Angular Momentum and Spin
Rotational invariance is the most productive symmetry in quantum mechanics, and this chapter extracts everything it gives. The generators of rotations satisfy \(\comm{J_{i}}{J_{j}}=\ii\hbar\varepsilon_{ijk}J_{k}\), and that single commutator—taken alone, with no reference to any particular system—fixes the entire spectrum: eigenvalues \(\hbar^{2}j(j+1)\) and \(\hbar m\), multiplets of \(2j+1\) states, and half-integer \(j\) alongside integer \(j\). The derivation is the ladder-operator argument already rehearsed for the oscillator in Elementary Quantum Systems, carried out here in the abstract, so that the orbital angular momentum of an electron, its spin, and the total angular momentum of a nucleus are one subject and not three. The chapter then treats the orbital realisation by the spherical harmonics, the intrinsic realisation—spin—that no function on a sphere can supply, the rule for adding two angular momenta with its Clebsch–Gordan coefficients, and the Wigner–Eckart theorem, which reduces every matrix element of a rotationally covariant operator to one number times a purely geometric factor.
Spin was forced on physics by data, not deduced. The doublet structure of the alkali spectra and the anomalous Zeeman effect [Zeeman:1897] [Paschen:1912] defeated every model built from orbital motion alone; Landé's empirical \(g\) factor [Lande:1921] fitted the splittings with a half-integer quantum number; Pauli's “two-valuedness” [Pauli:1925] named the extra degree of freedom without interpreting it; and Uhlenbeck and Goudsmit [Uhlenbeck:1925] [Uhlenbeck:1926] identified it as an intrinsic angular momentum of the electron, at the price of a factor two that Thomas [Thomas:1926] removed by a relativistic kinematic argument. The cleanest measurement—a silver beam splitting into exactly two [Gerlach:1922a]—is Experiment: Stern–Gerlach, the experiment chapter that follows this one. Pauli's spin theory [Pauli:1927b] is the formalism used below; The Dirac Equation later obtains spin-\(1/2\) from relativistic invariance rather than postulating it, and Identical Particles shows that the same half-integrality decides the statistics [Pauli:1940]. Standard treatments are [Sakurai:2017] [CohenTannoudji:1977] [Condon:1935].
Angular Momentum and Spin: all derivations of this chapter are pending.
The angular-momentum algebra
Rotations and their generators
[Reserved: rotations as a symmetry of the dynamics; the unitary representation \(U(R)=\ee^{-\ii\theta\,\vect{n}\cdot\vect{J}/\hbar}\) guaranteed by Wigner's theorem on symmetry operations [Bargmann:1964] (The Postulates of Quantum Mechanics); the commutator \(\comm{J_{i}}{J_{j}}=\ii\hbar\varepsilon_{ijk}J_{k}\) read off from the composition law of \(\SO(3)\) [Wigner:1931]; \(\vect{J}\) as the conserved charge of rotational invariance, the quantum counterpart of the classical theorem of Lagrangian Mechanics; the Casimir \(\vect{J}^{2}\) and the commuting pair \(\set{\vect{J}^{2},J_{z}}\) as the maximal set of simultaneously measurable rotational observables. Lie-algebraic setting: Lie Groups, Lie Algebras, and Fibre Bundles.]
The spectrum from the algebra alone
[Reserved: the raising and lowering operators \(J_{\pm}=J_{x}\pm\ii J_{y}\) with \(\comm{J_{z}}{J_{\pm}}=\pm\hbar J_{\pm}\); positivity of \(\vect{J}^{2}-J_{z}^{2}\) terminating the ladder at both ends; the resulting eigenvalues \(\hbar^{2}j(j+1)\) with \(j\in\set{0,\tfrac{1}{2},1,\tfrac{3}{2},\ldots}\) and \(m=-j,\ldots,j\); the matrix elements \(J_{\pm}\ket{j,m}=\hbar\sqrt{j(j+1)-m(m\pm1)}\ket{j,m\pm1}\). The algebraic derivation is that of matrix mechanics [Born:1926a], recast in the operator language of [Dirac:1930b]; it is the same argument that fixed the oscillator spectrum in Elementary Quantum Systems, and its conclusion—that half-integer \(j\) is allowed—is the opening this chapter needs for spin.]
