

 

For a spin-1/2 system, the 1/2 refers to the quantum number s.  Spin is spin angular momentum and is related to the intrinsic magnetic moment of a particle.  Unlike orbital angular momentum (l = 0, 1, 2,...), spin angular momentum can take on integer and half-integer values (s = 0, 1/2, 1, 3/2, 2,...).  Half-integer particles are called fermions (obeying Fermi-Dirac statistics) and integer spin particles are called bosons (obeying Bose-Einstein statistics).

The spin operators, Sx, Sy, and Sz represent the effect of a measurement of the spin in one of those three directions, while S2 describes the measurement of the spin squared of a particle.  It turns out that these four operators can be written in terms of Pauli 2 x 2 matrices

 

as Si = ħ/2 σi, where specifically

The spin operators also obey the following commutation relations

and [Si, S2] = 0, based on the commutation relations of the Pauli matrices. 

 

© Mario Belloni and Wolfgang Christian (2006).