Time depending quantum solution

[timgor]

Hello dear theorists!
Please help me to understand the following question:
I have slowly changing electric field that has a zero limits at t1=(- infinity) and t2=(+ infinity). All books write that the time dependent solution is sought in form of linear combination of static eigenfunctions solved at t=(- infinity) with coefficients depending on time. And the probability that system will have n-th eugenvalue will be proportional to square of n-th coefficient. But the last one is true only at t2=(+ infinity) when the field will be zero again and the system of eugenfunctions is unperturbed. At any intermediate time there will be nonzero field and another system of eigenfunctions and eugenvalues. I need to find the probabilities of these perturbed states with its perturbed eugenvalues but I have solution, constructed of nonperturbed functions at any time. I do not understand how to find it. Could you please explain me as for stupid guy? Thanks.

 

[azntoon]

The solution that you have is not necessarily the exact solution, but it is an approximation of the true solution. To find the exact solution at any intermediate time, you need to solve the Schrödinger equation with the changing electric field. This will give you a new set of eigenfunctions and eigenvalues which are perturbed from the static ones. The probabilities of finding the system in these states will be proportional to the square of the corresponding coefficients in the wavefunction expansion.