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 $0.25 + f(\alpha,t)$.

 

[hutchphd]

What is a "level"? Exact definition, please.

 

[Malamala]

What is a "level"? Exact definition, please.

In this case by "level" I mean one of the basis used. So being in the second level, corresponds to the square of the braket obtained from the actual wavefunction and (in the case of dimension 10): $(0,1,0,0,0,0,0,0,0,0)$.