We can also define the operator

(5.2) |

According to the quite general analysis of Section 4.1,

(5.3) |

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

(5.4) | ||

(5.5) |

According to the equally general analysis of Section 4.2, the (dimensionless) quantum number can, in principle, take (non-negative) integer or half-integer values, and the (dimensionless) 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 (4.1)-(4.3), because this identification depends on an analogy with classical mechanics, and the concept of spin is purely quantum mechanical. In other words, spin has no analogy in classical physics. Consequently, the restriction that the quantum number of the overall angular momentum must take integer values does not apply to spin angular momentum, because this restriction (found in Sections 4.3 and 4.4) ultimately depends on Equations (4.1)-(4.3). In other words, the spin quantum number is allowed to take half-integer values.

Consider a spin one-half particle (e.g., an electron, a proton, or a neutron), for which

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

(5.8) |

The kets are also properly normalized and complete, so that

(5.9) |

and

(5.10) |

It is easily verified that the Hermitian operators defined by

satisfy the commutation relations (4.8)-(4.10) (with the replaced by the ). (See Exercise 1.) The operator takes the form

It is also easily demonstrated that and , defined in this manner, satisfy the eigenvalue relations (5.6)-(5.7). Equations (5.11)-(5.14) constitute a realization of the spin operators and (for a spin one-half particle) in