# 5.2: Properties of Spin Angular Momentum

Let us denote the three components of the spin angular momentum of a particle by the Hermitian operators . We assume that these operators obey the fundamental commutation relations (297)-(299) for the components of an angular momentum. Thus, we can write

(417) |

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 *spin space* (i.e., the Hilbert sub-space consisting of kets which correspond to the different spin states of the particle).

### Contributors

- Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)