Approximation Methods

The list of quantum systems that can be solved exactly is short: the free particle, the square wells, the oscillator and the periodic chain of Elementary Quantum Systems, the rigid rotor of Angular Momentum and Spin, and the Coulomb problem of The Hydrogen Atom. Everything else in this treatise—helium, every molecule, every solid, every scattering cross-section, every decay rate—is reached by approximation, and the approximations are therefore not a technical appendix to quantum mechanics but the greater part of how it is compared with experiment. This chapter builds the four that carry that load: stationary perturbation theory, in the form Schrödinger gave it in his third communication [Schroedinger:1926c] on the model of Rayleigh's treatment of vibrating systems; the variational method [Ritz:1909], which bounds rather than expands; the semiclassical WKB approximation [Wentzel:1926] [Kramers:1926] [Brillouin:1926], which is the \(\hbar\to0\) limit of Hamilton–Jacobi Theory and the Optical–Mechanical Analogy made systematic; and time-dependent perturbation theory [Schroedinger:1926d] [Dirac:1927], whose leading term is the transition-rate formula that Fermi later christened the golden rule [Fermi:1950].

Two habits of the chapter deserve stating in advance. First, every method is presented with its failure mode named: perturbation series in quantum mechanics are generically asymptotic and divergent [Bender:1969] [Dyson:1952], the WKB form breaks down exactly at the classical turning points, and a variational bound says nothing about how close it is. Second, each method is anchored to a measurement it actually explains—the Stark [Stark:1914] and Zeeman [Zeeman:1897] splittings, the helium ground-state energy [Hylleraas:1929], alpha-decay lifetimes, spectral-line intensities—so that the accuracy claimed can be checked rather than asserted. The chapter closes with the two opposite limits of a time-dependent Hamiltonian: the adiabatic theorem [Born:1928], whose geometric remainder is Berry's phase [Berry:1984], observed directly in polarized light carried along a helical fibre [Tomita:1986]; and the sudden approximation at the other extreme. Standard treatments are [Sakurai:2017] [CohenTannoudji:1977] [Landau:1977].

Stationary perturbation theory

The Rayleigh–Schrödinger series

[Reserved: the split \(H=H_{0}+\lambda V\) with \(H_{0}\) solved; the expansion of eigenvalue and eigenvector in powers of \(\lambda\); the first-order shift \(E^{(1)}_{n}=\ev{V}{n^{(0)}}\) and the second-order shift \(\sum_{m\neq n}\abs{\bra{m}V\ket{n}}^{2}/(E_{n}-E_{m})\), always negative for the ground state; normalization conventions and the intermediate-normalization form; the small parameter is the ratio of a matrix element to a level spacing, not \(\lambda\) itself, which is what makes near-degenerate levels the dangerous case [Schroedinger:1926c]. Rigorous statement in terms of analytic families of operators [Kato:1966], with the functional-analytic setting of Hilbert Spaces.]

Degenerate and near-degenerate levels

[Reserved: why the naive series diverges term by term when \(E_{n}=E_{m}\), and the resolution—diagonalize \(V\) within the degenerate subspace first; the secular determinant, the resulting good quantum numbers, and the identification of the symmetry that the perturbation leaves unbroken; complete and partial lifting of a degeneracy; the crossing rule that levels of the same symmetry repel rather than cross, with avoided crossings as the observable signature (their dynamical version is Section 82.6.3); the worked cases used later are the Stark and Zeeman effects of Section 82.2 [Schroedinger:1926c].]

Does the series converge? Usually not

[Reserved: the honest statement that a perturbation series is generally asymptotic rather than convergent—optimal truncation, and error of the order of the first omitted term; the anharmonic oscillator whose coefficients grow factorially, computed exactly by Bender and Wu [Bender:1969]; Dyson's argument that the quantum-electrodynamic series cannot converge, because the theory is unstable for negative coupling [Dyson:1952] (see Quantum Electrodynamics and Renormalization); Borel and Padé resummation; and the cases where honest convergence does hold, for which the analytic perturbation theory of [Kato:1966] gives criteria.]

