Coupled quantum harmonic oscillators

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SUMMARY

The discussion centers on solving a problem involving coupled quantum harmonic oscillators with the Hamiltonian defined as H = 1/2 (m w^2 q1^2 + m mu^2 w^2 q2^2 + m lambda^2 w^2 q1 q2). The user initially struggled with applying the Adiabatic Method due to the Hamiltonian's time independence but later clarified that the Adiabatic Theorem is applicable because one eigenvalue of the Hamiltonian is time-dependent, influenced by parameters lambda and mu. This insight resolves the confusion regarding the application of the Adiabatic Method in this context.

PREREQUISITES
  • Understanding of quantum mechanics principles, specifically Hamiltonians.
  • Familiarity with the Adiabatic Theorem in quantum mechanics.
  • Knowledge of coupled oscillators and their mathematical representation.
  • Experience with unitary transformations in quantum systems.
NEXT STEPS
  • Study the Adiabatic Theorem in detail, focusing on its applications in quantum mechanics.
  • Explore the mathematical techniques for decoupling coupled quantum systems.
  • Investigate the role of time-dependent parameters in quantum Hamiltonians.
  • Review literature on quantum harmonic oscillators, particularly in the context of time-dependent perturbations.
USEFUL FOR

Quantum physicists, graduate students in quantum mechanics, and researchers exploring the dynamics of coupled quantum systems will benefit from this discussion.

dflake
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Hi folks,
I have to solve an exercise about two oscillators whose Hamiltonian is
H = 1/2 (m w^2 q1^2 + m mu^2 w^2 q2^2 + m lambda^2 w^2 q1 q2)
I successfully found the unitary transformation that decouples the problem, but I am also asked to use the Adiabatic Method to find approximate solutions of the eigenfunctions and eigenvalues of H, and find the values of lambda and mu that allow one to use such a method.
As the Hamiltonian is time independent, I don't understand how should I apply the adiabatic scheme here.
Does anyone have hints? Is this problem treated in any book you know?
Thanks a lot,
D
 
Physics news on Phys.org
avide. §§ COMUpdateWell, I found the answer to my own question. Adiabatic Theorem can be applied here as one of the eigenvalues of the Hamiltonian is time-dependent (the frequency of the oscillator which depends on lambda and mu).Hope this helps someone else. Davide.
 

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