Perturbation Theory · aiwiki.page

Perturbation theory is a collection of methods in mathematics and physics for approximating problems that differ slightly from simpler problems whose solutions are known. It introduces a parameter measuring the difference and calculates successive corrections to the reference solution. Applications include algebraic equations, differential equations, dynamical systems, and quantum mechanics. Its usefulness depends on whether these corrections remain controlled in the regime being studied. (math.ucdavis.edu)

Basic construction

A typical calculation writes an equation as $F(x,\varepsilon)=0$ , with the unperturbed problem obtained at $\varepsilon=0$ . One seeks an expansion such as

x(\varepsilon)=x_0+\varepsilon x_1+\varepsilon^2x_2+\cdots .

Substitution into the original equation and comparison of equal powers of $\varepsilon$ determines the corrections successively. The leading equation may be nonlinear, while higher-order equations are often linear in the new unknown correction. The parameter is commonly dimensionless, obtained by comparing relevant scales. (math.ucdavis.edu)

An elementary illustration is $x^2+\varepsilon x-1=0$ . Expanding the root near $x_0=1$ gives

x=1-\frac{\varepsilon}{2}+\frac{\varepsilon^2}{8} +O(\varepsilon^3).

The big-O notation specifies the order of the omitted remainder as $\varepsilon$ approaches zero. Such a power series provides an approximation without requiring the complete exact expression; the same coefficient-matching procedure extends to more complicated equations. (math.ucdavis.edu)

Regular and singular perturbations

In a regular perturbation problem, the small-parameter limit preserves the essential structure of the problem. In a singular perturbation, that limit changes its character—for example, by removing the highest derivative from a differential equation. A limiting equation may then be unable to satisfy all the original boundary conditions. (people.maths.ox.ac.uk)

Solutions can develop thin boundary layers, where variation occurs on a much shorter spatial or temporal scale than elsewhere. The method of matched asymptotic expansions constructs separate inner and outer approximations and connects them in an overlap region. A combined approximation can describe both the layer and the surrounding domain. (people.maths.ox.ac.uk)

Oscillatory problems present another difficulty: a small frequency correction can accumulate into a substantial phase error over long times. Terms growing with time, called secular terms, reveal this nonuniformity. The method of multiple scales introduces distinct fast and slow variables, while the Poincaré–Lindstedt method adjusts the frequency of periodic solutions. Averaging methods provide another way to describe slow evolution. (people.maths.ox.ac.uk)

Time-independent quantum theory

For a stationary quantum system, the Hamiltonian operator is separated into

H=H_0+\lambda V,

where $H_0$ has known eigenvalues and eigenstates, and $V$ is the perturbation. Both the energy levels and states are expanded in powers of $\lambda$ . This is commonly called Rayleigh–Schrödinger perturbation theory. (live.ocw.mit.edu)

For an isolated, nondegenerate level with normalized unperturbed state $|n^{(0)}\rangle$ , the first two energy corrections are

E_n^{(1)}=\langle n^{(0)}|V|n^{(0)}\rangle ,

E_n^{(2)}= \sum_{m\ne n} \frac{|\langle m^{(0)}|V|n^{(0)}\rangle|^2} {E_n^{(0)}-E_m^{(0)}}.

Thus $E_n=E_n^{(0)}+\lambda E_n^{(1)}+\lambda^2E_n^{(2)}+\cdots$ . The first correction is an expectation value; the second involves coupling to other unperturbed states. Corrections to the state describe changes in its wave function. (live.ocw.mit.edu)

Small energy denominators make the nondegenerate expansion unreliable near degeneracy. If several states share the same unperturbed energy, the perturbation’s matrix within their eigenspace must first undergo diagonalization. Its eigenvectors select suitable zeroth-order states, and its eigenvalues give the first-order shifts. (ocw.mit.edu)

Examples include an atom subjected to a weak external electric field or magnetic field, and corrections for anharmonicity in molecular vibrations. These problems illustrate that “small” refers to relevant couplings and spectral separations, not merely to the numerical size of a parameter. (ocw.mit.edu)

Time-dependent quantum theory

Time-dependent perturbation theory calculates transitions produced by $V(t)$ . For a system initially in state $|i\rangle$ , the first-order amplitude for another unperturbed state $|f\rangle$ is

c_f^{(1)}(t)= -\frac{i\lambda}{\hbar} \int_{t_0}^{t} \langle f|V(t')|i\rangle e^{i(E_f^{(0)}-E_i^{(0)})t'/\hbar}\,dt'.

Here $\hbar$ is the reduced Planck constant. The squared amplitude gives the leading transition probability. Higher orders involve time-ordered products of interaction operators. (ocw.mit.edu)

For weak coupling to a sufficiently dense set of final states, suitable long-time approximations yield Fermi’s golden rule, relating a transition rate to the squared coupling matrix element and the density of available states. Its derivation requires an appropriate time window; a low-order transition expansion is not necessarily valid indefinitely. (ocw.mit.edu)

Field theory and limits of validity

In quantum field theory, perturbative expansions organize interaction effects by coupling strength. Feynman diagrams represent contributions to scattering amplitudes and correlation functions. They are calculational representations of terms in an expansion, rather than literal pictures of particle trajectories. (damtp.cam.ac.uk)

A perturbation series need not converge to be useful. An asymptotic expansion can approximate a quantity increasingly well as the parameter decreases at any fixed truncation order, even when its infinite sum diverges. At a fixed nonzero parameter, adding terms beyond the smallest contribution may worsen the result. Convergence and asymptotic accuracy are therefore distinct questions. (math.ucdavis.edu)

Validity also depends on time, position, and proximity to resonances or degeneracies. A small residual in the governing equation does not automatically guarantee a small solution error; stability estimates may be needed. In dynamical systems, perturbative averaging and related constructions consequently specify the timescales and parameter regimes over which their approximations apply. (math.ucdavis.edu)