Question on time-independent perturbation theory

In summary, the conversation discusses time-independent perturbation theory and its application to the time-dependent Schrodinger equation. It is mentioned that the ground state wave function and energy can be written as a power series in the parameter lambda, and the question is raised if this can also be applied to the time-dependent equation. The concept of eigenvalues is brought up, but it is noted that they are meaningless in this context. The conversation concludes by mentioning that standard time-dependent perturbation theory does give the perturbed wave function as a power series in lambda.
  • #1
ani4physics
29
0
Hi all. I have been thinking about a very simple question, and I am a little confused. We know from time-independent perturbation theory that if the system is perturbed by the external perturbation λV which is much smaller compared to the unperturbed hamiltonian H0, we can write the ground state wave function and the ground state energy as a power series in the parameter λ. So that makes the time-independent Shcoringer to be:

[H0 + λV] [ψ0 + λψ1 + ...] = [E0 + λE1 + ...] [ ψ0(t) + λψ1 + ...]

where ψ0 and ψ1 are respectively the unperturbed and the first-order perturbed wave functions. Now my question is, can we also write the Time-dependent Schrodinger equation of this system as

(ih/2π) ∂/∂t [ψ0(t) + λψ1(t) + ...] = [H0 + λV] [ ψ0(t) + λψ1(t) + ...]
= [E0 + λE1 + ...] [ ψ0(t) + λψ1(t) + ...]

Thanks guys.
 
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  • #2
ani4physics said:
Hi all. I have been thinking about a very simple question, and I am a little confused. We know from time-independent perturbation theory that if the system is perturbed by the external perturbation λV which is much smaller compared to the unperturbed hamiltonian H0, we can write the ground state wave function and the ground state energy as a power series in the parameter λ. So that makes the time-independent Shcoringer to be:

[H0 + λV] [ψ0 + λψ1 + ...] = [E0 + λE1 + ...] [ ψ0(t) + λψ1 + ...]

where ψ0 and ψ1 are respectively the unperturbed and the first-order perturbed wave functions. Now my question is, can we also write the Time-dependent Schrodinger equation of this system as

(ih/2π) ∂/∂t [ψ0(t) + λψ1(t) + ...] = [H0 + λV] [ ψ0(t) + λψ1(t) + ...]
= [E0 + λE1 + ...] [ ψ0(t) + λψ1(t) + ...]

Thanks guys.

Why are you interested in eigenvalues of the time-dependent Schroedinger operator? They are meaningless.
 
  • #3
Standard time dependent perturbation theory does give the perturbed time-dependent wave function as a power series in lambda--the Dyson series.
 

1. What is time-independent perturbation theory?

Time-independent perturbation theory is a method used in quantum mechanics to approximate the behavior of a quantum system that is subjected to a small perturbation (a small change in the system's potential). It allows us to calculate the energy levels and corresponding wavefunctions of the perturbed system by treating the perturbation as a small correction to the unperturbed system.

2. How does time-independent perturbation theory work?

Time-independent perturbation theory works by breaking down the perturbed system into an unperturbed system and a small perturbation. The first step is to solve the unperturbed system to obtain its energy levels and wavefunctions. Then, the perturbation is added to the unperturbed system, and the energy levels and wavefunctions are recalculated using perturbation theory. The final solution is a combination of the unperturbed solution and the perturbation correction.

3. When is time-independent perturbation theory applicable?

Time-independent perturbation theory is applicable when the perturbation to the system is small compared to the unperturbed system. It is also most effective when the energy levels of the unperturbed system are well separated, and the perturbation only affects a small range of energies.

4. What are the limitations of time-independent perturbation theory?

One limitation of time-independent perturbation theory is that it only works well for small perturbations. If the perturbation is too large, the corrections calculated using perturbation theory may not accurately represent the true behavior of the system. Additionally, this method assumes that the perturbation is time-independent, meaning it does not change with time.

5. How is time-independent perturbation theory different from time-dependent perturbation theory?

Time-independent perturbation theory deals with systems that are subjected to a constant perturbation, while time-dependent perturbation theory deals with systems that are subjected to a time-varying perturbation. Time-dependent perturbation theory is more complex and can take into account the effects of the perturbation on the system's dynamics, while time-independent perturbation theory only considers its effects on the energy levels and wavefunctions.

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