Time-dep perturbation theory

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SUMMARY

The discussion centers on the implications of vanishing diagonal elements of the perturbation Hamiltonian \( H' \) in time-dependent perturbation theory, as outlined in Griffiths' textbook. Participants explore the conditions under which the matrix elements \( \langle a|H'|a\rangle = 0 \) hold true, particularly when the potential \( V(x) \) lacks angular dependence. The conversation highlights the significance of selection rules and their role in determining the physical meaning of these vanishing elements, emphasizing that non-zero diagonal elements would indicate a direct perturbation to the eigenvalue or eigenstate.

PREREQUISITES
  • Understanding of quantum mechanics principles, particularly perturbation theory.
  • Familiarity with matrix mechanics and Hamiltonians in quantum systems.
  • Knowledge of selection rules and their application in quantum transitions.
  • Basic comprehension of the implications of potential functions in quantum mechanics.
NEXT STEPS
  • Study Griffiths' "Introduction to Quantum Mechanics" for a deeper understanding of perturbation theory.
  • Research the implications of selection rules in quantum mechanics, focusing on angular momentum considerations.
  • Explore the physical significance of diagonal and off-diagonal matrix elements in quantum systems.
  • Investigate time-dependent perturbation theory applications in various quantum mechanical scenarios.
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Students and researchers in quantum mechanics, physicists interested in perturbation theory, and educators looking to clarify concepts related to Hamiltonians and selection rules.

omyojj
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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..
 
Physics news on Phys.org
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.
 

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