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 ${\bf S} \times {\bf S} = {\rm i}\,\hbar \, {\bf S}.$

$\ref{417}$

We can also define the operator

$[{\bf S}, S^2] = 0.$ $\ref{419}$

Thus, it is possible to find simultaneous eigenstates of $S_z$. These are denoted $S_z \,\vert s, s_z\rangle$

$S^2 \,\vert s, s_z\rangle$ $s$ can, in principle, take integer or half-integer values, and the quantum number $s, s-1 \cdots -s+1, -s$.

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 $\ref{290}$-$\ref{292}$, 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 $\ref{290}$-$\ref{292}$. In other words, the spin quantum number $S_z \,\vert\pm \rangle$

$S^2\, \vert\pm\rangle$ $\vert\pm \rangle$ denote eigenkets of the $\pm \hbar/2$. These kets are mutually orthogonal (since $\langle +\vert -\rangle = 0.$ $\ref{424}$

They are also properly normalized and complete, so that

$\vert+\rangle \langle +\vert + \vert-\rangle \langle -\vert = 1.$ $\ref{426}$

It is easily verified that the Hermitian operators defined by

$= \frac{\hbar}{2} \left(\, \vert+\rangle \langle -\vert + \vert-\rangle \langle +\vert\, \right),$ $\ref{427}$ $= \frac{{\rm i}\,\hbar}{2}\left(\, -\,\vert+\rangle \langle -\vert + \vert-\rangle \langle +\vert\,\right),$ $\ref{428}$ $= \frac{\hbar}{2}\left(\, \vert+\rangle \langle +\vert - \vert-\rangle \langle -\vert\,\right),$ $\ref{429}$

satisfy the commutation relations $\ref{297}$-$\ref{299}$ (with the $S_j$ ). The operator $S^2 = \frac{3\,\hbar^2}{4}.$

$\ref{430}$

It is also easily demonstrated that $S_z$, defined in this manner, satisfy the eigenvalue relations $\ref{422}$-$\ref{423}$. Equations $\ref{427}$-$\ref{430}$ constitute a realization of the spin operators $S^2$ (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)

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