Understanding Perturbation Theory: A Comprehensive Definition

[lordinfamous]

what is the perturbation theory? Thank you in advance.

 

[Doc Al]

Assuming you mean perturbation theory within quantum mechanics, start here: http://en.wikipedia.org/wiki/Perturbation_theory_(quantum_mechanics)

 

[chroot]

Some physical processes like composite particle decay can happen in an infinite number of different ways. In general, the simplest way occurs much more frequently than the other, more complex ways, but the total probability of decay is the sum of all of the ways. Perturbation theory is a mathematical technique to add together an infinite series of such mechanisms, each with a smaller contribution to the total probability than the previous.

- Warren

 

[arildno]

Perturbation theory has applications in a lot of different areas than quantum mechanics.



One well-known example is the free-surface problem in fluid mechanics (potential flow):

We seek a solution of Laplace's equation (i.e, the continuity equation rewritten in terms of the velocity potential) which satisfy the non-linear free surface conditions (fluid pressure equals air pressure, and kinematic condition) and horizontal bottom.

In particular, we seek a solution where the surface profile can be described as a dominantly monochromatic harmonic wave.
The non-linear corrections to the dominantly linear solution can be found by perturbation theory.


In short, perturbation theory is an indispensable tool for the analysis of systems whose behaviour is desribed by diff.eq's where we cannot find analytical solutions (i.e, most diff.eq's)

As chroot observes, (regular) perturbation theory is at its most effective where effects/behaviours lie in well-defined layers of importance (in an asymptotic limit). Then, we can peel off layer after layer by perturbation theory to gain a better approximation.
A host of specialized techniques has been developed, for example, WKBJ-method, Poincare-Lindstedt method, multiple scale analysis, and so on.