Perturbation theory at work

The Stark effect

[Reserved: the observation of electric-field splitting of the Balmer lines by Stark [Stark:1914], independently by Lo Surdo [LoSurdo:1913]; the quadratic shift for a non-degenerate level and the atomic polarizability it defines; the linear Stark effect, peculiar to hydrogen because states of opposite parity are degenerate (The Hydrogen Atom), computed by Schrödinger in the paper that introduced the method [Schroedinger:1926c]; parabolic coordinates; and the honest caveat that in a uniform field no bound state strictly exists—the levels are long-lived resonances that eventually field-ionize, as Oppenheimer showed [Oppenheimer:1928].]

The Zeeman effect

[Reserved: the splitting of spectral lines in a magnetic field [Zeeman:1897]; the weak-field regime, where the spin–orbit coupling dominates, \(\vect{J}\) is the good quantum number and the splitting is set by the Landé factor [Lande:1921] derived from the projection theorem of Angular Momentum and Spin; the strong-field Paschen–Back regime, where \(\vect{L}\) and \(\vect{S}\) decouple [Paschen:1912]; the crossover as a two-by-two degenerate problem solved exactly; why the “anomalous” effect was anomalous only before spin. Applications: stellar magnetic fields measured by Zeeman splitting (Stellar Structure and Nucleosynthesis), and magnetometry.]

Fine structure, hyperfine structure and isotope shifts

[Reserved: the standard worked example of degenerate perturbation theory—the \(\alpha^{2}\) corrections to hydrogen (relativistic kinetic, spin–orbit with the Thomas factor [Thomas:1926], and Darwin) collected in The Hydrogen Atom and derived exactly in The Dirac Equation; the hyperfine contact term [Fermi:1930]; nuclear volume and mass shifts. Each is a perturbation whose smallness is set by a physical ratio—\(\alpha^{2}\), the magneton ratio, the nuclear radius over \(a_{0}\)—which is the model for how the parameter should be identified in general.]

The Hellmann–Feynman and virial theorems

[Reserved: \(\pp E/\pp\lambda=\ev{\pp H/\pp\lambda}{\psi}\) for a normalized eigenstate depending on a parameter [Feynman:1939], and its use in converting a spectrum into forces—the electrostatic theorem for molecular binding (Atoms and Molecules); the quantum virial theorem for a homogeneous potential and its check \(\avg{T}=-\tfrac{1}{2}\avg{V}\) in hydrogen; the two theorems as exact constraints that any approximate calculation can be tested against, which is what makes them worth stating in a chapter about approximations.]

The variational method

The Rayleigh–Ritz principle

[Reserved: the theorem that \(\ev{H}{\psi}\ge E_{0}\) for any normalized trial state, with equality only for the ground state, and its stationarity form—the Schrödinger equation is the Euler–Lagrange equation of the energy functional (Calculus of Variations); the systematic method of minimizing over a finite basis, reducing the problem to a generalized eigenvalue determinant [Ritz:1909]; the min-max characterization of the higher eigenvalues (Hilbert Spaces), which turns the same construction into bounds on excited states; the practical warning that a good energy does not imply a good wavefunction, because the error in \(E\) is second order in the error in \(\psi\).]

Helium

[Reserved: the two-electron atom, the system on which the old quantum theory failed outright (Atomic Models and Spectra); the crude one-parameter trial function with an effective nuclear charge screening the second electron; and Hylleraas' explicit inclusion of the interelectronic distance \(r_{12}\) in the trial function, which reproduced the measured ionization energy to spectroscopic accuracy [Hylleraas:1929]—the first quantitative proof that quantum mechanics governs many-electron atoms. Ortho- and para-helium as the singlet–triplet structure of Angular Momentum and Spin and Identical Particles.]

Self-consistent fields

[Reserved: Hartree's self-consistent field, in which each electron moves in the averaged potential of the others and the equations are iterated to consistency [Hartree:1928]; Fock's antisymmetrized version, whose exchange term is a direct consequence of the statistics of Identical Particles [Fock:1930]; the variational character of the whole construction, and Koopmans' theorem identifying an orbital energy with an ionization energy [Koopmans:1934]; what is left out—electron correlation—and where it is recovered (Atoms and Molecules); density-functional methods as the working tool of Electrons in Solids: Band Theory.]

