Solution of Schrodinger Equation Evolve with Time-dependent Perturbation

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. The general solution of Schrodinger equation involving time dependent perturbation can be presented in compact and manageable form for periodic & non-periodic perturbation. On the basis of the solution of Schrodinger equation involving time-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 must perform 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 denoted by&for the simplicity we assumed Ho to be 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 real dimensionless parameter and much less than 1. The system is assumed to be initially in the stationary state an eigenstate 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 Schrodinger equation:

iħ = [H0 + λŴ(t)] (2)

The solution of is first order differential equation which corresponds to initial condition = is unique. The probability of finding the system in another eigenstate is, (t) = ||2 (3) Let (t) be the component of the ketin the basis then = (4) with = The closer 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) denote the matrix element of observable Ŵ(t) in the basis. When λ Ŵ(t) is zero, equation (5) is no longer coupled and their solution are very simple it can be written as: Cn(t) = bn (7) where bn is the constant depends 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. This equation is rigorously equivalent to Schrodinger equation. In general, we do 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 λq on 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 zeroth-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 < 0 system is in the initial state therefore, = . This relation is valid for all λ. Consequently, the coefficient of the expansion of must satisfy. = and Thus equation (10) immediately yields, for all positive t that, = which completely determined the zeroth-order solution. This permit us to write equation (11) for r =1 in the form: = = = (13) Then, we find the solution: = (14) Transition probability is equal to since and have the same modulus = where = + λ + …… Hence the probability of finding the system in the state ‘f ’ after time t (f ≠ i) is: = (15) Replacing λŴ(t) by W(t) we finally obtained: = (16) This result shows that is proportional to the square of the modulus of the Fourier transform of the perturbation matrix element.

This Fourier transform is evaluated at an angular frequency equal to the Bohr angular frequency associated with transition under consideration. Limitations Although time dependent Perturbation theory has wide applications while dealing with the small perturb in Hamiltonian it can not be valid for certain circumstances such as the interaction between quark and gluon in which the coupling constant is so high that the field cannot be treated with small energy. Similarly dealing to the convensional superconducting phenomena in which the strong correlated cooper pairs are formed should be treated with some other approximation called WKB such state is called non-adiabatic state.

Conclusion

In conclusion, the general solution of the Schrodinger equation evolving the time-dependent perturbation is express in manageable form. Here we discuss upto the first order solution. The second order solution also can be obtained in terms of first one. On the basis of this solution, we find the transition probability of system between two states after time ‘t’ in which the system is charecterized with small perturbation.