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Dependence of potential on only the *difference* between two variables

  1. Aug 5, 2009 #1
    I'm reading a paper that considers a diatomic molecule living in two dimensions in which the central nucleus is fixed at the origin. Ignore the electrons for the time being. Let [itex](r_1,\theta_1)[/itex] and [itex](r_2,\theta_2)[/itex] describe the locations of the nuclei, and let the molecule be subject to a potential [itex]V(r_1,r_2,\theta_1,\theta_2)[/itex]. The paper claims that, if the potential is the same when we rotate the entire molecule, we must have
    V(r_1,r_2,\theta_1,\theta_2) = V'(r_1,r_2,\theta_2-\theta_1);
    i.e., the potential only depends on the difference between [itex]\theta_1[/itex] and [itex]\theta_2[/itex]. So the potential really only depends on three variables: [itex]r_1[/itex], [itex]r_2[/itex], and [itex]\phi[/itex], where [itex]\phi = \theta_2 - \theta_1[/itex]. Can someone please explain why this is? I don't see it.
  2. jcsd
  3. Aug 8, 2009 #2
    what are the angle representative of?
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