The semiclassical approximation

The WKB expansion

[Reserved: the ansatz \(\psi=\ee^{\ii S/\hbar}\) with \(S=S_{0}+\hbar S_{1}+\cdots\), the leading order reproducing the Hamilton–Jacobi equation of Hamilton–Jacobi Theory and the Optical–Mechanical Analogy and the first order the amplitude \(1/\sqrt{p(x)}\) that conserves flux; the validity condition \(\abs{\dd\lambda/\dd x}\ll2\pi\) on the local de Broglie wavelength \(\lambda(x)=h/p(x)\), that is, a wavelength varying slowly on its own scale; the near-simultaneous papers of Wentzel [Wentzel:1926], Kramers [Kramers:1926] and Brillouin [Brillouin:1926], and the earlier purely mathematical treatment of the same equation by Jeffreys [Jeffreys:1925], which is why the method is also called JWKB.]

Turning points and connection formulae

[Reserved: the breakdown of the expansion where \(p(x)\to0\), since the assumed short wavelength diverges there; linearization of the potential about a turning point and the Airy-function solution (Ordinary Differential Equations and Sturm–Liouville Theory); the connection formulae joining oscillatory and exponential regions and the \(\pi/4\) phase they carry; the classically forbidden region and the exponentially small tail; and caustics as the same phenomenon in the optical language of Hamilton–Jacobi Theory and the Optical–Mechanical Analogy.]

Semiclassical quantization

[Reserved: two turning points and the quantization condition \(\oint p\,\dd x=(n+\tfrac{1}{2})h\), the Bohr–Sommerfeld rule [Sommerfeld:1916] corrected by the Maslov half-integer that the connection formulae supply; its exactness for the harmonic oscillator and its accuracy for high \(n\) as a form of the correspondence principle; the extension to non-separable systems and the general index counting [Keller:1958]; where semiclassical quantization fails altogether—classically chaotic systems, whose spectra are the subject of Nonlinear Dynamics and Chaos.]

Barrier penetration

[Reserved: the transmission exponential \(\exp\left(-\tfrac{2}{\hbar}\int\abs{p}\,\dd x\right)\) as the standard product of the method; the Gamow factor for the Coulomb barrier [Gamow:1928], which reproduces the empirical Geiger–Nuttall relation between alpha-decay half-life and energy over more than twenty orders of magnitude (Elementary Quantum Systems, Nuclear Forces and Nuclear Structure); field ionization of atoms [Oppenheimer:1928] and cold emission from metals; thermonuclear reaction rates as the same integral run inwards (Stellar Structure and Nucleosynthesis). The exponential sensitivity is what makes an approximate exponent adequate and an approximate prefactor irrelevant.]

Time-dependent perturbation theory

Variation of constants and the interaction picture

[Reserved: the expansion of the state in eigenstates of \(H_{0}\) with time-dependent coefficients, introduced for wave mechanics in Schrödinger's fourth communication [Schroedinger:1926d] and cast in the form used ever since by Dirac [Dirac:1927]; the interaction picture and the Dyson series [Dyson:1949], whose field-theoretic use is Quantum Electrodynamics and Renormalization; first-order amplitude as the Fourier transform of the perturbation at the Bohr frequency; unitarity and the limits of validity—the formulas below fail as soon as the depletion of the initial state matters.]

Fermi's golden rule

[Reserved: the transition rate \(\Gamma=(2\pi/\hbar)\abs{\bra{f}V\ket{i}}^{2}\rho(E_{f})\) into a continuum, with the \(\sin^{2}\) kernel becoming a delta function of energy at long times and the resulting rate constant in time—the origin of exponential decay. The result is Dirac's [Dirac:1927]; the name is Fermi's, whose Chicago lecture notes list it as the second of two “golden rules” [Fermi:1950]. The chapter says which is which rather than repeating the customary attribution of both to Fermi. Second-order form for transitions through virtual intermediate states; applications to beta decay (Nuclear Forces and Nuclear Structure), spontaneous and stimulated emission (Quantum Optics and the Photon), and scattering rates (Scattering Theory).]