Representation matrices and Euler angles
[Reserved: the \((2j+1)\)-dimensional irreducible representations \(D^{(j)}_{m'm}(\alpha,\beta,\gamma)\), Wigner's small \(d\) matrices and their construction [Wigner:1931]; unitarity, orthogonality and the composition law; the character \(\chi^{(j)}(\theta)=\sin\!\left[(j+\tfrac{1}{2})\theta\right]/ \sin(\theta/2)\) used below to decompose products; the Condon–Shortley phase convention [Condon:1935], which is what makes published tables of coefficients comparable at all, and which this treatise adopts.]
$\SU(2)$, $\SO(3)$ and the double cover
[Reserved: \(\SU(2)\) as the simply connected group with the same algebra as \(\SO(3)\) and the two-to-one homomorphism onto it; integer \(j\) as the representations of \(\SO(3)\) proper and half-integer \(j\) as the genuinely double-valued ones, so that a \(2\pi\) rotation returns a fermionic state to minus itself; the topological reason (\(\pi_{1}(\SO(3))=\Z_{2}\)), developed in Topological and Metric Spaces and Lie Groups, Lie Algebras, and Fibre Bundles. The sign is observable: neutron interferometry shows the interference pattern restored only after \(4\pi\) [Rauch:1975] [Werner:1975]. The relativistic completion, where the same double cover appears as \(\Spin(1,3)\), is Particles as Poincaré Representations.]
Split a neutron beam coherently, precess the spin of one branch through an angle \(\chi\) by sending it through a magnetic field, and recombine the two branches. The interference pattern at the exit does not repeat with period \(2\pi\) in \(\chi\): it repeats with period \(4\pi\), and at \(\chi=2\pi\) — one full turn of the spin — it is inverted, maxima standing where minima stood. Rauch and collaborators in Vienna and Werner and collaborators in Michigan measured the period independently, in neutron interferometers cut from single silicon crystals, and both found \(4\pi\) [Rauch:1975] [Werner:1975]. A rotation by \(2\pi\) is the identity on every classical object and is not the identity on a neutron.
Derivation. A rotation by \(\chi\) about the unit vector \(\vect{n}\) acts on a spin-\(\tfrac{1}{2}\) state as \(U=\ee^{-\ii\chi\,\vect{n}\cdot\vect{\sigma}/2}\). From the algebra \(\sigma_{i}\sigma_{j}=\delta_{ij}\identity +\ii\varepsilon_{ijk}\sigma_{k}\) one gets \(\left(\vect{n}\cdot\vect{\sigma}\right)^{2}=\identity\), so the exponential series splits into even and odd parts exactly as for a scalar:
Hence \(U(2\pi)=-\identity\) and \(U(4\pi)=+\identity\): the state, not merely the direction of the spin, changes sign after one turn and is restored only after two.
That sign is observable because the interferometer compares the rotated branch with an unrotated one. If each branch carries amplitude \(A\) and the geometry contributes a relative phase \(\delta\), the recombined intensity for an incident beam of spin state \(\ket{s}\) is
For an unpolarized beam the spin factor is the average over any orthonormal pair, that is \(\tfrac{1}{2}\tr U\), and since \(\tr\identity=2\) while \(\tr\sigma_{i}=0\), Equation (79.1) gives \(\tfrac{1}{2}\tr U=\cos(\chi/2)\) and
The fringe contrast is periodic in \(\chi\) with period \(4\pi\) and reverses sign at \(\chi=2\pi\), as observed. The same double-valuedness is why the rotation group of space has a double cover: half-integer \(j\) represents \(\SU(2)\) faithfully and \(\SO(3)\) only up to a sign.
