### 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

The nonnegative integer is called the azimuthal quantum number.

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

The integer is called the magnetic quantum number.

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

The spin azimuthal quantum number is usually called the spin for short. Note that while the orbital azimuthal quantum number 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 cannot not exist according to the theory.

For the spin angular momentum in the -​direction

Note that if the spin is half integer, then so are all the spin magnetic quantum numbers . If the nature of the angular momentum is self-evident, the subscript or of the magnetic quantum numbers 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 . (Excited proton and neutron states can have spin .) 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.