Spin Angular Momentum

The type of angular momentum discussed above is associated with the movement of particles in orbits. However, it turns out that even stationary particles can possess angular momentum. This is called spin angular momentum. The spin quantum number $s$ plays a role analogous to $l$ for spin angular momentum, i. e., the square of the spin angular momentum vector of a particle is

$$L_s^2 = \hbar^2 s (s + 1) .$$ (10.33)

The spin orientation quantum number $m_s$ is similarly related to $s$:

$$L_{zs} = \hbar m_s , ~~~~~ m_s = -s, -s + 1, \ldots , s - 1,s .$$ (10.34)

The spin angular momentum for an elementary particle is absolutely conserved, i. e., it can never change. Thus, the value of $s$ is an intrinsic property of a particle. The major difference between spin and orbital angular momentum is that the spin quantum number can take on more values, i. e., $s = 0, 1/2, 1, 3/2, 2, 5/2, \ldots$.

Particles with integer spin values $s = 0, 1, 2, \ldots$ are called bosons after the Indian physicist Satyendra Nath Bose. Particles with half-integer spin values $s = 1/2, 3/2, 5/2, \ldots$ are called fermions after the Italian physicist Enrico Fermi. As we shall see later in the course, bosons and fermions play very different roles in the universe.