12.1 Introduction

The standard eigenfunctions of orbital angular momentum are the so called “spherical harmonics” of chapter 4.2. They show that the square orbital angular momentum has the possible values

$$L^2 \equiv L_x^2+L_y^2+L_z^2 = l(l+1) \hbar^2 \quad \mbox{where $l$\ is one of 0, 1, 2, 3, \ldots}$$

The nonnegative integer $l$ is called the azimuthal quantum number.

Further, the orbital angular momentum in any arbitrarily chosen direction, taken as the $z$-direction from now on, comes in multiples $m$ of Planck's constant $\hbar$:

$$L_z = m_l\hbar \quad \mbox{where $m_l$\ is one of $-l$, $-l{+}1$, $-l{+}2$, \ldots, $l{-}1$, $l$.}$$

The integer $m_l$ is called the magnetic quantum number.

The possible values of the square spin angular momentum can be written as

$$S^2 \equiv S_x^2+S_y^2+S_z^2 = s(s+1)\hbar^2 \quad \quad... ...rn-.2em\lower.4ex\hbox{\the\scriptfont0 2}\kern.05em, \ldots}$$

The “spin azimuthal quantum number” $s$ is usually called the “spin” for short. Note that while the orbital azimuthal quantum number $l$ had to be an integer, the spin can be half integer. But one important conclusion of this chapter will be that the spin cannot be anything more. A particle with, say, spin $\frac13$ cannot not exist according to the theory.

For the spin angular momentum in the $z$ direction

$$S_z = m_s\hbar \quad \mbox{where $m_s$\ is one of $-s$, $-s{+}1$, $-s{+}2$, \ldots, $s{-}1$, $s$.}$$

Note that if the spin $s$ is half integer, then so are all the spin magnetic quantum numbers $m_s$. If the nature of the angular momentum is self-evident, the subscript $l$ or $s$ of the magnetic quantum numbers $m$ will be omitted.

Particles with half-integer spin are called fermions. That includes electrons, as well as protons and neutrons and their constituent quarks. All of these critically important particles have spin $\leavevmode\kern.03em \raise.7ex\hbox{\the\scriptfont0 1}\kern-.2em /\kern-.2em\lower.4ex\hbox{\the\scriptfont0 2}\kern.05em$. (Excited proton and neutron states can have spin $\leavevmode\kern.03em \raise.7ex\hbox{\the\scriptfont0 3}\kern-.2em /\kern-.2em\lower.4ex\hbox{\the\scriptfont0 2}\kern.05em$.) Particles with integer spin are bosons. That includes the particles that act as carriers of fundamental forces; the photons, intermediate vector bosons, gluons, and gravitons. All of these have spin 1, except the graviton which supposedly has spin 2.