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Homework Help: Diatomic molecule Hamiltonian

  1. Apr 27, 2012 #1
    1. The problem statement, all variables and given/known data
    I have to find the hamiltonian for a diatomic molecule, where the molecule can only rotate and translate and we supose that potencial energy doesn't change.


    2. Relevant equations



    3. The attempt at a solution

    Okey so I used Spherical coordinate system such as the kinetic energy of the molecule is
    [tex]T=\frac{1}{2}(m\dot{r}^2+I(\dot{\phi}^2+I(\dot{θ})^2)=L[/tex]

    To find the Hamiltonian i've considered:

    [tex]P_r=m\dot{r} \Rightarrow \dot{r}=\frac{P_r}{m}[/tex]

    And so on for the other momentum so the final solution for the hamiltonian is:

    [tex]H=\frac{1}{2}(\frac{P_r^2}{m}+\frac{P_\phi^2}{I}+ \frac{P_θ^2}{I})[/tex]

    BUT the correct solution given by my professor is:
    [tex]H=\frac{1}{2}(\frac{P_r^2}{m}+\frac{P_\phi^2}{I \sin^2\theta}+ \frac{P_θ^2}{I})[/tex]

    So I don't know why is there a sin^2\theta factor.
     
    Last edited: Apr 27, 2012
  2. jcsd
  3. Apr 27, 2012 #2

    vela

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    The problem lies in where you started. How'd you come up with this?
     
  4. Apr 27, 2012 #3
    Yep I already solved the problem. You were right Vela i started wrong. I divided the Hamiltonian between the lagrangian for the translation kinetic energy and the rotation kinetic energy so:
    [tex]T_t=\frac{1}{2}M(\dot{x}^2+\dot{y}^2+\dot{z}^2)= \frac{1}{2M} (P_x^2+P_y^2+P_z^2)[/tex]

    And then the rotation one:
    [tex]T_{rot}=\frac{1}{2}Mr^2[\dot{\theta^2}+\dot{\phi}^2\sin^2(\theta)][/tex]

    And then just like before i get the hamiltonian.

    Thank you for the response and sorry for my english ;)
     
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