Perturbation theory with two parameters?

[Amentia]

Hello,

I am looking for a reference which describe perturbation theory with two parameters instead of one. So far, I did not find anything on the topic. It might have a specific name and I am using the wrong keywords. Any help is appreciated.

To be clear, I mean I have $H = H_{0}+\lambda_{1}V_{1}+\lambda_{2}V_{2}$, where $\lambda_{1}$ and $\lambda_{2}$ are small vs. the terms in the unperturbed Hamiltonian but comparable to each other.

Thank you!

 

[hilbert2]

If you assume that the system changes like in 1st order perturbation approximation, then you can just do one 1st order calculation with perturbation term $\lambda_1 V_1$ and then do another one for the resulting wave functions with $\lambda_2 V_2$. You can try applying this process in two different orders, first $\lambda_1 V_1$ and then $\lambda_2 V_2$, and then the other way around, to estimate how correct this approximation is. If the 1st order PT is really a good approximation, then the order of successive perturbations shouldn't matter.

 

[vanhees71]

Another trick is to write
$$\hat{H}_{\text{pert}}=\epsilon (\lambda_1 V_1 + \lambda_2 V_2)$$
and take $\epsilon$ as the "expansion parameter" and do perturbation theory, setting $\epsilon=1$ at the end of the calculation. Effectively, of course, you get an expansion in powers of $\lambda_1$ and $\lambda_2$.

 

[Amentia]

Thank you for your answers. If I understand correctly, the second method is to use the already established formulas such as the ones given here:

https://en.wikipedia.org/wiki/Pertu...cs)#Second-order_and_higher-order_corrections

e.g.:

$$E_{n}(\epsilon) = E_{n}^{(0)}+\epsilon\langle n^{(0)}|V|n^{(0)}\rangle+\epsilon^{2}\sum_{k\ne n} \frac{|\langle k^{(0)}|V|n^{(0)}\rangle|^{2}}{E_{n}^{(0)}-E_{k}^{(0)}}$$

and use $\epsilon=1$ and $V=\lambda_{1}V_{1}+\lambda_{2}V_{2}$ so that at first order, we just need to add the contributions of each perturbation but at higher order we may get terms proportional to $\lambda_{1}\lambda_{2}$.

And the first method is also to start with the same equations but using the energies and states corrected by the first perturbation as the unperturbed ones for the second one? And the two methods should at least agree at first order independently of the order in which we apply the perturbations as there cannot be any cross term such as $\lambda_{1}\lambda_{2}$.