Electron Spin and Spin Quantum Numbers

Spin quantum numbers are not a historical curiosity confined to introductory atomic theory — they are the operating principle behind entire classes of modern chemical instrumentation and technology. Nuclear magnetic resonance (NMR) spectroscopy, the workhorse structural tool of organic and biomolecular chemistry, exploits the analogous nuclear spin quantum number (for $^1$H, $^{13}$C, $^{31}$P, etc.) splitting into Zeeman levels in a magnetic field, with chemical shifts and coupling patterns read out from transitions between those spin states — the same $\Delta E = g\mu B$ physics worked out in Example 2 for electrons. Electron paramagnetic resonance (EPR/ESR) spectroscopy directly probes unpaired electron spins in radicals, transition-metal complexes, and defect centers in materials, using the electron's own $g_e \approx 2$ and the $\alpha/\beta$ manifold exactly as introduced here. In quantum computing, electron and nuclear spin states are leading candidates for physical qubits precisely because they are naturally two-level systems — the same two-valuedness responsible for the Stern–Gerlach splitting is now engineered into spin qubits in silicon, diamond nitrogen-vacancy centers, and trapped ions. In materials science, spin underlies magnetism itself: ferromagnetism, antiferromagnetism, and the entire field of spintronics rely on controlling and reading out electron spin populations and spin currents. In photochemistry and photophysics, the distinction between singlet ($S=0$) and triplet ($S=1$) states of electronically excited molecules — built directly from how two individual electron spins ($m_s=\pm\tfrac12$ each) combine — governs phosphorescence lifetimes, photodynamic therapy mechanisms, and the efficiency of OLED materials, foreshadowing the coupling schemes formalized in Russell–Saunders term symbols. Finally, in fundamental atomic and molecular structure, spin is the non-negotiable ingredient that, via the Pauli Exclusion Principle, forces electrons to occupy successively higher orbitals rather than collapsing into the lowest one — the single fact that gives atoms their size, and gives the periodic table its structure of shells, periods, and groups.

Ask the tutor: How do individual electron spins ($m_s=\pm\tfrac12$) combine into a total atomic spin quantum number $S$, and how does that feed into Russell–Saunders term symbols?

Every electron in the universe carries exactly the same intrinsic angular momentum magnitude, characterized by the spin quantum number

This is not something you solve for from a Schrödinger equation the way you solve for $n$, $l$, and $m_l$ from the hydrogen-atom Hamiltonian. It is a fixed, built-in property of the particle, the way charge $-e$ and mass $m_e$ are fixed properties. Just as orbital angular momentum has magnitude $\sqrt{l(l+1)}\hbar$ and a $z$-projection $m_l\hbar$ with $m_l = -l, \dots, +l$, spin angular momentum has magnitude $\sqrt{s(s+1)}\hbar = \frac{\sqrt3}{2}\hbar$ and a $z$-projection $m_s\hbar$ where

Because $s=\tfrac12$ is fixed, there are only ever two possible spin states for a single electron — not the $2l+1$ states you get for varying $l$. These two states are conventionally called spin-up ($m_s = +\tfrac12$) and spin-down ($m_s = -\tfrac12$), and are represented by two orthonormal abstract functions, $\alpha$ and $\beta$, that live in an internal "spin space" rather than in ordinary position space. There is no coordinate $\omega$ you can integrate $\alpha$ or $\beta$ over the way you integrate $Y_l^{m_l}(\theta,\phi)$ over solid angle — the analogy is only algebraic, and that is the essential subtlety of the entire topic. The historical motivation is the Stern–Gerlach experiment. A beam of silver atoms — which have a single unpaired electron in an outer $5s$ orbital and thus zero net orbital angular momentum in the ground state — is passed through a strongly inhomogeneous magnetic field. A field gradient is needed because a uniform field only exerts a torque on a magnetic dipole, not a net force; the force on a dipole $\boldsymbol\mu$ requires $\mathbf{F} = \nabla(\boldsymbol\mu\cdot\mathbf{B})$, which is nonzero only where $\mathbf{B}$ varies in space. Classically, if the silver atom carried some continuously-orientable magnetic moment, you would expect the beam to fan out into a continuous vertical smear on the detector screen, corresponding to every possible orientation of the moment relative to the field. Instead, the beam splits cleanly into two spots, symmetric about the undeflected position. Two, not a continuum, and two, not three (as would follow if the deflection came from some integer orbital quantum number with $2l+1$ states, e.g. $2l+1 = 1$ for $l=0$ giving no splitting at all). This is the experimental signature of $s = \tfrac12$: a two-valued, quantized magnetic moment attached to the electron itself, entirely independent of its orbital motion.

