Perturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem. Perturbation theory is applicable if the problem at hand can be formulated by adding a "small" term to the mathematical description of the exactly solvable problem.

Perturbation theory leads to an expression for the desired solution in terms of a power series in some "small" parameter that quantifies the deviation from the exactly solvable problem. The leading term in this power series is the solution of the exactly solvable problem, while further terms describe the deviation in the solution, due to the deviation from the initial problem. Formally, we have for the approximation to the full solution A a series in the small parameter (here called ε), like the following: In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ...

In this example, A₀ would be the known solution to the exactly solvable initial problem and represent the "higher orders" which are found iteratively by some systematic procedure. For small ε these higher orders become successively more unimportant.

Examples

Examples for the "mathematical description" are: an algebraic equation, a differential equation (e.g., the equations of motion in celestial mechanics or a wave equation), a free energy (in statistical mechanics), a Hamiltonian operator (in quantum mechanics). Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... Graph of a differential equation In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ... In advanced physics, equations of motion usually refer to the Euler-Lagrange equations, differential equations derived from the Lagrangian. ... Celestial mechanics is a division of astronomy dealing with the motions and gravitational effects of celestial objects. ... The wave equation is an important partial differential equation that describes a variety of waves, such as sound waves, light waves and water waves. ... The free energy is a measure of the amount of mechanical (or other) work that can be extracted from a system, and is helpful in engineering applications. ... Statistical mechanics is the application of statistics, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ... Please wikify (format) this article as suggested in the Guide to layout and the Manual of Style. ... For a non-technical introduction to the topic, please see Introduction to Quantum mechanics. ...

Examples for the kind of solution to be found perturbatively: the solution of the equation (e.g., the trajectory of a particle), the statistical average of some physical quantity (e.g., average magnetization), the ground state energy of a quantum mechanical problem. A trajectory is an imagined trace of positions followed by an object moving through space. ... In mathematics, there are numerous methods for calculating the average or central tendency of a list of n numbers. ... In physics, the ground state of a quantum mechanical system is its lowest-energy state. ...

Examples for the exactly solvable problems to start with: Linear equations, including linear equations of motion (harmonic oscillator, linear wave equation), statistical or quantum-mechanical systems of non-interacting particles (or in general, Hamiltonians or free energies containing only terms quadratic in all degrees of freedom). A linear equation is an equation involving only the sum of constants or products of constants and the first power of a variable. ... A harmonic oscillator is a system which, when displaced from its equilibrium position, experiences a restoring force proportional to the displacement : where is a positive constant. ...

Examples of "perturbations" to deal with: Nonlinear contributions to the equations of motion, interactions between particles, terms of higher powers in the Hamiltonian/Free Energy. Interaction is a kind of action which occurs as two or more objects have an effect upon one another. ...

For physical problems involving interactions between particles, the terms of the perturbation series may be displayed (and manipulated) using Feynman diagrams. In this Feynman diagram, an electron and positron annihilate and become a quark-antiquark pair. ...

History

Perturbation theory has its roots in 17th century celestial mechanics, where the theory of epicycles was used to make small corrections to the predicted paths of planets. Curiously, it was the need for more and more epicycles that eventually lead to the Copernican revolution in the understanding of planetary orbits. The development of basic perturbation theory for differential equations was fairly complete by the middle of the 19th century. It was at that time that Charles Delaunay was studying the perturbative expansion for the Earth-Moon-Sun system, and discovered the so-called "problem of small denominators". Here, the denominator appearing in the n 'th term of the perturbative expansion could become arbitrarily small, causing the n 'th correction to be as large or larger than the first-order correction. At the turn of the 20th century, this problem lead Henri Poincare to make one of the first deductions of the existence of chaos, or what is prosaically called the "butterfly effect": that even a very small perturbation can have a very large effect on a system. Celestial mechanics is a division of astronomy dealing with the motions and gravitational effects of celestial objects. ... In the Ptolemaic system of astronomy, the epicycle (literally: on the cycle in Greek) was a geometric model to explain the variations in speed and direction of the apparent motion of the Moon, Sun, and planets. ... The Copernican principle is the philosophical statement that no special observers should be proposed. ... Graph of a differential equation In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ... Charles-Eugène Delaunay (April 9, 1816 – August 5, 1872) was a French astronomer and mathematician. ... The n-body problem is the problem of finding, given the initial positions, masses, and velocities of n bodies, their subsequent motions as determined by classical mechanics, i. ... Henri Poincaré, photograph from the frontispiece of the 1913 edition of Last Thoughts Jules Henri Poincaré (April 29, 1854 – July 17, 1912) was one of Frances greatest mathematicians, theoretical scientists and a philosopher of science. ... A plot of the trajectory Lorenz system for values r = 28, σ = 10, b = 8/3 In mathematics and physics, chaos theory describes the behavior of certain nonlinear dynamical systems that under certain conditions exhibit a phenomenon known as chaos. ... Point attractors in 2D phase space. ...

