- #1

Runei

- 193

- 17

I just want to make sure that I have understood the following argument the correct way:

For a given quantum system we take the hamiltonian to be a time-independent (and soluble) part, and a time-dependent part.

## \hat{H} = \hat{H_0} + H'(t) ##

Now, the solutions to the

*un*perturbed system are given by

## \Psi_n(x,t) = \psi_n(x) e^{-iE_nt/\hbar} ##

And any solution to the system can be written as

## \Psi(x,t) = \sum\limits_n c_n \psi_n(x) e^{-iE_nt/\hbar} ##

Argument:

When we go ahead and introduce the perturbed system, we can still write the solution to the system as a sum of the eigenfunctions of the unperturbed system, since these eigenfunctions represent a basis to Hilbert-space. And the solution to the perturbed system must also belong to Hilbert-space. Therefore:

## \Psi'(x,t) = \sum\limits_n c_n(t) \psi_n(x) e^{-iE_nt/\hbar} ##

The expansion coefficients in this case will depend on time, since the full hamiltonian now also depends on time, but the

*central*argument is that we can still write the solution we are looking for, as an sum of the solutions to the unperturbed system.

Is this correct or am I missing something essential? :)

Thanks in advance!