Question about perturbation theory

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Discussion Overview

The discussion revolves around the application of perturbation theory to a time-dependent Hamiltonian, specifically how to obtain corrections to the wavefunction due to a smaller, time-independent Hamiltonian. The focus is on understanding the effects of a small parameter on the probabilities associated with the system's states.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • One participant describes their situation involving a time-dependent Hamiltonian, ##H_0(t)##, which they can solve exactly to obtain the wavefunction ##\psi_0##.
  • The same participant introduces a smaller, time-independent Hamiltonian, ##H_1##, and seeks to understand how to analytically treat its effects on the wavefunction ##\psi_0## using perturbation theory.
  • Another participant requests clarification on the definitions of the Hamiltonians involved and the form of the solution.
  • The original poster explains that their Hamiltonian ##H_0(t)## is a large matrix (10 x 10) and that they can extract probabilities from the numerical solution of the time-dependent Schrödinger equation (TDSE).
  • The original poster expresses a desire to express the probability change in a form that highlights the contribution of the small parameter ##\alpha##, rather than just noting a small numerical change.
  • Several participants ask for a precise definition of what is meant by a "level" in this context.
  • The original poster clarifies that "level" refers to one of the basis states used in their calculations, providing an example of a specific state vector.

Areas of Agreement / Disagreement

Participants have not reached a consensus on the specifics of the perturbation approach or the definitions being discussed. There are multiple requests for clarification and definitions, indicating that some aspects of the discussion remain unresolved.

Contextual Notes

The discussion includes assumptions about the size of the Hamiltonians and the nature of the perturbation, which may not be fully articulated. The dependence on the parameter ##\alpha## and its implications for the analysis are also not completely resolved.

Malamala
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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##?
 
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I believe you will need to be explicit here. What are the two Hamiltonians and what does the solution look like?
 
hutchphd said:
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)##.
 
What is a "level"? Exact definition, please.
 
hutchphd said:
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)##.
 

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