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

AI Thread Summary
The discussion centers on a diatomic molecule in two dimensions, with a fixed nucleus at the origin, and explores the implications of rotational symmetry on the potential energy function. The paper asserts that if the potential remains unchanged under rotation, it must depend solely on the difference between the angles of the two nuclei, represented as V(r_1, r_2, θ_2 - θ_1). This leads to the conclusion that the potential can be expressed in terms of three variables: r_1, r_2, and φ, where φ is the angular difference. The question raised seeks clarification on the significance of these angles and their representation in the potential function. Understanding this relationship is crucial for grasping the underlying physics of the molecular interactions.
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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 (r_1,\theta_1) and (r_2,\theta_2) describe the locations of the nuclei, and let the molecule be subject to a potential V(r_1,r_2,\theta_1,\theta_2). The paper claims that, if the potential is the same when we rotate the entire molecule, we must have
<br /> V(r_1,r_2,\theta_1,\theta_2) = V&#039;(r_1,r_2,\theta_2-\theta_1);<br />
i.e., the potential only depends on the difference between \theta_1 and \theta_2. So the potential really only depends on three variables: r_1, r_2, and \phi, where \phi = \theta_2 - \theta_1. Can someone please explain why this is? I don't see it.
 
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