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Quantum Mechanics: Coupled Electric Harmonic Oscillators

  1. Jul 26, 2013 #1

    VVS

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    Hi

    I am doing this completely out of self interest and it is not my homework to do this.
    I hope somebody can help me.

    1. The problem statement, all variables and given/known data
    In the book Biological Coherence and Response to External Stimuli Herbert Fröhlich wrote a chapter on Resonance Interaction. Where he considers the interaction of two electric harmonic oscillations with frequencies [itex]\omega_{1}[/itex] and [itex]\omega_{2}[/itex]at a distance R with a coupling constant [itex]\gamma[/itex] which is proportional to [itex]\frac{1}{R^{3}}[/itex].
    I want to analyze this system quantum mechanically.


    2. Relevant equations

    The equations of motions are

    [itex]\ddot{q_1}+\omega_{1}^2 q_{1}=-\gamma q_{2}[/itex]

    [itex]\ddot{q_2}+\omega_{1}^2 q_{2}=-\gamma q_{1}[/itex]

    The Hamiltonian is given by

    [itex]H=\frac{1}{2}(\dot{q_1}^2+\omega_{1}^2 q_{1}^2)+\frac{1}{2}(\dot{q_2}^2+\omega_{2}^2 q_{2}^2)+\gamma q_{1}q_{2}[/itex]

    3. The attempt at a solution

    For a start we know Schrödinger's Equation:

    [itex]i \overline{h} \frac{\partial \Psi}{\partial t}=H\Psi[/itex]

    My problem is: Can I substitute the Hamiltonian into Schrödingers Equation?
    Because q1 and q2 are charges, aren't they?
     
  2. jcsd
  3. Jul 26, 2013 #2

    hilbert2

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    Science Advisor
    Gold Member

    The variables ##q_{1}## and ##q_{2}## are not charges, they are position coordinates of particles. The hamiltonian should be a function of canonical momenta ##p_{i}=\frac{\partial L}{\partial \overset{.}{q}_{i}}## and position coordinates ##q_{i}##, not a function of ##\overset{.}{q}_{i}## and ##q_{i}##. In this case, however it does not matter because here ##p_{i} = \overset{.}{q}_{i}##.

    To solve the problem, you first find the normal modes of the oscillation. You don't need any reference to quantum mechanics to do this. Then when you know the normal modes and can decouple the equations, you can quantize the modes as harmonic oscillators, changing classical position and momentum variables to corresponding operators.
     
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