Time independent perturbation theory in atom excitation

[Malamala]

Hello! In Griffiths chapter on Time independent perturbation theory, he has a problem (9.20) in which he asks us to calculate the first order contribution to the electron Hamiltonian in an atom if one takes into account the magnetic dipole/electric quadrupole excitations, beside the electric dipole, which he derives in the main text. As far as I understand, these excitations are smaller compared to the dipole excitation. So if I want to include them in the calculations (to first order in the perturbation theory), do I need to go to higher order in the perturbation theory for the dipole part, or first order is still enough? Basically my question is, can second (or higher) order (in perturbation theory) approximation of the electric dipole transitions, give a similar contribution to first order (in perturbation theory) approximation coming from the magnetic dipole/electric quadrupole transitions? For example (assuming random units), if the first order correction to the dipole part is $H^1_d=1$ and the second order is $H^2_d=0.01$ while the first order correction to the quadrupole part is also $H^1_q=0.01$, I would expect that one needs to go to second order in the dipole part, such that the perturbing hamiltonians of the same order of magnitude to be all included. Is this right? Thank you!

 

[A. Neumaier]

if the first order correction to the dipole part is $H^1_d=1$ and the second order is $H^2_d=0.01$ while the first order correction to the quadrupole part is also $H^1_q=0.01$, I would expect that one needs to go to second order in the dipole part, such that the perturbing Hamiltonians of the same order of magnitude to be all included. Is this right? Thank you!

Yes, but that's not part of Griffith's exercise. He asks about first order contributions, independent of an assessment of their accuracy.