On the Time-dependent Perturbation Theory

Published: 2021-06-23 16:25:04
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Time-dependent perturbation theory is the approximation method treating Hamiltonians that depends explicitly on time. It is most useful for studying process of absorption and emission of radiation by atoms or, more generally, for treating the transitions of quantum systems from one energy level to another energy level. Introduction We have dealt so far with Hamiltonian that do not depend explicitly on time. In nature, however most of the quantum phenomena are governed by time dependent Hamiltonian. Thegeneral solution of Schrodinger equation involving time dependent perturbation can be presented in compact and manageable form for periodic & nonperiodic perturbation.Onthe basis of the solution ofSchrodinger equation involvingtime dependent perturbation probability for various process including the interaction of electromagnetic field with matter can be calculated. The most satisfactory time-dependent perturbation theory is the method of variation of constraints developed by Dirac. This is basically the power expansion in term of the strength of the perturbation just as the Rayleigh Schrodinger perturbation theory in case of the time dependent perturbation.Method of variation of constant is useful only when the perturbation is weak. If the perturbation is strong then we mustperform up to higher term. However, in practice this is impossible & the result may diverge.
This technique is particularly useful for the clarification of resonance or transition phenomena of the system due to interaction with external perturbation Mathematical formulation Let us consider the physical system with an (unperturbed) Hamiltonian Ho, the eigenvalue and eigenfunction is denotedby&|for the simplicity we assumedHoto be discrete and nondegenerate Ho= | (1) At t = 0, a small perturbation of the system is introduced so that the new Hamiltonian is: H(t) = +λŴ(t) Whereλ is real dimensionless parameter and much less than 1. The system is assumed to be initially in the stationary state an eigen state of of eigenvalue Starting at t = 0 when the perturbation is applied, system evolves and can be found in different state. Between times 0 and t the system evolves in accordance with Schrodingerequation: iħ = [H0 + λŴ(t)](2) The solution of this first order differential equation which corresponds to initial condition = is unique.
The probability of finding the system in anothereigenstate|is, (t) = |2(3) Let (t) be the component of the ketin the basisthen = (4) with (t) = The closer relation is: =1 (5) Usin equation (4)&(5), (2) becomes iħ = iħ= iħ= Ek+(t) iħEkδnk+ Ck(t) iħEk Cn(t) + λ iħ=EnCn(t) + λ iħ =EnCn(t) + λ (6) Here Ŵnk(t)denote the matrix element of observable Ŵnk(t) in the basis. When λ Ŵ(t) is zero equation (5) are no longer coupled, and their solution are very simple it can be written as: Cn(t) = bn(7) where bn is the constant depend on the initial condition. For the nonzero perturbation we look the solution of the form, Cn(t) = bn(t)(8) Then from equation (5) iħ + En bn(t) = En bn(t)+ λbk(t) iħ = λbk(t)(9) where = is the Bohr angular frequency. Thisequation is rigorously equivalent to Schrodinger equation. In general, wedo not know how to find its exact solution. We look for the solution in the following form: = (10) Using equation (9) in (8). iħ = If we set equal the coefficients of λqon both side of the equation we find: For 0th order: = 0 (11) Thus, if λ is zero reduces to constant. For higher order: = (12)
Thus,we see that, with the zeroeth-order solution determined by above equation and the initial condition this equation enable us to find the first-order solution. Then we also find the second-order solution in terms of first one. Since at t

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