Perturbation theory saw a particularly dramatic expansion and evolution with the arrival of quantum mechanics. Although perturbation theory was used in the semi-classical theory of the Bohr atom, the calculations were monstrously complicated, and subject to somewhat ambiguous interpretation. The discovery of Heisenberg's matrix mechanics allowed a vast simplification of the application of perturbation theory. Notable examples are the Stark effect and the Zeeman effect, which have a simple enough theory to be included in standard undergraduate textbooks in quantum mechanics. Other early applications include the fine structure and the hyperfine structure in the hydrogen atom. For a non-technical introduction to the topic, please see Introduction to Quantum mechanics. ... The Bohr model of the atom The Bohr Model is a physical model that depicts the atom as a small positively charged nucleus with electrons in orbit at different levels, similar in structure to the solar system. ... Matrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg in 1925, it is also known as The Heisenberg Picture Matrix mechanics involves associating the properties of matter with matrices. ... The Stark effect is the splitting of a spectral line into several components in the presence of an electric field. ... The Zeeman effect (IPA ) is the splitting of a spectral line into several components in the presence of a magnetic field. ... 1. ... In atomic physics, hyperfine structure is a small perturbation in the energy levels (or spectra) of atoms or molecules due to the magnetic dipole-dipole interaction, arising from the interaction of the nuclear magnetic dipole with the magnetic field of the electron. ... depiction of a hydrogen-1 atom showing the Van der Waals radius and the proton nucleus. ...

In modern times, perturbation theory underlies almost all of quantum chemistry and quantum field theory. In chemistry, perturbation theory was used to obtain the first solutions for the helium atom. The earliest use of perturbation theory for molecular physics was the development of the linear combination of atomic orbitals molecular orbital method (LCAO-MO) by Ugo Fano and others in the 1930's. Linus Pauling, as a pioneer of the valence bond theory, is one of the first quantum chemists. ... Quantum field theory (QFT) is the application of quantum mechanics to fields. ... General Name, Symbol, Number helium, He, 2 Chemical series noble gases Group, Period, Block 18, 1, s Appearance colorless Atomic mass 4. ... Molecular physics is the study of the physical properties of molecules and of the chemical bonds between atoms that bind them into molecules. ... This article may be too technical for most readers to understand. ... Ugo Fano, 1912 - 2001 Ugo Fano, a leader in theoretical physics in the 20th century was born in Torino, Italy, in 1912 and passed away on February 13, 2001 in Chicago, Illinois, at the age of 88. ...

In the middle of the 20'th century, Richard Feynman realized that the perturbative expansion could be given a dramatic and beautiful graphical representation in terms of what are now called Feynman diagrams. Although originally applied only in quantum field theory, such diagrams now find increasing use in any area where perturbative expansions are studied. Richard Phillips Feynman (May 11, 1918 – February 15, 1988) (surname pronounced FINE-man; in IPA) was an influential American physicist known for expanding greatly on the theory of quantum electrodynamics, quark theory, and the physics of the superfluidity of supercooled liquid helium. ... A Feynman diagram is a bookkeeping device for performing calculations in quantum field theory, invented by American physicist Richard Feynman. ... Quantum field theory (QFT) is the application of quantum mechanics to fields. ...