∎Orbital angular momentum
Spherical harmonics
[Reserved: \(\vect{L}=\vect{r}\times\vect{p}\) and its differential form on the sphere; single-valuedness of a wavefunction of position restricting \(\ell\) to integers; the spherical harmonics \(Y_{\ell m}(\theta,\varphi)\) as the simultaneous eigenfunctions of \(\vect{L}^{2}\) and \(L_{z}\), their orthonormality, completeness and parity \((-1)^{\ell}\); the associated Legendre functions and the addition theorem; the connection to the Laplace equation and to Sturm–Liouville theory Ordinary Differential Equations and Sturm–Liouville Theory. Conventions and tables follow [Condon:1935]; standard modern treatments [Sakurai:2017] [CohenTannoudji:1977].]
Central potentials and the radial reduction
[Reserved: separation of the Schrödinger equation in a central potential, the effective radial equation with the centrifugal term \(\hbar^{2}\ell(\ell+1)/2mr^{2}\), and the \((2\ell+1)\)-fold degeneracy that rotational invariance guarantees for every central problem; why that degeneracy is exactly what an external field lifts (Approximation Methods); the worked solutions—free particle in spherical waves, spherical well, isotropic oscillator—and the Coulomb case, whose extra degeneracy is the subject of The Hydrogen Atom.]
Rotational spectra of molecules
[Reserved: the rigid rotor, energies \(E_{J}=\hbar^{2}J(J+1)/2I\) and the equally spaced \(\Delta J=\pm1\) lines in the far infrared and microwave, the direct spectroscopic reading of the \(J(J+1)\) law and hence of the algebra of Section 79.1.2 [Herzberg:1950]; moments of inertia and bond lengths obtained from line spacings; centrifugal distortion and rotation–vibration coupling; nuclear-spin statistics of homonuclear molecules, where ortho and para intensity alternation is a direct measurement of the spin–statistics connection (Identical Particles); the classical rotor is Rigid Bodies and Rotating Frames, the molecular structure Atoms and Molecules.]
The pure rotational spectrum of a heteronuclear diatomic molecule, observed in the far infrared and the microwave, is a series of very nearly equally spaced lines. Two things about it are sharp. The spacing is constant, to a slow correction that grows with the rotational quantum number and is the centrifugal stretching of the bond; and the series extrapolates to zero at zero, so that the first line stands one full spacing above the origin rather than half a spacing [Herzberg:1950]. For carbon monoxide the first line lies near \(115\,\mathrm{GHz}\) and the following ones near two and three times that frequency. The rotational energies therefore go as \(J(J+1)\) and not as \(J^{2}\): the spectrum reads the eigenvalue of \(\vect{J}^{2}\) off the sky.
Derivation. A rigid rotor has \(\Ham=\vect{J}^{2}/2I\) with \(I\) its moment of inertia, so by the multiplet argument used for Phenomenon 79.3 its levels are \(E_{J}=\hbar^{2}J(J+1)/2I\) with \(J\) a non-negative integer, integral rather than half-integral because the rotor's angular momentum is orbital (Section 79.2.1). Electric-dipole radiation carries one unit of angular momentum, so \(\Delta J=\pm1\) (Section 79.2.4), and absorption from \(J\) to \(J+1\) occurs at
that is \(\nu_{J}=\nu_{0}\left(J+1\right)\) with \(\nu_{0}=\hbar/2\pi I\). The lines are equally spaced by \(\nu_{0}\) and the first of them lies at \(\nu_{0}\), one whole spacing from the origin, as observed.
The alternative worth excluding is the naive \(E_{J}\propto J^{2}\), which a reading of \(\vect{J}^{2}\) as the square of a vector of length \(\hbar J\) would give. It predicts \(E_{J+1}-E_{J}\propto2J+1\): lines in the ratios \(1:3:5:\ldots\), still equally spaced but with the first line at half the spacing. The two laws are distinguished by one measurement of the lowest line against the spacing, and the measurement selects \(J(J+1)\). Since \(\nu_{0}\) fixes \(I\), the same spectrum then returns the bond length; and centrifugal stretching enters only as a correction of relative order \(\left(\hbar J/I\omega_{\mathrm{vib}}\right)^{2}\), which is why the leading pattern survives it.
