Asymptotic behaviour after perturbation

[Gerenuk]

Is it possible to write down a statement about the asymptotic behaviour of the wave function after a small perturbation has been switched on?

So I have and initial wave function $\psi_0$ and the Hamiltonian
$$H=\begin{cases} H_0 & t<0\\ H_0+V+R & t\geq 0 \end{cases}$$
where V is a small perturbation and R and even smaller random extra perturbation. It would be nice to have the wavefunction $\psi(t=\infty)$ in terms of the eigenstates of V.

 

[James Leighe]

Is it possible to write down a statement about the asymptotic behaviour of the wave function after a small perturbation has been switched on?

So I have and initial wave function $\psi_0$ and the Hamiltonian
$$H=\begin{cases} H_0 & t<0\\ H_0+V+R & t\geq 0 \end{cases}$$
where V is a small perturbation and R and even smaller random extra perturbation. It would be nice to have the wavefunction $\psi(t=\infty)$ in terms of the eigenstates of V.

The wave-function is asymptotic regardless AFAIK. So your particle at infinite time could be literally anywhere at any speed or not even existent anymore (as the same particle) etc. So the eigenstates could be whatever you wanted to chose but the idea is that we would now know nothing at all about the particle whatsoever (and it would be extremely unpredictable even after a short amount of time, even with ALL the knowledge of the particles initial position and momentum).

 

[Gerenuk]

I need an formal solution in terms of V, R and so on. The particle wouldn't be everywhere since if H0 were an infinite well, the particle of course would be within the well. Anyway, I'm looking for *an equation* as the solution.