All chemists are familiar with the yellow color imparted to a flame by sodium atoms. The strongest yellow line (the D line) in the sodium spectrum is actually two closely spaced lines. The sodium D line arises from a transition from the excited configuration $1 s^{2} 2 s^{2} 2 p^{6} 3 p$ to the ground state. The doublet nature of this and other lines in the Na spectrum indicates a doubling of the expected number of states available to the valence electron.
To explain this fine structure of atomic spectra, Uhlenbeck and Goudsmit proposed in 1925 that the electron has an intrinsic (built-in) angular momentum in addition to the orbital angular momentum due to its motion about the nucleus. If we picture the electron as a sphere of charge spinning about one of its diameters, we can see how such an intrinsic angular momentum can arise. Hence we have the term spin angular momentum or, more simply, spin. However, electron "spin" is not a classical effect, and the picture of an electron rotating about an axis has no physical reality. The intrinsic angular momentum is real, but no easily visualizable model can properly explain its origin. We cannot hope to understand microscopic particles based on models taken from our experience in the macroscopic world. Other elementary particles besides the electron have spin angular momentum.
In 1928, Dirac developed the relativistic quantum mechanics of an electron, and in his treatment electron spin arises naturally.
In the nonrelativistic quantum mechanics to which we are confining ourselves, electron spin must be introduced as an additional hypothesis. We have learned that each physical property has its corresponding linear Hermitian operator in quantum mechanics. For such properties as orbital angular momentum, we can construct the quantum-mechanical operator from the classical expression by replacing $p{x}, p{y}, p_{z}$ by the appropriate operators. The inherent spin angular momentum of a microscopic particle has no analog in classical mechanics, so we cannot use this method to construct operators for spin. For our purposes, we shall simply use symbols for the spin operators, without giving an explicit form for them.
Analogous to the orbital angular-momentum operators $\hat{L}^{2}, \hat{L}{x}, \hat{L}{y}, \hat{L}{z}$, we have the spin angular-momentum operators $\hat{S}^{2}, \hat{S}{x}, \hat{S}{y}, \hat{S}{z}$, which are postulated to be linear and Hermitian. $\hat{S}^{2}$ is the operator for the square of the magnitude of the total spin angular
momentum of a particle. $\hat{S}_{z}$ is the operator for the $z$ component of the particle's spin angular momentum. We have
\(
\begin{equation}
\hat{S}^{2}=\hat{S}{x}^{2}+\hat{S}{y}^{2}+\hat{S}_{z}^{2} \tag{10.1}
\end{equation}
\)
We postulate that the spin angular-momentum operators obey the same commutation relations as the orbital angular-momentum operators. Analogous to $\left[\hat{L}{x}, \hat{L}{y}\right]=$ $i \hbar \hat{L}{z},\left[\hat{L}{y}, \hat{L}{z}\right]=i \hbar \hat{L}{x},\left[\hat{L}{z}, \hat{L}{x}\right]=i \hbar \hat{L}_{y}$ [Eqs. (5.46) and (5.48)], we have
\(
\begin{equation}
\left[\hat{S}{x}, \hat{S}{y}\right]=i \hbar \hat{S}{z}, \quad\left[\hat{S}{y}, \hat{S}{z}\right]=i \hbar \hat{S}{x}, \quad\left[\hat{S}{z}, \hat{S}{x}\right]=i \hbar \hat{S}_{y} \tag{10.2}
\end{equation}
\)
From (10.1) and (10.2), it follows, by the same operator algebra used to obtain (5.49) and (5.50), that
\(
\begin{equation}
\left[\hat{S}^{2}, \hat{S}{x}\right]=\left[\hat{S}^{2}, \hat{S}{y}\right]=\left[\hat{S}^{2}, \hat{S}_{z}\right]=0 \tag{10.3}
\end{equation}
\)
Since Eqs. (10.1) and (10.2) are of the form of Eqs. (5.107) and (5.108), it follows from the work of Section 5.4 (which depended only on the commutation relations and not on the specific forms of the operators) that the eigenvalues of $\hat{S}^{2}$ are [Eq. (5.142)]
\(
\begin{equation}
s(s+1) \hbar^{2}, \quad s=0, \frac{1}{2}, 1, \frac{3}{2}, \ldots \tag{10.4}
\end{equation}
\)
and the eigenvalues of $\hat{S}_{z}$ are [Eq. (5.141)]
\(
\begin{equation}
m{s} \hbar, \quad m{s}=-s,-s+1, \ldots, s-1, \quad s \tag{10.5}
\end{equation}
\)
The quantum number $s$ is called the spin of the particle. Although nothing in Section 5.4 restricts electrons to a single value for $s$, experiment shows that all electrons do have a single value for $s$, namely, $s=\frac{1}{2}$. Protons and neutrons also have $s=\frac{1}{2}$. Pions have $s=0$. Photons have $s=1$. However, Eq. (10.5) does not hold for photons. Photons travel at speed $c$ in vacuum. Because of their relativistic nature, it turns out that photons can have either $m{s}=+1$ or $m{s}=-1$, but not $m{s}=0$ (see Merzbacher, Chapter 22). These two $m{s}$ values correspond to left circularly polarized and right circularly polarized light.