∎Parity and selection rules
[Reserved: parity as a discrete symmetry commuting with rotations, the parity of \(Y_{\ell m}\), and the resulting electric-dipole selection rules \(\Delta\ell=\pm1\), \(\Delta m=0,\pm1\) with Laporte's rule as their empirical form [Condon:1935]; the intensity consequences observed in every atomic spectrum (Atomic Models and Spectra); higher multipoles and the metastable states that survive because the dipole matrix element vanishes—the \(2s\) level of hydrogen and the forbidden nebular lines. The general statement is the Wigner–Eckart theorem of Section 79.5.2; parity as a symmetry in its own right, and its violation by the weak interaction, is Discrete Symmetries and CPT.]
Spin
The evidence for an intrinsic angular momentum
[Reserved: the four independent lines of evidence, each with the measurement that carries it—alkali doublets and the anomalous Zeeman effect [Zeeman:1897] [Paschen:1912] with Landé's \(g\) factor [Lande:1921]; Pauli's exclusion principle and its fourth quantum number [Pauli:1925]; the Stern–Gerlach splitting of a ground-state silver beam into exactly two components [Stern:1921] [Gerlach:1922a] [Gerlach:1922b], treated in full in Experiment: Stern–Gerlach; and the gyromagnetic ratio measured mechanically by magnetization on rotation [Barnett:1915]. The proposal itself [Uhlenbeck:1925] [Uhlenbeck:1926], and why a literal spinning sphere is excluded by the equatorial speed it would require.]
A collimated beam of ground-state silver atoms crossing a strongly inhomogeneous magnetic field emerges neither as one broadened trace, which is what a classical moment of arbitrary orientation would give, nor as an odd number of traces, which is what any orbital angular momentum would give, but as exactly two, placed symmetrically about the undeflected position [Stern:1921] [Gerlach:1922a]. The size of the deflection gives a moment of one Bohr magneton [Gerlach:1922b]. The full apparatus, its systematics and its sequential versions are Experiment: Stern–Gerlach; what is used here is the bare count. An even number of components cannot come from orbital motion, so the silver atom carries an angular momentum that is not orbital and whose projection quantum number is half-integral.
Derivation. An atom of magnetic moment \(\vect{\mu}\) in a field \(\vect{B}\) has energy \(-\vect{\mu}\cdot\vect{B}\) and therefore feels the force \(\vect{F}=\nabla\left(\vect{\mu}\cdot\vect{B}\right)\); with the field along \(z\) and varying along \(z\), this is \(F_{z}=\mu_{z}\,\pp B_{z}/\pp z\). The beam is split into one trace for each distinct value of \(\mu_{z}\), so counting traces counts the spectrum of \(\mu_{z}\), and since \(\vect{\mu}\) is proportional to \(\vect{J}\), the spectrum of \(J_{z}\).
That spectrum follows from the commutator \(\comm{J_{i}}{J_{j}}=\ii\hbar\varepsilon_{ijk}J_{k}\) and nothing else. Let \(\ket{\lambda,m}\) satisfy \(\vect{J}^{2}\ket{\lambda,m}=\hbar^{2}\lambda\ket{\lambda,m}\) and \(J_{z}\ket{\lambda,m}=\hbar m\ket{\lambda,m}\), and put \(J_{\pm}=J_{x}\pm\ii J_{y}\). Then \(\comm{J_{z}}{J_{\pm}}=\pm\hbar J_{\pm}\), so \(J_{\pm}\) carries an eigenvector of \(J_{z}\) to one with \(m\) raised or lowered by one, while \(\comm{\vect{J}^{2}}{J_{\pm}}=0\) leaves \(\lambda\) alone. From \(J_{\mp}J_{\pm}=\vect{J}^{2}-J_{z}^{2}\mp\hbar J_{z}\),
so \(m\) is bounded above and below and both ladders must terminate. Call the largest value \(j\) and the smallest \(j'\). Termination means \(J_{+}\ket{\lambda,j}=0\) and \(J_{-}\ket{\lambda,j'}=0\), which by Equation (79.4) give \(\lambda=j(j+1)\) and \(\lambda=j'(j'-1)\); eliminating \(\lambda\) leaves \(j'=-j\), the root \(j'=j+1\) being excluded by \(j'\leq j\). The two ends are joined by a whole number of unit steps, so \(2j\) is a non-negative integer and the multiplet contains
Two traces therefore mean \(2j+1=2\), that is \(j=\tfrac{1}{2}\). This is the point at which the observation forces the theory's hand: Equation (79.5) permits half-integer \(j\) on purely algebraic grounds, no function of position on a sphere realizes it (Section 79.2.1), and the silver beam shows Nature using it anyway.
