Potential Energies: Same Value, Different Minima?

[Physicslad78]

Hi guys... I have a small question on potential energies:
I have got two potential energies: $$\begin{equation}<br /> U_1=-\frac{k^2}{2}+\frac{w\sqrt{3}}{2}\sin^2\theta \cos 2 \phi<br /> \end{equation}$$

and $$\begin{equation}<br /> <br /> U_2==-\frac{k^2}{2}+\frac{w\sqrt{3}}{2}\sin^2\theta \sin 2 \phi<br /> \end{equation}$$
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

 

[alxm]

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.

 

[Physicslad78]

Oh Yeah..True! thanks a lot alxm...But i presume they would not be equivalent to
$$\begin{equation}<br /> U_3=-\frac{k^2}{2}+\frac{w\sqrt{3}}{2}\sin 2\theta \cos\phi\end{equation}$$
?

 

[alxm]

Well, then you've scaled a coordinate. Could be either a different physical system or a different coordinate system.

 

[Physicslad78]

yep..thanks a lot..One final question..If I want to write $$\sin^2\theta \sin 2 \phi$$ in terms of spherical Harmonics..I think these are related to the $$Y_{2,-2}$$ and $$Y_{2,2}$$ spherical Harmonics but there will be an $$i$$ 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 :
$$\begin{equation}<br /> H=i~w~\sqrt{\frac{2\pi}{5}}~ (Y_{2,-2}-Y_{2,2}).<br /> \end{equation}$$

Thanks again