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 dependant 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.

 

[denian]

Hello,

Thank you for your question. It seems like you are asking about the time-dependent quantum solution for a system with a slowly changing electric field. Let me try to explain it in simpler terms.

In quantum mechanics, the state of a system is described by a wave function. This wave function can change with time, and the time-dependent Schrödinger equation is used to describe this change. The solution to this equation is a linear combination of static eigenfunctions, which are solutions to the time-independent Schrödinger equation.

The probability of a system having a certain eigenvalue is proportional to the square of the coefficient of the corresponding eigenfunction in the linear combination. However, this is only true when the system is in an unperturbed state, i.e. when there is no external influence on the system.

In your case, the slowly changing electric field is perturbing the system and causing it to be in a different state. This means that the eigenfunctions and eigenvalues of the system will also be perturbed. To find the probabilities of these perturbed states, you will need to use a different approach.

One way to do this is by using perturbation theory, which allows you to approximate the perturbed eigenfunctions and eigenvalues. Another approach is to use numerical methods to solve the time-dependent Schrödinger equation for the perturbed system.

I hope this helps to clarify your question. If you have any further doubts, please let me know.