With $s=\frac{1}{2}$, the magnitude of the total spin angular momentum of an electron is given by the square root of (10.4) as
\(
\begin{equation}
\left[\frac{1}{2}\left(\frac{3}{2}\right) \hbar^{2}\right]^{1 / 2}=\frac{1}{2} \sqrt{3} \hbar \tag{10.6}
\end{equation}
\)
For $s=\frac{1}{2}$, Eq. (10.5) gives the possible eigenvalues of $\hat{S}{z}$ of an electron as $+\frac{1}{2} \hbar$ and $-\frac{1}{2} \hbar$. The electron spin eigenfunctions that correspond to these $\hat{S}{z}$ eigenvalues are denoted by $\alpha$ and $\beta$ :
\(
\begin{align}
& \hat{S}{z} \alpha=+\frac{1}{2} \hbar \alpha \tag{10.7}\
& \hat{S}{z} \beta=-\frac{1}{2} \hbar \beta \tag{10.8}
\end{align}
\)
Since $\hat{S}{z}$ commutes with $\hat{S}^{2}$, we can take the eigenfunctions of $\hat{S}{z}$ to be eigenfunctions of $\hat{S}^{2}$ also, with the eigenvalue given by (10.4) with $s=\frac{1}{2}$ :
\(
\begin{equation}
\hat{S}^{2} \alpha=\frac{3}{4} \hbar^{2} \alpha, \quad \hat{S}^{2} \beta=\frac{3}{4} \hbar^{2} \beta \tag{10.9}
\end{equation}
\)
$\hat{S}{z}$ does not commute with $\hat{S}{x}$ or $\hat{S}{y}$, so $\alpha$ and $\beta$ are not eigenfunctions of these operators. The terms spin $u p$ and spin down refer to $m{s}=+\frac{1}{2}$ and $m{s}=-\frac{1}{2}$, respectively. See Fig. 10.1. We shall later show that the two possibilities for the quantum number $m{s}$ give the doubling of lines in the spectra of the alkali metals.
FIGURE 10.1 Possible orientations of the electron spin vector with respect to the $z$ axis. In each case, $\mathbf{S}$ lies on the surface of a cone whose axis is the $z$ axis.
The wave functions we have dealt with previously are functions of the spatial coordinates of the particle: $\psi=\psi(x, y, z)$. We might ask: What is the variable for the spin eigenfunctions $\alpha$ and $\beta$ ? Sometimes one talks of a spin coordinate $\omega$, without really specifying what this coordinate is. Most often, one takes the spin quantum number $m_{s}$ as being the variable on which the spin eigenfunctions depend. This procedure is quite unusual as compared with the spatial wave functions; but because we have only two possible electronic spin eigenfunctions and eigenvalues, this is a convenient choice. We have
\(
\begin{equation}
\alpha=\alpha\left(m{s}\right), \quad \beta=\beta\left(m{s}\right) \tag{10.10}
\end{equation}
\)
As usual, we want the eigenfunctions to be normalized. The three variables of a one-particle spatial wave function range continuously from $-\infty$ to $+\infty$, so normalization means
\(
\int{-\infty}^{\infty} \int{-\infty}^{\infty} \int_{-\infty}^{\infty}|\psi(x, y, z)|^{2} d x d y d z=1
\)
The variable $m_{s}$ of the electronic spin eigenfunctions takes on only the two discrete values $+\frac{1}{2}$ and $-\frac{1}{2}$. Normalization of the one-particle spin eigenfunctions therefore means
\(
\begin{equation}
\sum{m{s}=-1 / 2}^{1 / 2}\left|\alpha\left(m{s}\right)\right|^{2}=1, \quad \sum{m{s}=-1 / 2}^{1 / 2}\left|\beta\left(m{s}\right)\right|^{2}=1 \tag{10.11}
\end{equation}
\)
Since the eigenfunctions $\alpha$ and $\beta$ correspond to different eigenvalues of the Hermitian operator $\hat{S}_{z}$, they are orthogonal:
\(
\begin{equation}
\sum{m{s}=-1 / 2}^{1 / 2} \alpha^{}\left(m{s}\right) \beta\left(m{s}\right)=0 \tag{10.12}
\end{}
\)
Taking $\alpha\left(m{s}\right)=\delta{m{s} 1 / 2}$ and $\beta\left(m{s}\right)=\delta{m{s},-1 / 2}$, where $\delta_{j k}$ is the Kronecker delta function, we can satisfy (10.11) and (10.12).
When we consider the complete wave function for an electron including both space and spin variables, we shall normalize it according to
\(
\begin{equation}
\sum{m{s}=-1 / 2}^{1 / 2} \int{-\infty}^{\infty} \int{-\infty}^{\infty} \int{-\infty}^{\infty}\left|\psi\left(x, y, z, m{s}\right)\right|^{2} d x d y d z=1 \tag{10.13}
\end{equation}
\)
The notation
\(
\int\left|\psi\left(x, y, z, m_{s}\right)\right|^{2} d \tau
\)
will denote summation over the spin variable and integration over the full range of the spatial variables, as in (10.13). The symbol $\int d v$ will denote integration over the full range of the system's spatial variables.
An electron is currently considered to be a pointlike elementary particle with no substructure. High-energy electron-positron collision experiments show no evidence for a nonzero electron size and put an upper limit of $3 \times 10^{-19} \mathrm{~m}$ on the radius of an electron [D. Bourilkov, Phys. Rev. D, 62, 076005 (2000); arxiv.org/abs/hep-ph/0002172]. Protons and neutrons are made of quarks, and so are not elementary particles. The proton rms charge radius is $0.88 \times 10^{-15} \mathrm{~m}$.