Coupled quantum harmonic oscillators

[dflake]

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

 

[azntoon]

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.

 

[denian]

imitri

Hi Dimitri,

Thank you for sharing your exercise with us. Coupled quantum harmonic oscillators are a common problem in quantum mechanics and have many applications in various fields such as quantum computing and quantum chemistry.

The Hamiltonian you have provided represents the total energy of the two coupled oscillators, with q1 and q2 being the position operators and m, w, lambda, and mu being constants. The unitary transformation you have found helps in decoupling the problem, making it easier to solve. However, as you have mentioned, you are also asked to use the Adiabatic Method to find approximate solutions for the eigenfunctions and eigenvalues of the Hamiltonian.

The Adiabatic Method is a technique used to solve time-independent problems in quantum mechanics. It involves slowly changing the parameters of the Hamiltonian while keeping the system in its ground state. In your case, the parameters are lambda and mu, and the Hamiltonian is time-independent. This means that the Adiabatic Method can be applied to find approximate solutions for the eigenfunctions and eigenvalues.

To use the Adiabatic Method, you can start with a simple Hamiltonian where lambda and mu are equal to zero. Then, slowly increase the values of lambda and mu while keeping the other parameters constant. As you increase the values, the Hamiltonian will start to resemble the original one, and the system will remain in its ground state. This will give you an approximate solution for the eigenfunctions and eigenvalues of the original Hamiltonian.

The values of lambda and mu that allow you to use the Adiabatic Method can be determined by looking at the energy gap between the ground state and the first excited state. If this gap is significantly larger than the rate at which you are changing the parameters, then the Adiabatic Method can be applied.

I hope this helps. If you need further assistance, I would recommend looking into books on quantum mechanics that cover the Adiabatic Method, such as "Quantum Mechanics" by David J. Griffiths or "Quantum Mechanics: Concepts and Applications" by Nouredine Zettili.

Best of luck with your exercise!