### 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.