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
We can also define the operator
According to the quite general analysis of Section 4.1,
Thus, it is possible to find simultaneous eigenstates of and . These are denoted , where
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
They are also properly normalized and complete, so that
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).
- Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)