PerturbationTheory

PerturbationTheory - PHYS4221 Quantum Mechanics I Fall 2010...

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PHYS4221 Quantum Mechanics I Fall 2010 Supplementary notes on “Perturbation Theory” More often than not, the vast majority of problems in quantum mechanics cannot be solved rigorously in closed form with the present resources of mathematics. For such problems we are then inevitably forced to resort to some form of approximation methods. If the given problem differs from an exactly solvable problem by only a small amount, one can often apply the perturbation theory . The perturbation theory consists in splitting up the Hamiltonian of the given system into two parts , one of which must be simple and the other small. The first part may then be considered as the Hamiltonian of a simplified or unperturbed system, which can be solved exactly, and the addition of the second will then require small corrections, of the nature of a perturbation , in the solution of the unperturbed system. The requirement that the first part shall be simple usually requires in practice that it shall not involve the time explicitly. If the second part contains a small numerical factor λ , we can obtain the solution of our equations for the perturbed system in the form of a power series in λ , which, provided it converges, will give the answer to our problem with any desired accuracy. 1 Of course, if the method is to be of practical interest, good approximations can better be obtained by taking only one or two terms in the expansion series. There are two distinct perturbation methods, namely the time-independent per- turbation theory and the time-dependent perturbation theory . The former method is applicable only when the perturbation term does not involve the time explicitly, whereas the latter applies to the case with a time-dependent perturbation. With the time-independent perturbation theory, one compares the stationary states of the per- turbed system with those of the unperturbed system; on the other hand, with the time-dependent perturbation theory, one takes a stationary state of the unperturbed system and see how it varies with time under the influence of the perturbation. 1 This means that we implicitly assume the analyticity of the energy eigenvalues and eigenkets in a complex λ -plane around λ = 0. 1
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1. Time-independent Perturbation Theory There are two distinct versions of time-independent perturbation theory. The more widely used is known as the Rayleigh-Schr¨ odinger Perturbation Theory (RSPT), in which both the energy eigenkets and eigenvalues are expressed in power series of the small perturbation parameter λ . The less popular approach, known as the Brillouin- Wigner Perturbation Theory
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