Review of Angular Momentum in Quantum Mechanics

As we learned earlier, angular momentum is quantized in quantum mechanics. We can simultaneously measure only the magnitude of the angular momentum vector and one component, usually taken to be the $z$ component. Measurement of the other two components simultaneously with the $z$ component is forbidden by the uncertainty principle.

The magnitude of the orbital angular momentum of an object can take on the values $\vert \mbox{\bf L} \vert = [ l ( l + 1) ]^{1/2} \hbar$ where $l = 0,1,2, \ldots$. The $z$ component can likewise equal $L_z = m \hbar$ where $m = -l, -l + 1 , \ldots , l$.

Particles can have an intrinsic spin angular momentum as well as an orbital angular momentum. The possible values for the magnitude of the spin angular momentum are $\vert \mbox{\bf S} \vert = [ s ( s + 1 ) ]^{1/2} \hbar$ and the $z$ component of the spin angular momentum $S_z = m_s \hbar$ where $m_s = -s , -s + 1 , \ldots , s$. Spin differs from orbital angular momentum in that the spin can take on half-integer as well as integer values: $s = 0 , 1/2 , 1 , 3/2 , \ldots$ are possible spin quantum numbers.

Spin is an intrinsic, unchangeable quantity for an elementary particle. Particles with half-integer spins, $s = 1/2, 3/2, 5/2, \ldots$, are called fermions, while particles with integer spins, $s = 0, 1, 2, \ldots$ are called bosons. Fermions can only be created or destroyed in particle-antiparticle pairs, whereas bosons can be created or destroyed singly.