$$\require{cancel}$$

# 9.1: Spin Operators

Because spin is a type of angular momentum, it is reasonable to suppose that it possesses similar properties to orbital angular momentum. Thus, by analogy with Section [s8.2], we would expect to be able to define three operators—$$S_x$$, $$S_y$$, and $$S_z$$—that represent the three Cartesian components of spin angular momentum. Moreover, it is plausible that these operators possess analogous commutation relations to the three corresponding orbital angular momentum operators, $$L_x$$, $$L_y$$, and $$L_z$$. [See Equations ([e8.6])–([e8.8]).] In other words,

\begin{aligned} \label{e10.1x} [S_x, S_y]&= {\rm i}\,\hbar\,S_z,\\[0.5ex] [S_y, S_z]&= {\rm i}\,\hbar\,S_x,\\[0.5ex] [S_z,S_x]&= {\rm i}\,\hbar\,S_y.\label{e10.2x}\end{aligned} We can represent the magnitude squared of the spin angular momentum vector by the operator $S^2 = S_x^{\,2} + S_y^{\,2}+ S_z^{\,2}.$ By analogy with the analysis in Section [s8.2], it is easily demonstrated that $[S^2, S_x] = [S^2, S_y] = [S^2,S_z] = 0.$ We thus conclude (see Section [smeas]) that we can simultaneously measure the magnitude squared of the spin angular momentum vector, together with, at most, one Cartesian component. By convention, we shall always choose to measure the $$z$$-component, $$S_z$$.

By analogy with Equation ([e8.13]), we can define raising and lowering operators for spin angular momentum: $S_\pm = S_x \pm {\rm i}\,S_y.$ If $$S_x$$, $$S_y$$, and $$S_z$$ are Hermitian operators, as must be the case if they are to represent physical quantities, then $$S_\pm$$ are the Hermitian conjugates of one another: that is,

$\label{e10.7} (S_\pm)^\dag = S_\mp.$ Finally, by analogy with Section [s8.2], it is easily demonstrated that \begin{aligned} S_+\,S_- &= S^2-S_z^{\,2}+\hbar\,S_z,\label{e10.7a}\\[0.5ex] S_-\,S_+&= S^2-S_z^{\,2}-\hbar\,S_z,\label{e10.8}\\[0.5ex] [S_+,S_z]&= - \hbar\,S_+,\label{e10.9}\\[0.5ex] [S_-,S_z]&= +\hbar\,S_-.\label{e10.10}\end{aligned}

# Contributors

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