∎Pauli matrices and spinors
[Reserved: the \(j=\tfrac{1}{2}\) representation written out, \(\vect{S}=\tfrac{\hbar}{2}\vect{\sigma}\), with the algebra \(\sigma_{i}\sigma_{j}=\delta_{ij}\identity+ \ii\varepsilon_{ijk}\sigma_{k}\), tracelessness, and completeness of \(\set{\identity,\sigma_{i}}\) on \(2\times2\) matrices; two-component spinors, the Pauli equation with its \(g=2\) magnetic coupling [Pauli:1927b]; the Bloch-sphere parametrization of a pure spin-\(1/2\) state and its density matrix (The Postulates of Quantum Mechanics); why the mean of \(\vect{\sigma}\) behaves classically while its components never commute. Reobtained from the Dirac equation in The Dirac Equation.]
Magnetic moments and the $g$ factor
[Reserved: the Bohr magneton \(\mu_{\mathrm{B}}=e\hbar/2m_{\mathrm{e}}\) and the nuclear magneton smaller by the mass ratio; \(\vect{\mu}=g\,\mu_{\mathrm{B}}\vect{J}/ \hbar\) with \(g_{\ell}=1\) for orbital motion and \(g_{s}\approx2\) for spin [Pauli:1927b]; the anomaly \(g_{s}-2\), first measured by Kusch and Foley from the ratio of atomic \(g\) values [Kusch:1948], now the most precisely tested prediction in physics (Experiment: The Electron Anomalous Magnetic Moment, with the radiative theory in Quantum Electrodynamics and Renormalization); nuclear moments in nuclear magnetons (Nuclear Forces and Nuclear Structure); the magnetism of matter as the bulk consequence (Magnetism in Matter).]
In a weak magnetic field a spectral line of an atom splits into several components, equally spaced in frequency [Zeeman:1897]. The pattern that a purely orbital moment predicts — a symmetric triplet whose splitting unit is \(\mu_{\mathrm{B}}B/h\) and is the same for every line of every atom — is the exception, not the rule. Most lines split into more components than three, with a unit peculiar to the levels involved and in general not an integer multiple of \(\mu_{\mathrm{B}}B/h\); the sodium D lines, the classic case, split into four and six [Paschen:1912]. Landé found that the whole body of data is reproduced by attaching to each level a single number \(g\), computable from its three quantum numbers, and that fitting the observed patterns requires those quantum numbers to take half-integer values [Lande:1921].
Landé's \(g\) factor from the projection theorem: that the matrix elements of any vector operator inside one multiplet are proportional to those of the total angular momentum, so that the observed splitting unit becomes a closed formula in the three quantum numbers of the level, and that fitting it to the data requires the electron's own gyromagnetic factor to be twice the orbital one
Thomas precession
[Reserved: the spin–orbit coupling computed naively in the electron rest frame is twice the observed one; Thomas' resolution [Thomas:1926]—the rest frame is not inertial, and the composition of two non-collinear boosts leaves a rotation, so the spin precesses at \(\vect{\omega}_{\mathrm{T}}\) and halves the coupling in the low-velocity limit. The kinematics is that of Minkowski Space and Its Symmetries, where the Wigner rotation is derived; the numerical consequence is the observed fine structure of The Hydrogen Atom, and the exact relativistic statement is the Bargmann–Michel–Telegdi equation for the polarization of a particle in an external field [Bargmann:1959], which is what the \(g-2\) storage rings of Experiment: The Electron Anomalous Magnetic Moment measure against. This is the worked case where a factor two decided between a right and a wrong theory.]
