We can also define the operator

(418) |

According to the quite general analysis of Section 4.1,

(419) |

Thus, it is possible to find simultaneous eigenstates of and . These are denoted , where

(420) | ||

(421) |

According to the equally general analysis of Section 4.2, the quantum number can, in principle, take integer or half-integer values, and the quantum number can only take the values .

Spin angular momentum clearly has many properties in common with
orbital angular momentum. However, there is one vitally important difference.
Spin angular momentum operators *cannot* be expressed in terms of
position and momentum operators, like in Equations (290)-(292), because this
identification depends on an analogy with classical mechanics, and the concept
of spin is purely quantum mechanical: i.e., it has no analogy in classical physics.
Consequently, the restriction that the quantum number of the overall angular
momentum must take *integer* values is lifted for spin angular momentum,
since this restriction (found in Sections 4.3 and 4.4) depends on Equations (290)-(292).
In other words, the spin quantum number
is allowed to take *half-integer* values.

Consider a spin one-half particle, for which

Here, the denote eigenkets of the operator corresponding to the eigenvalues . These kets are mutually orthogonal (since is an Hermitian operator), so

(424) |

They are also properly normalized and complete, so that

(425) |

and

(426) |

It is easily verified that the Hermitian operators defined by

satisfy the commutation relations (297)-(299) (with the replaced by the ). The operator takes the form

It is also easily demonstrated that and , defined in this manner, satisfy the eigenvalue relations (422)-(423). Equations (427)-(430) constitute a realization of the spin operators and (for a spin one-half particle) in