häiriöteorian

Häiriöteoria, also known as perturbation theory, is a mathematical method used to find approximate solutions to problems that cannot be solved exactly. It is particularly useful in physics and engineering when dealing with complex systems that are close to simpler, solvable systems. The core idea is to start with a known exact solution for a simplified version of the problem and then introduce a small "perturbation" or disturbance. This perturbation represents the neglected terms or complexities of the original problem.

The method involves expanding the solution in a series of terms, where each term represents a successive approximation to the true solution. The first term is the exact solution to the simplified problem. Subsequent terms account for the effects of the perturbation. As more terms are included in the series, the approximation generally becomes more accurate. However, the validity of perturbation theory relies on the perturbation being sufficiently small. If the perturbation is too large, the series may converge slowly or not at all, and the approximation will be poor.

In quantum mechanics, perturbation theory is widely used to calculate the energy levels and wave functions of atoms and molecules when external fields or interactions are present. In celestial mechanics, it helps in understanding the long-term evolution of planetary orbits, which are not perfectly elliptical due to the gravitational influence of other bodies. Engineering applications include analyzing the behavior of structures under stress or the dynamics of control systems with small uncertainties. The success of perturbation theory hinges on the smallness of the deviation from the idealized, exactly solvable case.