Coupled Quantum Harmonic Oscillator

In summary, the problem is to transform the Hamiltonian of a coupled Harmonic Oscillator into the sum of two decoupled Hamiltonians using a variable transformation. The equations used are H = H1 + H2 + qxy, where H1=0.5*m*omega^2*x^2+0.5m^-1P_x^2 and H2=0.5*m*omega^2*y^2+0.5m^-1P_y^2, and q is the coupling constant. The attempt at a solution involved trying various variable transformations and completing the square, but without success in getting rid of the coupled term. The correct approach is to use a rotation of coordinates. Any help in getting
  • #1
stumpedstuden
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Homework Statement


I need to transform the Hamiltonian of a coupled Harmonic Oscillator into the sum of two decoupled Hamiltonians (non-interacting oscillators).


Homework Equations


H = H1 + H2 + qxy, where H1=0.5*m*omega^2*x^2+0.5m^-1P_x^2 and H2=0.5*m*omega^2*y^2+0.5m^-1P_y^2, and q is the coupling constant


The Attempt at a Solution


I have tried a number of variable transformations, etc as well as attempted to complete the square to deive the proper variables that will allow me to rewrite the Hamiltonian properly. All without success. I thought completing the square and making the proper substitions would work but I still end up with a coupled term. Once this step is complete solviing the rest of the problem should be pretty straightforward.

Any help that gets me started would be greatly appreciated.
 
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  • #2
To get rid of the cross term, you don't want to complete the square; you need to use a rotation of the coordinates.
 

What is a coupled quantum harmonic oscillator?

A coupled quantum harmonic oscillator is a system in quantum mechanics where two or more quantum harmonic oscillators are linked together, meaning their motion and energy are dependent on each other.

How does the coupling between quantum harmonic oscillators affect their behavior?

The coupling between quantum harmonic oscillators can lead to interesting phenomena such as energy exchange, synchronization, and entanglement. It can also affect their individual energy levels and frequencies.

What are the applications of coupled quantum harmonic oscillators?

Coupled quantum harmonic oscillators have applications in various fields such as molecular physics, condensed matter physics, and quantum computing. They can also be used to model interactions between different particles in quantum systems.

What is the difference between coupled and uncoupled quantum harmonic oscillators?

In an uncoupled quantum harmonic oscillator system, each oscillator behaves independently and its energy levels and frequencies are not affected by other oscillators. In contrast, coupled quantum harmonic oscillators have a mutual influence on each other's motion and energy.

How are coupled quantum harmonic oscillators described mathematically?

Coupled quantum harmonic oscillators are typically described using Hamiltonian equations, which involve the position and momentum operators for each oscillator. The equations can be solved using perturbation theory or numerical methods to determine the behavior of the oscillators.

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