A partial resolution of the small-divisor problem was given by the statement of the KAM theorem in 1954. Developed by Andrey Kolmogorov, Vladimir Arnold and Jurgen Moser, this theorem stated the conditions under which a system of partial differential equations will have only mildly chaotic behaviour under small perturbations. The Kolmogorov-Arnold-Moser theorem is a theorem in non-linear dynamics that solves the small-divisor problem in classical perturbation theory. ... Andrey Kolmogorov Andrey Nikolaevich Kolmogorov (Андре́й Никола́евич Колмого́ров) (kahl-mah-GAW-raff) (April 25, 1903 in Tambov - October 20, 1987 in Moscow) was a Soviet mathematician who made major advances in the fields of probability theory and topology. ... Vladimir I. Arnold (Moscow, December 2001). ... Jurgen Moser (1928 – 1999) was an American mathematician who specialized in dynamical systems. ...

In the late 20th century, broad dissatisfaction with perturbation theory in the quantum physics community, including not only the difficulty of going beyond second order in the expansion, but also questions about whether the perturbative expansion is even convergent, has lead to a strong interest in the area of non-perturbative analysis, that is, the study of exactly solvable models. The prototypical model is the KdV equation, a highly non-linear equation for which the interesting solutions, the solitons, cannot be reached by perturbation theory, even if the perturbations were carried out to infinite order. Much of the theoretical work in non-perturbative analysis goes under the name of quantum groups and non-commutative geometry. In mathematics, a series is the sum of the terms of a sequence of numbers. ... In theoretical physics, an exactly solvable model or integrable model refers to a physical model, a physical theory, or set of differential equations whose exact solution may be calculated analytically in terms of elementary or special functions; the adjective integrable is therefore implies solvablility. ... The Korteweg-de Vries equation (KdV equation for short) is the following partial differential equation for a function φ of two real variables, x and t: Its solutions clump up into solitons. ... In mathematics and physics, a soliton is a self-reinforcing solitary wave caused by nonlinear effects in the medium. ... In abstract algebra, a Hopf algebra is a bialgebra H over a field K together with a K-linear map such that the following diagram commutes . (Here Δ is the comultiplication of the bialgebra, ∇ its multiplication, η its unit and ε its counit. ... In mathematics, there is a close relationship between spaces, which are geometric in nature, and the numerical functions on them. ...

Perturbation orders

The standard exposition of perturbation theory is given in terms the order to which the perturbation is carried out: first order perturbation theory or second order perturbation theory, and whether the perturbed states are degenerate (that is, singular), in which case extra care must be taken, and the theory is slightly more difficult.

This section needs to be expanded to include the standard textbook examples of each of the three expansions.

First-order non-singular perturbation theory

This section develops, in simplified terms, the general theory for the perturbative solution to a differential equation to the first order. In order to keep the exposition simple, a crucial assumption is made: that the solutions to the unperturbed system are not degenerate, so that the perturbation series can be inverted. There are ways of dealing with the degenerate (or "singular") case; these require extra care. Graph of a differential equation In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ...

Suppose one wants to solve a differential equation of the form

Dg(x) = λg(x)

where D is some specific differential operator, and λ is an eigenvalue. Many problems involving ordinary or partial differential equations can be cast in this form. It is presumed that the differential operator can be written in the form In mathematics, a differential operator is a linear operator defined as a function of the differentiation operator. ...

D = D⁽⁰⁾ + εD⁽¹⁾

where ε is presumed to be small, and that furthermore, the complete set of solutions for D⁽⁰⁾ are known. That is, one has a set of solutions , labelled by some arbitrary index n, such that

.

Furthermore, one assumes that the set of solutions form an orthonormal set: In linear algebra, two vectors v and w are said to be orthonormal if they are both orthogonal (according to a given inner product) and normalized. ...

with δ_mn the Kronecker delta function. In mathematics, the Kronecker delta or Kroneckers delta, named after Leopold Kronecker (1823-1891), is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise. ...

To zeroth order, one expects that the solutions g(x) are then somehow "close" to one of the unperturbed solutions . That is,

and

.

where denotes the relative size, in big-O notation. To solve this problem, one assumes that the solution g(x) can be written as a linear combination of the : The Big O notation is a mathematical notation used to describe the asymptotic behavior of functions. ...

with all of the constants except for n, where . Plugging this last expansion into the differential equation, and making use of orthogonality, one obtains

This can be trivially re-written as a simple linear algebra problem of finding the eigenvalue of a matrix, where Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear transformations, and systems of linear equations in finite dimensions. ... In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ... Look up matrix in Wiktionary, the free dictionary. ...

where the matrix elements A_nm are given by

Rather than solving this full matrix equation, one notes that, of all the c_m in the linear equation, only one, namely c_n, is not small. Thus, to the first order in ε, the linear equation may be solved trivially as

since all of the other terms in the linear equation are of order . The above gives the solution of the eigenvalue to first order in perturbation theory.

