Understanding Perturbation Theory: A Comprehensive Definition

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In summary, Perturbation theory is a mathematical technique used to analyze systems with complex behaviors that cannot be solved analytically. It is widely used in various fields, including quantum mechanics and fluid mechanics, to add together an infinite series of mechanisms and obtain a more accurate approximation of a system's behavior. Various specialized techniques have been developed to apply perturbation theory effectively, such as the WKBJ-method and Poincare-Lindstedt method.
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what is the perturbation theory? Thank you in advance.
 
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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.

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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.
 
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1. What is perturbation theory?

Perturbation theory is a mathematical method used to approximate solutions to problems that are too complex to solve exactly. It involves breaking down a complex problem into simpler ones and then using mathematical techniques to find an approximate solution.

2. Why is perturbation theory important?

Perturbation theory is important because it allows us to understand and solve complex problems in physics, engineering, and other fields. It provides a systematic approach to finding approximate solutions, which can then be refined as needed.

3. What are the key concepts in perturbation theory?

The key concepts in perturbation theory include the unperturbed system, the perturbation operator, and the perturbation parameter. The unperturbed system represents the simpler version of the problem, while the perturbation operator represents the effects of small changes on the system. The perturbation parameter is a small number that helps us control the level of approximation.

4. What are the limitations of perturbation theory?

Perturbation theory has limitations when applied to highly nonlinear problems, as the approximations made may not accurately capture the behavior of the system. It also assumes that the perturbation parameter is small, which may not always be the case. Additionally, perturbation theory may not work well for problems with multiple interacting perturbations.

5. How is perturbation theory applied in practical situations?

Perturbation theory is applied in practical situations by first identifying a problem that is too complex to solve exactly. The problem is then broken down into simpler ones, and perturbation theory is used to find an approximate solution. This solution can then be refined or compared to exact solutions if they are available. Perturbation theory is commonly used in fields such as quantum mechanics, fluid dynamics, and celestial mechanics.

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