The magnetic moment responsible for the deflection is related to the spin angular momentum by

where $g_e \approx 2.0023$ is the electron's dimensionless g-factor — itself a striking number, because the analogous relation for orbital angular momentum has $g_l = 1$ exactly. The fact that spin's gyromagnetic ratio is (almost) double the orbital value was, historically, another clue that spin could not be pictured as ordinary orbital motion of charge; it took Dirac's relativistic theory to derive $g_e = 2$ from first principles (with quantum electrodynamics later refining it to $2.00231930\ldots$). A second essential picture is the vector model of spin, exactly parallel to the vector model used for orbital angular momentum. Because $m_s = \pm\tfrac12$ while the total spin magnitude is $\sqrt{3}/2\,\hbar \approx 0.866\,\hbar$, the spin vector can never point exactly along $z$: it is tipped at an angle

for the $\alpha$ state (and the supplementary angle for $\beta$), and it precesses about the $z$-axis because $\hat S_x$ and $\hat S_y$ are not simultaneously well-defined with $\hat S_z$ — precisely as for orbital angular momentum, discussed under angular momentum operators.

The formal structure of spin is obtained by direct analogy with orbital angular momentum, but every operator now acts on the abstract spin functions $\alpha$ and $\beta$ rather than on spatial functions of $\theta,\phi$.

where: $\hat S^2$ is the total spin angular momentum-squared operator, and $s=\tfrac12$ is the same for both spin states — this eigenvalue never changes, since $s$ is fixed for the electron. Reading: both spin states carry the identical total spin magnitude; they differ only in orientation.

where: $\hat S_z$ is the $z$-component of spin angular momentum, and $\pm\tfrac12\hbar$ are the two allowed eigenvalues, indexed by $m_s = \pm\tfrac12$. Reading: this is the direct spin analogue of $\hat L_z Y_l^{m_l} = m_l\hbar\, Y_l^{m_l}$ — the sharp quantum number is the projection, not the magnitude.

where: these are the fundamental commutation relations that define an angular momentum in quantum mechanics. Reading: spin satisfies exactly the same algebra as orbital angular momentum ($\hat L_x,\hat L_y,\hat L_z$) — this is why spin behaves like an angular momentum in every observable respect, even though it has no classical rotational origin.

where: the inner products here are defined abstractly over "spin space," not by integration over ordinary spatial coordinates. Reading: $\alpha$ and $\beta$ form a complete orthonormal basis for the two-dimensional spin space of a single electron — any spin state of one electron is some linear combination $c_1\alpha + c_2\beta$ with $|c_1|^2+|c_2|^2=1$.

where: $\psi_{nlm_l}(\mathbf r)$ is the ordinary spatial hydrogenic orbital built from $n$, $l$, $m_l$, and $\omega$ is a formal "spin coordinate" with no classical meaning, introduced purely so that integrals over spin can be written in the same Dirac-notation formalism as spatial integrals. Reading: a complete description of one electron in an atom requires four quantum numbers total — $n, l, m_l, m_s$ — and the spin-orbital is the fundamental building block from which Slater determinants (and hence the Pauli principle) are constructed.

