perturbációelmélet

Perturbációelmélet, or perturbation theory, is a mathematical method used to find approximate solutions to problems that cannot be solved exactly. It is particularly useful when a problem is similar to one that can be solved exactly but differs by a small amount. This small difference is called a perturbation. The method works by assuming that the solution to the original, solvable problem is a good starting point and then systematically adding corrections to account for the perturbation.

The core idea of perturbation theory is to expand the solution in terms of a small parameter, often denoted by epsilon (ε). If the original problem can be solved exactly when ε = 0, perturbation theory seeks a solution of the form of a power series in ε: u = u₀ + εu₁ + ε²u₂ + ..., where u₀ is the exact solution to the unperturbed problem, and u₁, u₂, etc., are correction terms. These correction terms are found by substituting the series expansion into the original equations and solving for each term order by order.

Perturbation theory finds wide applications in various scientific and engineering disciplines. In physics, it is crucial for quantum mechanics, celestial mechanics, and fluid dynamics. For instance, in quantum mechanics, it is used to calculate the energy levels and wave functions of atoms and molecules when they are subjected to external fields or when electron-electron interactions are considered. In celestial mechanics, it helps to understand the long-term evolution of planetary orbits, which are slightly perturbed by the gravitational influence of other celestial bodies. In mathematics, it is employed in solving differential equations that are not analytically solvable. The success of perturbation theory relies on the smallness of the perturbation parameter; as the perturbation grows, the approximation may become less accurate, and other methods might be required.