# Question on time-independent perturbation theory

## Main Question or Discussion Point

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.

## Answers and Replies

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A. Neumaier
2019 Award
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.

Standard time dependent perturbation theory does give the perturbed time-dependent wave function as a power series in lambda--the Dyson series.