Time-dependent perturbation theory is the approximation method that deals with Hamiltonians that explicitly depend on time. It is particularly useful for studying the processes 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 So far we have dealt with Hamiltonians that do not explicitly depend on time. We say no to plagiarism. Get a tailor-made essay on "Why Violent Video Games Shouldn't Be Banned"? Get an original essayIn nature, however, most quantum phenomena are governed by the time-dependent Hamiltonian. The general solution of the Schrodinger equation involving time-dependent perturbation can be presented in compact and manageable form for periodic and non-periodic perturbations. Based on the solution of the Schrodinger equation involving the time-dependent perturbation probability for various processes, including the interaction of the electromagnetic field with matter, it is possible to calculate. The most satisfactory time-dependent perturbation theory is the constraint variation method developed by Dirac. This is basically power expansion in terms of perturbation intensity just like Rayleigh-Schrodinger perturbation theory in the case of time-dependent perturbation. The constant variation method is useful only when the perturbation is weak. If the disturbance is strong then we must operate up to the highest term. However, in practice this is impossible and the result can diverge. This technique is particularly useful for clarifying resonance or transition phenomena of the system due to the interaction with external perturbations. Mathematical formulation We consider the physical system with an (unperturbed) Hamiltonian Ho, the eigenvalue and eigenfunction are denoted by&for simplicity we have assumed Ho as discrete and non-degenerate Ho= (1) At t = 0, a small perturbation of the system is introduced so that the new Hamiltonian is:H(t) = +λ Ŵ(t)Where λ is a dimensionless real parameter and much less than 1. The system is assumed to be initially in the steady state an eigenstate of eigenvalue Starting from t = 0 when the perturbation is applied, the system evolves and can be found in different state. Between times 0 and t the system evolves according to the Schrodinger equation:iħ = [H0 + λŴ(t)] (2)The solution of is a first order differential equation which corresponds to the initial condition = is unique. The probability of finding the system in another eigenstate is, (t) = ||2 (3) Let (t) be the ketin component the basis then = (4) with = The closest relation is: =1 (5) Using equation (4) and (5) in (2); iħ = iħ= iħ= Ek+ iħEkδnk+ iħEk Cn(t) + iħ=EnCn(t) + λ iħ =EnCn(t) + λ (6) Here Ŵnk(t) denotes the matrix element of the observable Ŵ(t ) in the base. When λ Ŵ(t) is zero, equation (5) is no longer coupled and their solution is very simple and can be written as: Cn(t) = bn (7) where bn is the constant depends on the initial condition . For the non-zero perturbation we look for 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. This equation is strictly equivalent to the Schrodinger equation. In general, we don't know how to find the exact solution. We look for the solution in the following form: = (10) Using equation (9) in (8). iħ = If we set the coefficients of λq equal on both sides of the equation we find: For order 0: = 0 (11) Therefore, if λ is zero it reduces to a constant. For higher order: = (12) Therefore,.