Friday, July 16, 2010

Sum Rules

I just skimmed over the paper by Stavros Fallieros. In those 15 minutes, I learned something new, which I shared in an email to my students. Below is the email, with typos and all.

I have just read the papers sent to me by Shoresh on the rigid rotator. My analysis of those papers in light of our work is as follows.

These papers show that the sum rules cannot be violated. As we know, for a given potential, there can be both bound states and free states. Since the sum rules require one to sum over all these states, summing over just a subset gives the wrong result.

In the case of a rigid rotator, the wavefunctions are confined to a wire loop (or attached with a rigid rod, the problem is equivalent). If one starts with a wire loop that is represented by a non-infinite potential so that the wavefunction can spill out of the loop, there are free and bound states. In this case, it is simple to sum over all states and the sum rules are obeyed yielding unity for the (0,0) sum rule.

When the electron is confined to the wire, which can be approximated as the limiting case of a delta function shell, the free states are pushed away to infinite energy. If one includes these infinite-energy states in the sum rules, all is well and the (0,0) sum rule is still obeyed. However, when not accounting for the free states by use of the limiting case, the free states are missed and the sum rules give 2/3 instead of 1. So, the problem is not that the sum rules need to be generalized or that they are being violated, but that we are not taking into account all the states as required by the sum rules. The easiest way to deal with this issue is to just throw in the factor of 1/3 to account for the bound states at infinite energy, which are otherwise difficult determine when starting from the idealized quantum wire.

Now to Shoresh's problem. In our calculations, we get the result that just two terms exceed the sum rules. Since all of the terms are positive definitive, there is no way that missed states can fix the problem. They will only make the sum rule violation problem bigger. So, I still think that there is a problem with the calculation that needs to be traced. In the end, when applying the sum rules, we should get a result that is less than unity.

Once we are certain that our formulation is correct, it will be interesting to see how the issues with infinite energy states affect the off-diagonal sum rules. The upshot is that if the off-diagonal sum rules give nonzero sums, then the dipole-free expression will give the wrong result. However, if we can determine the effect of the infinite energy states on the sum rules, we can reformulate the dipole-free form to make it work in these special cases.

We still need to fix the betaDF and betaSOS code. Perhaps Julian can concentrate on this problem while Shoresh works on the sum rules.

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