The function g(x) to first order is obtained through similar reasoning. Substituting

so that

gives an equation for . It may be solved integrating with the partition of unity In mathematics, a partition of unity of a topological space X is a set of continuous functions {ρi} from X to the unit interval [0,1] such that every point has a neighbourhood where all but a finite number of the functions are identically zero, and the sum of all...

to give

which gives the exact solution to the perturbed differential equation to the first order in the perturbation ε.

Several important observations can be made about the form of this solution. First, the sum over functions with differences of eigenvalues in the denominator resembles the resolvent in Fredholm theory. This is no accident; the resolvent acts essentially as a kind of Green's function or propagator, passing the perturbation along. Higher order perturbations resemble this form, with an additional sum over a resolvent appearing at each order. In mathematics, the resolvent formalism is a technique for applying concepts from complex analysis to the study of the spectrum of operators on Hilbert spaces and more general spaces. ... In mathematics, Fredholm theory is a theory of integral equations. ... In mathematics, a Greens function is a type of function used to solve inhomogeneous differential equations subject to boundary conditions. ... In quantum mechanics and quantum field theory, the propagator gives the probability amplitude for a particle to travel from one place to another in a given time, or to travel with a certain energy and momentum. ...

The form of this solution is sufficient to illustrate the idea behind the small-divisor problem. If, for whatever reason, two eigenvalues are close so that difference become small, the corresponding term in the sum will become disproportionately large. In particular, if this happens in higher-order terms, the high order perturbation may become as large or larger in magnitude than the first-order perturbation. Such a situation calls into question the validity of doing a perturbation to begin with. This can be understood to be a fairly catastrophic situation; it is frequently encountered in chaotic dynamical systems, and requires the development of techniques other than perturbation theory to solve the problem. A plot of the trajectory Lorenz system for values r = 28, σ = 10, b = 8/3 In mathematics and physics, chaos theory describes the behavior of certain nonlinear dynamical systems that under certain conditions exhibit a phenomenon known as chaos. ...

Curiously, the situation is not at all bad if two or more eigenvalues are exactly equal. This case is referred to as singular or degenerate perturbation theory. The degeneracy of eigenvalues indicates that the unperturbed system has some sort of symmetry, and that the generators of the symmetry commute with the unperturbed differential equation. Typically, the perturbing term does not possess the symmetry; one says the perturbation lifts or breaks the degeneracy. In this case, the perturbation can still be performed; however, one must be careful to work in a basis for the unperturbed states so that these map one-to-one to the perturbed states, rather than being a mixture. Square with symmetry group D4 Symmetry is a characteristic of geometrical shapes, equations, and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. ...

Example of second-order singular perturbation theory

Consider the following equation for the unknown variable x:

x = 1 + εx⁵

For the initial problem with ε = 0, the solution is x₀ = 1. For small ε the lowest order approximation may be found by inserting the ansatz Ansatz is a term (from German) often used by physicists. ...

into the equation and demanding the equation to be fulfilled up to terms that involve powers of ε higher than the first. This yields x₁ = 1. In the same way, the higher orders may be found. However, even in this simple example it may be observed that for (arbitrarily) small ε > 0 there are four other solutions to the equation (with very large magnitude). The reason we don't find these solutions in the above perturbation method is because these solutions diverge when while the ansatz assumes regular behavior in this limit.

The four additional solutions can be found using the methods of singular perturbation theory. In this case this works as follows. Since the four solutions diverge at ε = 0, it makes sense to rescale x. We put

x = yε ^{− ν}

such that in terms of y the solutions stay finite. This means that we need to choose the exponent ν to match the rate at which the solutions diverge. In terms of y the equation reads:

ε ^{− ν}y = 1 + ε^{1 − 5ν}y⁵

The 'right' value for ν is obtained when the exponent of ε in the prefactor of the term proportional to y is equal to the exponent of ε in the prefactor of the term proportional to y⁵, i.e. when ν = 1 / 4. This is called 'significant degeneration'. If we choose ν larger then the four solutions will collapse to zero in terms of y and they will become degenerate with the solution we found above. If we choose ν smaller then the four solutions will still diverge to infinity.

