Time-dependent perturbation theory

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SUMMARY

The discussion centers on time-dependent perturbation theory in quantum mechanics, specifically addressing the representation of perturbed wavefunctions as linear combinations of unperturbed eigenstates. Participants clarify that while the perturbation may alter energy levels, the coefficients in the linear combination represent the probability amplitudes for finding a particle in specific states. The conversation highlights the practical applications of this theory, such as calculating transition rates in systems influenced by oscillating electromagnetic fields, and emphasizes the importance of understanding the time-dependent nature of eigenfunctions in quantum systems.

PREREQUISITES
  • Understanding of quantum mechanics principles, particularly wavefunctions and eigenstates.
  • Familiarity with perturbation theory in quantum mechanics.
  • Knowledge of Hamiltonians and their role in quantum systems.
  • Basic concepts of probability amplitudes and quantum state measurements.
NEXT STEPS
  • Study the derivation and applications of Fermi's Golden Rule in quantum transitions.
  • Explore the implications of time-dependent Hamiltonians in quantum mechanics.
  • Learn about the mathematical formulation of perturbation theory and its practical applications.
  • Investigate the effects of oscillating electromagnetic fields on atomic electron orbitals.
USEFUL FOR

Quantum physicists, graduate students in physics, and researchers interested in the dynamics of quantum systems and perturbation theory applications.

  • #31
Niles said:
...Why is it always that is says a small time-dependent perturbation?

Actually the external force or field can be strong, not small. The time-dependent solution exists and can still be represented as a superposition of old basis vectors with time-dependent coefficients.

In case of small perturbation these exact coefficients are well approximated with the perturbation series (a la Taylor series). The perturbation theory technique is developing these coefficients in Taylor series in powers of perturbation. Thus smallness of V(t) guarantees good accuracy of a truncated series (practical convergence).
 

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