A mathematical technique frequently used to obtain approximate solutions to equations describing physical systems that are too complicated to solve exactly. The problem is rewritten in two portions: one which can be solved exactly, and a smaller part (the perturbation) which allows the calculation of corrections to the first answer in terms of a sequence of ever-decreasing terms. The technique is essential in many branches of physics, particularly quantum theory.
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. Formally, we have for the approximation to the full solution A a series in the small parameter (here called ε), like the following:
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.
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).
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.
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).
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.
For physical problems involving interactions between particles, the terms of the perturbation series may be displayed (and manipulated) using Feynman diagrams.
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. The development of basic perturbation theory for differential equations was fairly complete by the middle of the 19th century.
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.
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.
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.
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.
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.
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.
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.
Furthermore, one assumes that the set of solutions form an orthonormal set:
with δmn the Kronecker delta function.
To zeroth order, one expects that the solutions g(x) are then somehow "close" to one of the unperturbed solutions . That is,
and
. To solve this problem, one assumes that the solution g(x) can be written as a linear combination of the :with all of the constants except for n, where . Substituting 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
where the matrix elements Anm are given by
Rather than solving this full matrix equation, one notes that, of all the cm in the linear equation, only one, namely cn, 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 . Substituting
so that
gives an equation for . It may be solved integrating with the partition of unity
to give
which gives the exact solution to the perturbed differential equation to the first order in the perturbation ε. 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. it is frequently encountered in chaotic dynamical systems, and requires the development of techniques other than perturbation theory to solve the problem.
Example of second-order singular perturbation theory
Consider the following equation for the unknown variable x:
x = 1 + εx5For 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
into the equation and demanding the equation to be fulfilled up to terms that involve powers of ε higher than the first. 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 terms of y the equation reads:
ε y5The '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.
Putting ν = 1 / 4 in the above equation yields:
y = εThis 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 ε1 / 4 we put:
There are 5 solutions for y0: 0, 1, -1, i and -i. 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. 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).
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).
Perturbation techniques can be also used to find approximate solutions to non-linear differential equations.
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.
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