Time independent perturbation theory in atom excitation

In summary, Griffiths chapter on Time independent perturbation theory discusses the calculation of first order contributions to the electron Hamiltonian in an atom, taking into account magnetic dipole and electric quadrupole excitations in addition to electric dipole excitations. While these additional excitations may be smaller in magnitude, they can still contribute significantly to the overall result. Therefore, it may be necessary to go to higher order in perturbation theory for the dipole part in order to accurately include these contributions. However, this is not addressed in Griffiths' exercise and is left up to the reader's discretion.
  • #1
Malamala
299
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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!
 
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  • #2
Malamala said:
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.
 

Related to Time independent perturbation theory in atom excitation

What is time independent perturbation theory?

Time independent perturbation theory is a mathematical method used in quantum mechanics to calculate the effects of small perturbations on the energy levels of a system. It allows us to approximate the energy levels of a perturbed system by using the known energy levels of an unperturbed system.

How is time independent perturbation theory used in atom excitation?

In atom excitation, time independent perturbation theory is used to calculate the energy levels and transition probabilities of an atom when it is perturbed by an external electric or magnetic field. This allows us to understand and predict how atoms will behave in different environments.

What are the assumptions made in time independent perturbation theory?

There are several assumptions made in time independent perturbation theory, including the assumption that the perturbation is small compared to the unperturbed system, and that the perturbation is turned on and off slowly. Additionally, it assumes that the perturbation is time independent, meaning that it does not change over time.

What are the limitations of time independent perturbation theory?

While time independent perturbation theory is a useful tool in understanding the effects of small perturbations on a system, it is limited in its ability to accurately predict the behavior of highly perturbed systems. It also does not account for the effects of time-dependent perturbations or non-linear systems.

How does time independent perturbation theory relate to other perturbation methods?

Time independent perturbation theory is one of several perturbation methods used in quantum mechanics, including time-dependent perturbation theory and variational perturbation theory. These methods all use different approaches and assumptions to approximate the behavior of a perturbed system, and can be used in conjunction with each other to improve accuracy.

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