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Perturbation theory develops an expression for the desired solution in terms of a formal power series known as a perturbation 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 ...
Rabi problem in time-dependent perturbation theory [ edit ] In the quantum approach, the periodic driving force can be considered as periodic perturbation and, therefore, the problem can be solved using time-dependent perturbation theory, with
Perturbation theory is applicable if the problem at hand cannot be solved exactly, but can be formulated by adding a "small" term to the mathematical description of the exactly solvable problem.
The method removes secular terms—terms growing without bound—arising in the straightforward application of perturbation theory to weakly nonlinear problems with finite oscillatory solutions. [1] [2] The method is named after Henri Poincaré, [3] and Anders Lindstedt. [4]
Action-angle coordinates are also useful in perturbation theory of Hamiltonian mechanics, especially in determining adiabatic invariants. One of the earliest results from chaos theory, for dynamical stability of integrable dynamical systems under small perturbations, is the KAM theorem, which states that the invariant tori are partially stable.
In mathematics, an eigenvalue perturbation problem is that of finding the eigenvectors and eigenvalues of a system = that is perturbed from one with known eigenvectors and eigenvalues =. This is useful for studying how sensitive the original system's eigenvectors and eigenvalues x 0 i , λ 0 i , i = 1 , … n {\displaystyle x_{0i},\lambda _{0i ...
This category deals with topics in perturbation theory and variational principles, which commonly occur in the theory of differential equations, with problems in quantum mechanics forming an important subset thereof.
In mathematics and physics, multiple-scale analysis (also called the method of multiple scales) comprises techniques used to construct uniformly valid approximations to the solutions of perturbation problems, both for small as well as large values of the independent variables. This is done by introducing fast-scale and slow-scale variables for ...