Say you have a Yukawa potential (a.k.a. screened coulomb potential) [tex]V(r) = -\frac{e^2}{r}e^{-rq}[/tex] where q is the inverse screening length, how would you find the critical q for having bound states? I'm working on reproducing N.F. Mott's argument about the critical spacing of a lattice of hydrogen atoms for a metal-insulator transition.(adsbygoogle = window.adsbygoogle || []).push({});

I realize any negative potential will have bound states, and the potential just as I have written it will have at least one bound state (i.e. if q goes to infinity the potential becomes a delta function well which always has one bound state), but my professor told me to derive the condition for no bound states (he said it was an 'elementary' quantum mechanics exercise).

I found one site http://farside.ph.utexas.edu/teaching/qm/lectures/node69.html that says that the criterion for a bound state is [tex] \frac{2m}{hbar^2}\frac{|V_0|}{q^2} > 2.7[/tex] and this gives the same answer that Mott got for the lattice spacing. I just don't know how to get this criterion in the first place.

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# Bound states of Yukawa potential

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