Spin angular momentum

Up to now, we have tacitly assumed that the state of a particle in quantum mechanics can be completely specified by giving the wave-function $\psi$ as a function of the spatial coordinates $x$, $y$, and $z$. Unfortunately, there is a wealth of experimental evidence which suggests that this simplistic approach is incomplete.

Consider an isolated system at rest, and let the eigenvalue of its total angular momentum be $j (j+1) \hbar^2$. According to the theory of orbital angular momentum outlined in Sects. 5.4 and 5.5, there are two possibilities. For a system consisting of a single particle, $j=0$. For a system consisting of two (or more) particles, $j$ is a non-negative integer. However, this does not agree with observations, because we often find systems which appear to be structureless, and yet have $j\neq 0$. Even worse, systems where $j$ has half-integer values abound in nature. In order to explain this apparent discrepancy between theory and experiments, Gouldsmit and Uhlenbeck (in 1925) introduced the concept of an internal, purely quantum mechanical, angular momentum called spin. For a particle with spin, the total angular momentum in the rest frame is non-vanishing.

Let us denote the three components of the spin angular momentum of a particle by the Hermitian operators $(S_x, S_y, S_z)\equiv {\bf S}$. We assume that these operators obey the fundamental commutation relations (304)-(306) for the components of an angular momentum. Thus, we can write

$${\bf S} \times {\bf S} = {\rm i} \hbar {\bf S}.$$ (421)

We can also define the operator

$$S^2 = S_x^{ 2}+S_y^{ 2} + S_z^{ 2}.$$ (422)

According to the quite general analysis of Sect. 5.1,

$$[{\bf S}, S^2]= 0.$$ (423)

Thus, it is possible to find simultaneous eigenstates of $S^2$ and $S_z$. These are denoted $\vert s, s_z\rangle$, where

$$S_z \vert s, s_z\rangle = s_z \hbar \vert s, s_z\rangle,$$ (424)

$$S^2 \vert s, s_z\rangle = s (s+1) \hbar^2 \vert s, s_z\rangle.$$ (425)

According to the equally general analysis of Sect. 5.2, the quantum number $s$ can, in principle, take integer or half-integer values, and the quantum number $s_z$ can only take the values $s, s-1 \cdots -s+1, -s$.

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 Eqs. (297)-(299), since 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 Sects. 5.3 and 5.4) depends on Eqs. (297)-(299). In other words, the quantum number $s$ is allowed to take half-integer values.

Consider a spin one-half particle, for which

$$S_z \vert\pm \rangle = \pm \frac{\hbar}{2} \vert\pm \rangle,$$ (426)

$$S^2 \vert\pm\rangle = \frac{3 \hbar^2}{4} \vert\pm \rangle.$$ (427)

Here, the $\vert\pm \rangle$ denote eigenkets of the $S_z$ operator corresponding to the eigenvalues $\pm \hbar/2$. These kets are orthonormal (since $S_z$ is an Hermitian operator), so

$$\langle +\vert -\rangle = 0.$$ (428)

They are also properly normalized and complete, so that

$$\langle +\vert + \rangle=\langle -\vert - \rangle = 1,$$ (429)

and

$$\vert+\rangle \langle +\vert + \vert-\rangle \langle -\vert = 1.$$ (430)

It is easily verified that the Hermitian operators defined by

$$S_x = \frac{\hbar}{2} \left( \vert+\rangle \langle -\vert + \vert-\rangle \langle +\vert \right),$$ (431)

$$S_y = \frac{{\rm i} \hbar}{2}\left( - \vert+\rangle \langle -\vert + \vert-\rangle \langle +\vert \right),$$ (432)

$$S_z = \frac{\hbar}{2}\left( \vert+\rangle \langle +\vert - \vert-\rangle \langle -\vert \right),$$ (433)

satisfy the commutation relations (304)-(306) (with the $L_j$ replaced by the $S_j$). The operator $S^2$ takes the form

$$S^2 = \frac{3 \hbar^2}{4}.$$ (434)

It is also easily demonstrated that $S^2$ and $S_z$, defined in this manner, satisfy the eigenvalue relations (426)-(427). Equations (431)-(434) constitute a realization of the spin operators ${\bf S}$ and $S^2$ (for a spin one-half particle) in spin space (i.e., that Hilbert sub-space consisting of kets which correspond to the different spin states of the particle).