Putting ν = 1 / 4 in the above equation yields:

y = ε^{1 / 4} + y⁵

This equation can be solved using ordinary perturbation theory in the same way as regular expansion for x was obtained. Since the expansion parameter is now ε^{1 / 4} we put:

There are 5 solutions for y₀: 0, 1, -1, i and -i. We must disregard the solution y = 0. The case y = 0 corresponds to the original regular solution which appears to be at zero for ε = 0, because in the limit we are rescaling by an infinite amount. The next term is y₁ = − 1 / 4. In terms of x the four solutions are thus given as:

Commentary

Both regular and singular perturbation theory are frequently used in physics and engineering. Regular perturbation theory may only be used to find those solutions of a problem that evolve smoothly out of the initial solution when changing the parameter (that are "adiabatically connected" to the initial solution). A well known example from physics where regular perturbation theory fails is in fluid dynamics when one treats the viscosity as a small parameter. Close to a boundary, the fluid velocity goes to zero, even for very small viscosity (the no-slip condition). For zero viscosity, it is not possible to impose this boundary condition and a regular perturbative expansion amounts to an expansion about an unrealistic physical solution. Singular perturbation theory can, however, be applied here and this amounts to 'zooming in' at the boundaries (boundary layer theory, solvable using the method of matched asymptotic expansions). This article covers adiabatic processes in thermodynamics. ... In fluid dynamics, the no-slip condition states that fluids stick to surfaces past which they flow. ... In mathematics, in particular in solving differential equations with perturbation theory, method of matched asymptotic expansions is an approach to finding an approximate solution to a problem when a naïve perturbation approach fails. ...

Perturbation theory can fail when the system can go to a different "phase" of matter, with a qualitatively different behaviour that cannot be understood by perturbation theory (e.g., a solid crystal melting into a liquid). In some cases this failure manifests itself by divergent behavior of the perturbation series. Such divergent series can sometimes be resummed using techniques such as Borel resummation. In mathematics, a Borel summation is a generalisation of the usual notion of summation of a series. ...

Perturbation techniques can be also used to find approximate solutions to non-linear differential equations. Examples of techniques used to find approximate solutions to these types of problems are the Lindstead-Poincaré technique, the method of harmonic balancing, and the method of multiple time scales.

There is absolutely no guarantee perturbative methods would result in a convergent solution. In fact, asymptotic series are the norm. For a discussion of convergence and convergent series, see limit (mathematics). ... In mathematics and applications, particularly the analysis of algorithms, asymptotic analysis is a method of classifying limiting behaviour, by concentrating on some trend. ...

Perturbation theory in chemistry

Many of the ab initio quantum chemistry methods use perturbation theory directly or are closely related methods. Møller-Plesset perturbation theory uses the difference between the Hartree-Fock Hamiltonian and the exact non-relativistic Hamiltonian as the perturbation. The zero order energy is the sum of orbital energies. The first-order energy is the Hartree-Fock energy and electron correlation is included at second-order or higher. Calculations to second, third or forth order are very common and the code is included in most ab initio quantum chemistry programs. A related but more accurate method is the coupled cluster method. Ab initio quantum chemistry methods are computational chemistry methods based on quantum chemistry. ... Møller-Plesset perturbation theory is an implementation of perturbation theory in quantum chemistry, which provides a method for adding excitations to the Hartree-Fock wavefunction and therefore including the effect of electron correlation. ... In computational physics and computational chemistry, the Hartree-Fock (HF) or self-consistent field (SCF) calculation scheme is a self-consistent iterative variational procedure to calculate the Slater determinant (or the molecular orbitals which it is made of) for which the expectation value of the electronic molecular Hamiltonian is minimum. ... Computational chemistry is a branch of chemistry that uses the results of theoretical chemistry incorporated into efficient computer programs to calculate the structures and properties of molecules and solids, applying these programs to real chemical problems. ... This article needs to be cleaned up to conform to a higher standard of quality. ...

See also

• Structural stability

Structural stability is a mathematical concept concerning whether a given function is sensitive to a small perturbation. ...

External links

• Introduction to Perturbation Methods by Mark H. Holmes
• Chapter II: Introduction to perturbation methods by Johan Byström, Lars-Erik Persson, and Fredrik Strömberg