The Facts

I will begin by stating the bare facts briefly, at the cost of some precision.

Any quantum-mechanical system possesses angular momentum, sometimes called spin¹. Angular momentum is quantized, that is it is always an integer multiple of $\hbar/2$. (Here, $\hbar=h/2\pi$, where $h$ of course is Planck's constant.) Traditionally, $j$ stands for angular momentum². If we use so-called natural units of measurement (and we will), then $\hbar=1$ and possible values of $j$ are $0, \frac{1}{2}, 1, \frac{3}{2}, \ldots$.

Say we measure the component of angular momentum along some axis, say the $z$-axis. Let $j_z$ stand for this component. ³ Then $j_z$ is also quantized, and possible values for $j_z$ are

$$-j,-j+1,\ldots,j-1,j$$

A more precise definition of $j$ is the maximum possible value for $j_z$. For example, a system with angular momentum one-half (called a ``spin-$\frac{1}{2}$ system'' for short) will have $j_z=\pm\frac{1}{2}$; a system with angular momentum 1 (spin 1) will have $j_z=-1,0$, or $+1$. The same result holds for the component of angular momentum measured along any axis.

The total magnitude of angular momentum is given by $\sqrt{j_x^2+j_y^2+j_z^2}$; if this is measured somehow⁴, the result will be $\sqrt{j(j+1)}$. In classical mechanics, we would get $j$ and not $\sqrt{j(j+1)}$ (as we will see in a moment).

Suppose we combine two systems together, one with angular momentum $j$, the other with angular momentum $j'$. The resulting composite system will have angular momentum equal to one of the values

$$\vert j-j'\vert,\vert j-j'\vert+1,\ldots,j+j'$$

For example, combining two spin-$\frac{1}{2}$ systems can result in a spin-0 system or a spin-1 system.

This addition rule for angular momenta lies behind a wealth of physical phenomena. A simple example: suppose an atom absorbs a photon. A photon always has spin 1. Suppose the atom starts out in a state with $j=1$. After absorbing the photon, the atom must make a transition to a state with $j=0,1$, or 2. The same conclusion holds when an atom emits a photon. This three-way choice ultimately manifests itself as a triplet of spectral lines.

Now let's be a little more precise. In classical mechanics, angular momentum is a vector, say j, with components $j_x,j_y,j_z$ in some coordinate system, and magnitude $j=\vert{\bf j}\vert=\sqrt{j_x^2+j_y^2+j_z^2}$. Any component, say $j_z$, can range in value from $-j$ to $+j$.

In quantum mechanics, we have a (complex) Hilbert space of state-vectors (say $H$) for any system. A state-vector $v\in H$ specifies a state of the system, and $v$ and $w$ specify the same state if and only if $v=cw$ for some non-zero complex number $c$. Let $Q$ be some classical, real-valued variable that you can measure (like energy or momentum). The ``quantum version'' of $Q$ is a Hermitian operator on $H$. The eigenvalues of this operator are the possible values you can get from measuring $Q$. The quantum system has a definite value for $Q$ if and only if the system is in an eigenstate of the operator. If not, the act of measurement will serve to cast the system into such an eigenstate, with probabilities that can be computed by the rules of quantum mechanics.

Applying this prescription to angular momentum, we see that $j_x,j_y,j_z$ all must be Hermitian operators. It turns out now that the Hilbert space of quantum states decomposes, in the most general case, into a direct sum:

$$H = H_0 \oplus H_{1/2} \oplus \ldots \oplus H_j \oplus \ldots$$

where each summand in turn decomposes:

$$H_j = \sum_{m=-j}^j K_{jm}$$

and each element of $K_{jm}$ is an eigenvector of $j_z$ with eigenvalue $m$. (Here, $m$ ranges in steps of 1 from $-j$ to $+j$.) In particular cases, some of these summands may be missing. For example, if the quantum system has definite angular momentum $j$, then $H=H_j$.

Each $K_{jm}$ is of course invariant under $j_z$, i.e., we have an invariant direct-sum decomposition of $H$ for the operator $j_z$. The $K_{jm}$ are not invariant under $j_x$ or $j_y$, but it turns out that the $H_j$ are invariant under all three operators $j_x$, $j_y$, and $j_z$. Put another way, the invariant direct-sum decompositions for $j_x$ and $j_y$ have the same $H_j$'s but different $K_{jm}$'s.

Finally, the operator $j_x^2+j_y^2+j_z^2$ is itself a Hermitian operator; each $H_j$ is an eigenspace, with eigenvalue $j(j+1)$.

This excursion into Hilbert spaces should make mathematically-minded folk more comfortable with the catalog of ``spin facts''. The space $H_j$ is just the space of states that have ``spin $j$''; the subspace $K_{jm}$ is the space with component $m$ along the $z$-axis. I haven't discussed the combination rules; this translates into statements about the direct-sum decomposition of tensor products. Nor have I explained the curious appearence of the term $j(j+1)$; this is bound up with the non-commutativity of $j_x$, $j_y$, $j_z$.

Historically, the rules for spin came from playing with experimental data. The rules worked but remained mysterious. The birth of quantum mechanics came later, at the hands of Heisenberg, Schrödinger, and Dirac. Spin fell into place soon after that.

Quantum mechanics is mysterious, of course, as numerous philosophical treatises attest. But during the crucial period from Bohr's first great work (in 1913) to Heisenberg's discovery of matrix mechanics (in 1925), certain technical facts sowed confusion above and beyond the general quantum ``spookiness''. I will single out one of these for special attention: the spin of the electron. In 1926, Goudsmit and Uhlenbeck proposed (correctly) that the electron has spin one-half. Only integer spin quantum systems have classical counterparts, as we will see. The fractional spin of the electron lurked amid the general confusion during the heyday of the ``old quantum theory'' (1913-1925), bedevilling physicists. (Goudsmit and Uhlenbeck actually made their proposal in the language of the old quantum theory, not the newly minted quantum mechanics. The next year Pauli showed how to incorporate spin into quantum mechanics.)

In these notes, I will roam through the history of the old quantum theory, zeroing in on the adumbrations of modern quantum spin. (I will intersperse remarks on what the pioneers were ``really doing''.) I will not lay out the whole array of modern mathematical toys that makes Everything Clear. The reader whose appetite has been whetted must turn to standard textbooks for that. But I will try paint an impressionistic picture of the luminous synthesis, with a few broad brush strokes and one or two detailed pointillistic patches.

© 2001 Michael Weiss

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