Spin beyond one half
[Reserved: spin-\(1\) and the photon, whose two observed polarizations rather than three follow from masslessness (Particles as Poincaré Representations) and whose angular momentum was measured mechanically as a torque on a birefringent plate [Beth:1936]; spin-\(1\) massive bosons (Electroweak Unification and the Higgs Boson); spin-\(3/2\) and spin-\(2\) states in nuclei and in the hadron spectrum [Navas:2024]; the spin-\(2\) graviton polarizations of Gravitational-Wave Theory. What the evidence base does and does not contain about higher spins is Higher-Spin Wave Equations.]
A half-wave plate hung on a fine quartz fibre and traversed along its axis by circularly polarized light feels a steady torque about that axis, and the torque reverses when the handedness of the light is reversed. Beth built the apparatus, with the beam passed twice through the suspended plate to double the effect and a resonant detection scheme to pull the deflection out of the Brownian motion of the fibre, and measured a torque per unit incident power equal to \(2/\omega\) [Beth:1936]. Angular momentum is carried by a beam of light independently of any orbital motion of anything, and the amount per photon is one unit of \(\hbar\).
Derivation. A half-wave plate converts circularly polarized light of one handedness into the other and transmits it. Suppose each photon of angular frequency \(\omega\) carries a spin projection \(s\hbar\) along its direction of propagation. Each transmitted photon has that projection reversed, so it gives up \(2s\hbar\) to the plate, and a beam of power \(P\) delivers \(P/\hbar\omega\) photons per second. The torque is therefore
Two things are worth noting about Equation (79.6). First, \(\hbar\) cancels: the predicted torque is a classical quantity, which is why a torsion fibre can measure it, and the experiment therefore determines the integer \(s\) without needing to know \(\hbar\). Second, the measured value \(\tau=2P/\omega\) gives \(s=1\). The electromagnetic field carries spin one, a fact read here off a suspended crystal and recovered from the transformation properties of the field in Particles as Poincaré Representations; that the photon nevertheless shows two polarization states rather than the three of a massive spin-\(1\) particle is a consequence of its masslessness, established there.
∎Addition of angular momenta
The coupling problem and the Clebsch–Gordan series
[Reserved: two commuting angular momenta \(\vect{J}_{1}\), \(\vect{J}_{2}\) and their sum; the uncoupled basis \(\ket{j_{1}m_{1}}\ket{j_{2}m_{2}}\) against the coupled basis \(\ket{j_{1}j_{2};JM}\); the decomposition \(j_{1}\otimes j_{2}=\abs{j_{1}-j_{2}}\oplus\cdots\oplus(j_{1}+j_{2})\), proved by counting states or by multiplying characters [Wigner:1931]; dimensional check \((2j_{1}+1)(2j_{2}+1)=\sum_{J}(2J+1)\); why the coupled basis is the useful one whenever the interaction is rotationally invariant but the individual momenta are not separately conserved—spin–orbit coupling, \(LS\) and \(jj\) coupling schemes in atoms (Atoms and Molecules).]
Clebsch–Gordan coefficients
[Reserved: the coefficients \(\braket{j_{1}m_{1}j_{2}m_{2}}{JM}\) as the unitary matrix between the two bases; their construction by repeated application of \(J_{-}\) to the stretched state, the selection rules \(M=m_{1}+m_{2}\) and the triangle inequality, reality in the Condon–Shortley convention, orthogonality and recursion relations [Wigner:1931] [Condon:1935]; the closed Racah form of the general coefficient [Racah:1942]; the tabulated special cases \(\tfrac{1}{2}\otimes\tfrac{1}{2}\) and \(\ell\otimes\tfrac{1}{2}\) that every atomic calculation reuses.]
