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Potential energies

  1. Jun 2, 2009 #1
    Hi guys.... I have a small question on potential energies:
    I have got two potential energies: [tex]
    \begin{equation}
    U_1=-\frac{k^2}{2}+\frac{w\sqrt{3}}{2}\sin^2\theta \cos 2 \phi
    \end{equation}
    [/tex]

    and [tex]
    \begin{equation}

    U_2==-\frac{k^2}{2}+\frac{w\sqrt{3}}{2}\sin^2\theta \sin 2 \phi
    \end{equation}
    [/tex]
    where k is a constant and 0<theta<pi and 0<phi<2 pi. I minimized both of these and found that say for k=1, w=0.5 both U1 and U2 have the SAME value (-0.9333 I guess) but DIFFERENT minima....Does it mean that the two potentials represent the same physics or could the physical situations corresponding to both be different?


    Thanks
     
    Last edited: Jun 2, 2009
  2. jcsd
  3. Jun 2, 2009 #2

    alxm

    User Avatar
    Science Advisor

    A cosine changes to a sine, so that could be viewed as corresponding to identical physical systems, with a coordinate system rotated by 90 degrees.
     
  4. Jun 2, 2009 #3
    Oh Yeah..True!! thanks a lot alxm...But i presume they would not be equivalent to
    [tex]
    \begin{equation}
    U_3=-\frac{k^2}{2}+\frac{w\sqrt{3}}{2}\sin 2\theta \cos\phi\end{equation}[/tex]
    ?
     
  5. Jun 2, 2009 #4

    alxm

    User Avatar
    Science Advisor

    Well, then you've scaled a coordinate. Could be either a different physical system or a different coordinate system.
     
  6. Jun 2, 2009 #5
    yep..thanks a lot..One final question..If I want to write [tex] \sin^2\theta \sin 2 \phi [/tex] in terms of spherical Harmonics..I think these are related to the [tex] Y_{2,-2} [/tex] and [tex] Y_{2,2} [/tex] spherical Harmonics but there will be an [tex] i [/tex] appearing and this term will be a part of a Hamiltonian so I will end up with complex energies! Is there a way out of this. In fact the Hamiltonian I get is :
    [tex]
    \begin{equation}
    H=i~w~\sqrt{\frac{2\pi}{5}}~ (Y_{2,-2}-Y_{2,2}).
    \end{equation}
    [/tex]

    Thanks again
     
    Last edited: Jun 2, 2009
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