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