Two spins one half: singlet and triplet
[Reserved: the explicit decomposition \(\tfrac{1}{2}\otimes\tfrac{1}{2}=0\oplus1\), the antisymmetric singlet and the symmetric triplet, and the exchange operator written as \(\tfrac{1}{2}(\identity+\vect{\sigma}_{1}\cdot\vect{\sigma}_{2})\); the singlet–triplet splitting observed as ortho- and para-helium (Atoms and Molecules) and as the hydrogen hyperfine splitting of The Hydrogen Atom; the exchange interaction behind ferromagnetic order (Magnetism in Matter); and the singlet as the state whose correlations violate the Bell inequalities (Entanglement and Bell Tests and Experiment: Bell Tests) [Bell:1964].]
$3j$, $6j$ and $9j$ symbols
[Reserved: Wigner's \(3j\) symbol as the symmetrized Clebsch–Gordan coefficient, with the full permutation symmetry that the coefficient hides [Wigner:1931]; the recoupling of three angular momenta and the \(6j\) symbol relating the two coupling orders, of four and the \(9j\) symbol [Racah:1942] [Racah:1943]; sum rules and orthogonality relations; why this apparatus is not bookkeeping but the practical content of atomic and nuclear structure calculations (Atoms and Molecules and Nuclear Forces and Nuclear Structure), and the same recoupling algebra reappears for isospin.]
Tensor operators and the Wigner–Eckart theorem
Spherical tensor operators
[Reserved: the definition of an irreducible tensor operator \(T^{(k)}_{q}\) by its commutators with \(\vect{J}\), equivalently by its transformation under \(U(R)\) with the matrices \(D^{(k)}\); scalars, vectors and rank-two tensors as the cases \(k=0,1,2\); the decomposition of a Cartesian tensor into irreducible parts—trace, antisymmetric part, symmetric traceless part—and its physical reading as monopole, vector and quadrupole [Racah:1942] [Condon:1935]; products of tensor operators coupled with Clebsch–Gordan coefficients.]
The Wigner–Eckart theorem
[Reserved: the theorem \(\bra{j'm'}T^{(k)}_{q}\ket{jm}= \braket{jm\,kq}{j'm'}\,\bra{j'}\!\left\|T^{(k)}\right\|\!\ket{j}/ \sqrt{2j+1}\), separating a geometric factor fixed entirely by rotational invariance from one reduced matrix element carrying all the dynamics, stated by Eckart [Eckart:1930] and by Wigner [Wigner:1931]; the immediate corollaries—selection rules \(\abs{j-k}\le j'\le j+k\) and \(m'=m+q\), the proportionality of the matrix elements of any vector operator within a multiplet to those of \(\vect{J}\) (the projection theorem), and hence the derivation of Landé's empirical \(g\) factor [Lande:1921] from first principles.]
What the theorem buys
[Reserved: worked consequences, each attached to a measurement— intensity ratios within a spectral multiplet, fixed by geometry alone and independent of the atomic model [Condon:1935]; the vanishing of a static electric dipole moment for a state of definite parity, and the searches for the electron electric dipole moment as tests of Discrete Symmetries and CPT; nuclear electric quadrupole moments requiring \(I\ge1\), so that a spin-\(1/2\) nucleus has none (Nuclear Forces and Nuclear Structure); multipole expansions of radiation and their selection rules (Radiation and Scattering of Electromagnetic Waves).]
Spin dynamics and magnetic resonance
Larmor precession
[Reserved: the Hamiltonian \(-\vect{\mu}\cdot\vect{B}\), the exact solution for a spin in a static field, and the precession of \(\avg{\vect{S}}\) at \(\omega_{\mathrm{L}}=g\mu_{\mathrm{B}}B/\hbar\); Ehrenfest's theorem (Elementary Quantum Systems) making the quantum mean obey the classical torque equation exactly for this system, one of the few cases where it does so with no correction whatever; the Zeeman splitting as the energy form of the same statement [Zeeman:1897]; the observed precession of a free neutron's spin measured interferometrically [Werner:1975]; the muon precession that the \(g-2\) experiments of Experiment: The Electron Anomalous Magnetic Moment count.]
