Classical Perturbation Theory-Time Dep. vs. Time Indep (Goldstein).

[Bosh]

Classical Perturbation Theory--Time Dep. vs. Time Indep (Goldstein).

Hi,

I'm going through Goldstein, and I'm a little confused on the distinction between time dependent and time independent perturbation theory. In section 12.2, they do the case of a simple harmonic perturbation on force free motion. I would have thought that the perturbation $\Delta H = \frac{m\omega^2 x^2}{2}$ would not be considered time-dependent. Is the key that when you plug in the unperturbed solution for x, i.e., $\frac{\alpha t}{m} + \beta$, the perturbation hamiltonian is now time-dependent?

If so, it would seem that the example they treat in the section on time-independent perturbation theory, the anharmonic oscillator with $\Delta H = \frac{\epsilon m^2 \omega_0^2 q^3}{q_0}$, could also have been treated as a time-dependent problem.

Any insight would be appreciated! Thanks!

Dan

 

[eljose79]

Dear Dan,

Thank you for your question regarding the distinction between time dependent and time independent perturbation theory. I can understand your confusion, as the two cases may seem similar at first glance.

The key difference between the two is in the nature of the perturbation itself. In time dependent perturbation theory, the perturbation term is explicitly time dependent, as in your example of \frac{m\omega^2 x^2}{2}. This means that the perturbation is changing with time and cannot be separated from the time variable. As you correctly noted, when the unperturbed solution is plugged in for x, the perturbation Hamiltonian becomes time dependent.

On the other hand, in time independent perturbation theory, the perturbation term is not explicitly time dependent. This means that the perturbation can be separated from the time variable, as in the case of the anharmonic oscillator with \frac{\epsilon m^2 \omega_0^2 q^3}{q_0}. This allows for the use of the time-independent perturbation theory approach, which is often simpler and more straightforward than time-dependent perturbation theory.

In summary, the key distinction between time dependent and time independent perturbation theory lies in the nature of the perturbation term itself. I hope this helps clarify the difference between the two cases. Please let me know if you have any further questions.