Bound states of Yukawa potential

[drummerguy]

Say you have a Yukawa potential (a.k.a. screened coulomb potential) $$V(r) = -\frac{e^2}{r}e^{-rq}$$ 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.

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 $$\frac{2m}{hbar^2}\frac{|V_0|}{q^2} > 2.7$$ 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.

 

[SpectraCat]

Say you have a Yukawa potential (a.k.a. screened coulomb potential) $$V(r) = -\frac{e^2}{r}e^{-rq}$$ 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.

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 $$\frac{2m}{hbar^2}\frac{|V_0|}{q^2} > 2.7$$ 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.

So can't you just solve the TISE for the potential you gave, and figure out the ZPE in terms of the particle mass and the parameters of the potential? You may have to make some simplifying assumptions that make sense in the limit you want to test, but I think it is doable. Then it should be trivial to find the minimum value of q that allows a single bound state.