Example 1 — Foundational: eigenvalues and the vector-model tilt angle for a single electron's spin. Consider one electron in an arbitrary atomic orbital, spin state $\beta$ ($m_s = -\tfrac12$). Find (a) the eigenvalue of $\hat S^2$ acting on $\beta$, (b) the eigenvalue of $\hat S_z$ acting on $\beta$, and (c) the angle the spin vector makes with the $+z$ axis in the vector model. (a) Using $\hat S^2 \chi = s(s+1)\hbar^2 \chi$ with $s = \tfrac12$ (fixed, regardless of $m_s$):

(b) Using $\hat S_z \chi = m_s \hbar\, \chi$ with $m_s = -\tfrac12$ for $\beta$:

(c) The tilt angle follows from $\cos\theta = m_s/\sqrt{s(s+1)}$:

The spin vector for $\beta$ points into the lower hemisphere, tilted $125.3^\circ$ from $+z$ — never fully aligned along $-z$, exactly mirroring the $54.7^\circ$ result for $\alpha$ found in the physical-picture section. This asymmetric (non-round) angle, rather than a "clean" $180^\circ$, is itself evidence that $S_z$ is only one component of a vector whose $x$- and $y$-components remain fundamentally uncertain.

Example 2 — Applied: population of spin states in a magnetic field via the Boltzmann distribution (basis of NMR/EPR signal). An electron placed in a magnetic field of strength $B = 1.20\ \text{T}$ (a typical value inside an EPR spectrometer's cavity) has its two spin states split in energy by the Zeeman interaction,

where $\mu_B = 9.274\times10^{-24}\ \text{J/T}$ is the Bohr magneton and $g_e = 2.0023$. Compute $\Delta E$ and the ratio of populations of the upper ($\beta$-like, higher energy) to lower ($\alpha$-like, lower energy) spin state at $T = 298\ \text{K}$. Step 1 — Compute the Zeeman splitting:

Step 2 — Convert to a frequency via $\Delta E = h\nu$, to see the microwave resonance condition used in EPR:

(consistent with Q-band EPR microwave frequencies). Step 3 — Apply the Boltzmann distribution at $T = 298\ \text{K}$, using $k_B T = (1.381\times10^{-23}\ \text{J/K})(298\ \text{K}) = 4.116\times10^{-21}\ \text{J}$:

Only about a 0.54% population excess in the lower spin state at room temperature — this tiny imbalance is precisely why magnetic resonance signals (NMR, EPR) are intrinsically weak and require sensitive detection, and why lowering temperature or raising field strength (larger $\Delta E$) improves signal-to-noise. As $T\to\infty$, $N_\beta/N_\alpha \to 1$ (equal populations, no net magnetization, no signal) — the expected limiting behavior, confirming the formula.

Example 3 — Exam/competition level: constructing a properly antisymmetrized (Pauli-compliant) two-electron spin-orbital product and checking orthogonality. Two electrons occupy the same spatial $1s$ orbital of a helium-like ion, $\psi_{1s}(\mathbf r)$. Write the correctly antisymmetrized two-electron wavefunction consistent with the Pauli principle, and show explicitly that a naive symmetric product would (incorrectly) allow both electrons to have identical quantum numbers. Step 1 — A naive (wrong) attempt might write both electrons in the same spin-orbital:

Here both electrons share all four quantum numbers $n=1,l=0,m_l=0,m_s=+\tfrac12$ — this is forbidden. Moreover this wavefunction is symmetric under electron exchange ($1\leftrightarrow2$), whereas electrons (fermions) require overall antisymmetry. Step 2 — Build the antisymmetrized combination using a Slater determinant over the two spin-orbitals $\psi_{1s}\alpha$ and $\psi_{1s}\beta$:

Step 3 — Verify antisymmetry directly: swap labels $1 \leftrightarrow 2$ everywhere,

confirming the required antisymmetry. Step 4 — Verify normalization, assuming $\psi_{1s}$ is normalized ($\langle\psi_{1s}|\psi_{1s}\rangle = 1$) and $\alpha,\beta$ are orthonormal:

The cross terms vanish because they contain $\langle\alpha|\beta\rangle = 0$. This factor of $\tfrac{1}{2}(1+1)=1$ confirms the $1/\sqrt2$ prefactor was chosen correctly — this is the general normalization for a two-electron Slater determinant with no other overlaps, and generalizes to the $1/\sqrt{N!}$ prefactor for $N$-electron determinants. The resulting state has one electron effectively "$\alpha$" and one "$\beta$" in a way that respects indistinguishability — precisely the ground-state helium wavefunction (spatial part times singlet spin function), and the direct machinery behind the Pauli Exclusion Principle's textbook statement that the $1s$ orbital holds at most two electrons, of opposite spin.