Time-dep perturbation theory

[omyojj]

while I`m reading the griffiths` textbook..

got my curiosity from "Typically, the diagonal matrix elements of H` vanish"

i.e. <a|H`|a>=0 in general..

If V(x) does not have an angular dependence..

the selection rule implies <a|H`|a>=0 (since Δl=0)..yes..

but what if it does?

what it(vanishing diagonal element of H`) means
physically (in view of the perturbation theory)

sorry for my bad english..

 

[Dr Transport]

If the diagonal elements of the perturbation didn't vanish, then they wouldn't perturb the system but be a small addtion to the unperturbed eigenvalue or another eigenstate.

 

[denian]

Time-dependent perturbation theory is a powerful tool used in quantum mechanics to study the behavior of a system under the influence of a time-dependent external potential. It allows us to calculate the changes in the system's energy levels and wavefunctions as a result of this external perturbation.

In the context of your question, the statement "the diagonal matrix elements of H' vanish" means that the perturbation operator H' has no effect on the diagonal elements of the Hamiltonian matrix. This is because the diagonal elements represent the energy of the system in its unperturbed state, and the perturbation operator only causes changes in the off-diagonal elements.

If the potential V(x) does not have an angular dependence, then the selection rule <a|H'|a>=0 applies, meaning that the perturbation will not cause any changes in the energy levels of the system. However, if the potential does have an angular dependence, then the selection rule may not apply and the diagonal elements of H' may not necessarily vanish.

In terms of the physical interpretation, a vanishing diagonal element of H' means that the perturbation does not affect the energy of the system in its unperturbed state. This could be due to symmetries in the system or in the perturbation potential, which prevent any changes in the energy levels.

I hope this helps clarify the concept of time-dependent perturbation theory and its applications. Your English is perfectly fine, and keep up the curiosity and interest in physics!