This discussion focuses on applying perturbation theory to a time-dependent Hamiltonian, ##H_0(t)##, which is solved numerically to obtain the wavefunction ##\psi_0(t)##. A smaller, time-independent Hamiltonian, ##H_1##, is introduced to analyze its effects on the system's probability levels. The user seeks to derive corrections to the wavefunction due to ##H_1##, aiming for a functional representation of the probability as ##0.25 + f(\alpha,t)##, where ##\alpha## is a small parameter. The discussion emphasizes the need for explicit definitions and analytical treatment of perturbations in quantum mechanics.
PREREQUISITESQuantum physicists, graduate students in physics, and researchers working on time-dependent quantum systems and perturbation theory applications.
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).hutchphd said:I believe you will need to be explicit here. What are the two Hamiltonians and what does the solution look like?
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)##.hutchphd said:What is a "level"? Exact definition, please.