A sample of matter, or a beam of atoms or molecules, placed in a static magnetic field \(B_{0}\) and irradiated with a weak transverse field oscillating at frequency \(\nu\), exchanges energy with the radiation only in a narrow band of frequencies, centred on a value strictly proportional to \(B_{0}\) and independent of the intensity of the radiation. Rabi's molecular beams turned the resonance into the first precise measurements of nuclear magnetic moments [Rabi:1938]; Purcell, Torrey and Pound found the same absorption in a block of paraffin and Bloch, Hansen and Packard found it by nuclear induction in water, both in 1946 [Purcell:1946a] [Bloch:1946]. The proportionality constant is a property of the species and of nothing else, which is what makes the resonance an instrument.
Derivation. Write the moment as \(\vect{\mu}=\gamma\vect{S}\), so that the Hamiltonian in the static field is \(\Ham=-\gamma\,\vect{S}\cdot\vect{B}\). For any observable the Ehrenfest relation of Elementary Quantum Systems gives \(\dd\avg{A}/\dd t=(\ii/\hbar)\avg{\comm{\Ham}{A}}\), and here the commutator can be evaluated in closed form from the algebra alone:
so that, with \(\varepsilon_{jik}B_{j}S_{k} =-\left(\vect{B}\times\vect{S}\right)_{i}\),
Equation (79.7) is exact — no expansion, no truncation — and it is the classical equation for a top of angular momentum \(\avg{\vect{S}}\) carrying the moment \(\gamma\avg{\vect{S}}\). Its solution is a precession about \(\vect{B}\) at the angular frequency
with the component along \(\vect{B}\) constant. A transverse driving field is therefore in step with the moment through every cycle when, and only when, it turns at \(\omega_{\mathrm{L}}\), which is the observed resonance.
The energy form of the same statement is the sharper one. The eigenvalues of \(\Ham\) are \(-\gamma\hbar mB_{0}\) with \(m=-j,\ldots,j\) by Equation (79.5), so adjacent levels are separated by \(\hbar\abs{\gamma}B_{0}\) and a photon is absorbed when \(h\nu=\hbar\abs{\gamma}B_{0}\), that is when \(2\pi\nu=\omega_{\mathrm{L}}\). Because the levels are equally spaced the whole multiplet resonates at one frequency, which is why the absorption line is single and why measuring it measures \(\gamma\) alone; the observed resonance frequency divided by \(B_{0}\) is the gyromagnetic ratio, with no model of the nucleus intervening.
∎Rabi's resonance method
[Reserved: a rotating transverse field added to a static one; the exact two-level solution and the Rabi formula for the transition probability, with its resonance at the Larmor frequency and its generalized frequency off resonance [Rabi:1937]; the molecular-beam magnetic-resonance method and the first precise nuclear moments [Rabi:1938]; Ramsey's method of separated oscillatory fields [Ramsey:1950] as the narrowing that made the caesium frequency standard possible, feeding the SI definition of the second (Measurement, SI Units, and the Theory of Errors); the perturbative version of the same result appears as the golden rule in Approximation Methods, and the exact solution here is the check on it.]
Nuclear magnetic resonance
[Reserved: resonance in bulk matter rather than a beam, detected by absorption [Purcell:1946a] and by induction [Bloch:1946]; relaxation times \(T_{1}\) and \(T_{2}\) and the Bloch equations, the open-system description of which is Open Quantum Systems and Decoherence; the chemical shift and its use in molecular structure (Atoms and Molecules); spatial encoding by field gradients and the resulting imaging [Lauterbur:1973]—an angular-momentum commutator turned into a clinical instrument; electron spin resonance as the analogue at the electron magneton.]
Spin as the elementary two-state system
[Reserved: spin-\(1/2\) as the minimal quantum system and the template for every two-level problem—ammonia inversion, neutral-kaon and neutrino flavour oscillation (Flavour Physics and Neutrinos and Experiment: CP Violation), the qubit; observables that do not commute, so no state is dispersion-free in all three components, and the resulting spin-projection statistics that the Stern–Gerlach chapter measures directly (Experiment: Stern–Gerlach); the singlet correlations of Entanglement and Bell Tests as the sharpest statement that these statistics are not classical [Bell:1964].]