Periodic perturbations and the two-level check

[Reserved: a harmonic perturbation, absorption and stimulated emission at \(\omega\approx\omega_{fi}\), the resonance denominators and the rotating-wave approximation; oscillator strengths and the Thomas–Reiche–Kuhn sum rule [Kuhn:1925], which fixes the total absorption of an atom, summed over lines, independently of its structure; the exactly solvable two-level system in a rotating field [Rabi:1937] used as the check on the perturbative answer, showing where first order is right and where Rabi flopping takes over (Angular Momentum and Spin, Quantum Optics and the Photon).]

The Born approximation

[Reserved: the golden rule applied to scattering by a potential; the first Born amplitude as the Fourier transform of the potential in the momentum transfer [Born:1926b], the Rutherford cross-section [Rutherford:1911] recovered from a Coulomb potential, and the form factor as the Fourier transform of a charge distribution—which is how nuclear and nucleon radii are measured (Nuclear Forces and Nuclear Structure and Experiment: Deep Inelastic Scattering). Validity at high energy and weak coupling, the second Born term, and the non-perturbative alternative—partial waves and phase shifts—in Scattering Theory.]

Adiabatic and sudden limits

The adiabatic theorem

[Reserved: Born and Fock's theorem [Born:1928]: a system in a non-degenerate eigenstate of a slowly varying Hamiltonian remains in the instantaneous eigenstate, with transition probability vanishing in the limit of infinitely slow change; the condition as a ratio of the rate of change to the square of the level spacing; the dynamical phase; the classical counterpart, adiabatic invariance of the action, in Hamilton–Jacobi Theory and the Optical–Mechanical Analogy; uses—adiabatic passage in spectroscopy, adiabatic cooling, and the slow ramps of Experiment: Bose–Einstein Condensation.]

Berry's geometric phase

[Reserved: the extra, non-integrable phase left over when the Hamiltonian is carried around a closed loop in parameter space—equal to the flux of a curvature through the loop and independent of how slowly the circuit is traversed, so it is geometric and not dynamical [Berry:1984]; the spin-\(1/2\) example giving half the solid angle subtended, and the degeneracy point acting as a monopole of the curvature; the optical precursor in the interference of polarized beams [Pancharatnam:1956]; the non-adiabatic generalization [Aharonov:1987]. Observed: rotation of the plane of polarization of light in a helically wound optical fibre, matching the predicted solid angle [Tomita:1986]. The fibre-bundle language is Lie Groups, Lie Algebras, and Fibre Bundles; the same curvature governs the quantized Hall conductance of Experiment: The Quantum Hall Effect.]

Landau–Zener transitions

[Reserved: the failure of adiabaticity at an avoided crossing, and the exact two-level transition probability \(\exp(-2\pi\gamma)\) for a linear sweep through it [Zener:1932], with \(\gamma\) the ratio of the squared gap to the sweep rate; the two limits—slow sweep following the adiabatic branch, fast sweep diabatically crossing it; applications to atomic and molecular collisions (Atoms and Molecules), to the Mikheyev–Smirnov–Wolfenstein matter resonance [Wolfenstein:1978] that carries solar neutrinos through the Sun's density gradient (Flavour Physics and Neutrinos and Experiment: Neutrino Oscillations), and to control sequences in two-level systems.]

The sudden approximation

[Reserved: the opposite extreme—a change of Hamiltonian fast compared with every relevant period, so that the state vector has no time to evolve and the transition probabilities are simply the overlaps \(\abs{\braket{f_{\text{new}}}{i_{\text{old}}}}^{2}\); the validity criterion in terms of the switching time against \(\hbar/\Delta E\) [Landau:1977]; the worked case of nuclear beta decay, where the sudden change of nuclear charge leaves the atomic electrons excited or ionized—shake-up and shake-off, measurable in the electron spectrum (Nuclear Forces and Nuclear Structure).]

Born–Oppenheimer separation

[Reserved: the adiabatic theorem applied to nuclei and electrons, with the expansion parameter \((m_{\mathrm{e}}/M)^{1/4}\); electronic energies computed at fixed nuclei become the potential surface on which the nuclei then move [Born:1927]; the resulting separation of electronic, vibrational and rotational scales that organizes every molecular spectrum (Atoms and Molecules, Angular Momentum and Spin); where the separation fails—conical intersections between surfaces, at which the geometric phase of Section 82.6.2 becomes observable in the vibrational spectrum.]