Question about perturbation theory

[Malamala]

Hello! I have a situation where I have time dependent Hamiltonian, $H_0(t)$ which I can solve for exactly and thus get $\psi_0$ as its eigenfunction (given the initial conditions). Now, on top of this, I add a time independent Hamiltonian, $H_1$ much smaller than $H_0$. How can I get the corrections to the wavefunction $\psi_0$ as a function of time, due to $H_1$?

 

[hutchphd]

I believe you will need to be explicit here. What are the two Hamiltonians and what does the solution look like?

 

[Malamala]

I believe you will need to be explicit here. What are the two Hamiltonians and what does the solution look like?

My Hamiltonian, $H_0(t)$ is a large matrix (in principle I can truncate it, so for now let's say it's 10 x 10). I can solve the TDSE exactly using numerical methods and get the wavefunction $\psi_0(t)$. I don't have an explicit analytical form, just 10 numbers as a function of time. What I want to know, is the probability of the system to be in a given level as a function of time (and I can easily extract that by squaring the number associated to that level out of the 10 calculated).

In principle, I can easily solve the TDSE for $H_0(t) + H_1$ and get the probability for the new system. However, $H_1$ is much much smaller than $H_0$ (it depends on a given parameter, call it $\alpha$, which is much smaller than anything else in the problem). If I would solve the TDSE for $H_0(t) + H_1$ as a whole, it would be hard to see the effect of $\alpha$ on the probability I am interested in. So what I want is to somehow treat $H_1$ analytically (in some sort of perturbation theory), on top of the numerical solution obtained from $H_0$, such that I have a better understanding of the physical effect $\alpha$ has on my system.

Basically, I don't want to know that the probability changed, let's say, from $0.25$ to $0.250001$, but I want to have something like $0.25 + f(\alpha,t)$.