Everything about Perturbation Theory totally explained
» This article describes perturbation theory as a general mathematical method. For perturbation theory as applied to quantum mechanics, see perturbation theory (quantum mechanics).
Perturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which can't 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've for the approximation to the full solution A, a series in the small parameter (here called
), like the following:
»
In this example,
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
these higher orders are presumed to become successively less important.
Examples
Examples for the "mathematical description" are:
an
algebraic equation,
a
differential equation (for example, 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 (for example, the
trajectory of a particle),
the
statistical average of some
physical quantity (for example, 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. Curiously, it was the need for more and more epicycles that eventually led to the 16th century
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-Eugène 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 led
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.
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.
In modern times, perturbation theory underlies much of
quantum chemistry and
quantum field theory. In chemistry, perturbation theory was used to obtain the first solutions for the
helium atom.
In the middle of the 20th 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.
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.
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 led 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, can't 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.
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.
Suppose one wants to solve a differential equation of the form
»
where
D is some specific
differential operator, and
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
»
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 isn't 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 (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 can't be understood by perturbation theory (for example, 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.
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
Lindstedt-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.
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.
Further Information
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