For a 3D Yukawa potential

$$ V(r) = - \lambda { e^{-Mr} \over r}. $$

Bargmann's upper bound can be read as necessary condition for the existence of at least one bound state; we want $N_l>1$ and from $$\int r V(r) > (2 l +1) N_l$$ we had (for the $l=0$ wave)

$$\frac \lambda M > 1$$

(in units where $1=2m=\hbar$). And it looks a bit poor limit in this case. Is there a better bound, preferably a sufficient and necessary condition, or at least a good numerical approximation?

  • $\begingroup$ I thought that Bargmann's bound is the best bound one may get, the exact condition! $\endgroup$ – Luboš Motl Sep 25 '15 at 13:16
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    $\begingroup$ Hmm @LubosMotl I believe it is the best bound for general V(r), but that for specific potentials it can be possible to do better (or not; as some potentials saturate it). $\endgroup$ – arivero Sep 25 '15 at 13:20
  • $\begingroup$ @Qmechanic thanks for the edition. I am going to put the "Bohr model" answer too, to clarify further the question. $\endgroup$ – arivero Sep 25 '15 at 17:20

Let me do a draft of the "Bohr model" answer that I think is failing in some O(1) factor, but still clarifies further the question. It is unclear how to find a bound for the existence of a bound state, but at least we can try for onset of the existence of a local minimum in the potential.

In Bohr's way, model states are simply the equilibrium between force and centrifugal term $$ V'(r) = {L^2 \over m r^3}$$

For enough nice -not a technical term... just nice in a lot of ways!- potentials, as we vary their parameters, the LHS will become equal to the RHS in some point, but furthermore they will be tangent. If you prefer, the function we could call $f'(r) \equiv RHS - LHS$ is everywhere nonzero when there is no bound state, and then at the onset it not only has $f'(r)=0$ but also $f''(r)=0$. So we consider also this second condition $$ V''(r) = {-3 L^2 \over m r^4}$$

which solves to either $r=0$ (and nothing can be said then, if the bound state starts from zero radius) or $$ r V''(r) + 3 V'(r) =0$$

Note that the "jump-in radius" is going to be in any case independent of $L$ and of the coupling constant. Particularly for the Yukawa potential

$$V'(r)= \lambda {e^{-Mr}\over r^2} (Mr +1)$$ $$V''(r)=- 2 \lambda {e^{-Mr}\over r^2} (\frac 12 M^2r^2 + Mr +1)$$

So the equation is $M^2r^2 - Mr -1 =0$ and it solves to $$Mr= \frac 12 (1 +\sqrt 5 )$$

Now, imposing this in the first equation we get, if we believe Bohr's $L=n\hbar$ $$ \lambda {e^{-\phi}} (\phi +1) = {(n\hbar)^2 \over m r}$$ and the onset condition for the existence of this state is, setting $n=1$ and QM units $\hbar=1=2m$, $$ {\lambda\over M} = {2 e^{+\phi}\over \phi (\phi +1)} \approx 2.38$$

I am not sure if this is right; as I said, there could be the case that the bound state orbit originates at $r=0$ as we vary the parameters of the potential, or I could have missed some of the -huge- differences between Bohr models and Schrödinger solution. We could also believe to Pauli's $L=\sqrt{l(l+1)}\hbar$. Also, note that this is really the onset for the extreme of the radial effective potential; what happens really at this point is an inflexion point which as we move $\lambda$, separates in a maximum outwards and a minimum inwards; eventually the minimum becomes a resonance and after crossing to negative values of energy it can be a bound state. But as we have really solved only for $l > 0$ I am afraid the bound is for the existence of the second state, not for the lowest energy one.

  • $\begingroup$ Lurking in numerical literature, it seems that 1.68 is a well known bound happening during computer simulations, (and already from Hulthen times). So this calculation is out by an $O(\sqrt 2)$ factor. $\endgroup$ – arivero Sep 25 '15 at 18:35
  • $\begingroup$ Googling around: I have found $\lambda {e^{-\phi}} (\phi +1) = {L^2 \over m r}$ in slide 23 of a talk of Sean Tulin research.dsu.edu/cetup/documents/2013-talks/dark-matter/… about Dark Matter. And a role of $\phi$ in Yukawa onsets is mentioned in arxiv.org/abs/0811.0287v1 but it is unclear if it is the same method. $\endgroup$ – arivero Sep 26 '15 at 1:08

Bennett, Herbert S. "Upper Limits for the Number of Bound States Associated with the Yukawa Potential" Journal of Research of the National Bureau of Standards, Vol. 86, No.5, September-October 1981 (PDF):

The number of bound-state solutions of the Schrodinger equation for the screened Coulomb potential (Yukawa potential), $-(C/r) \exp(-\alpha > r)$, occurs frequently in theoretical discussions concerning, for example, gas discharges, nuclear physics, and semiconductor physics. The number of bound states is a function of $(C/\alpha)$. Three upper limits for the number of bound states associated with the Yukawa potential are evaluated and compared. These three limits are those given by Bargmann, Schwinger, and Lieb. In addition, the Sobolev inequality states that whenever $(C/\alpha) < 1.65$ no bound state occurs. This agrees to within a few percent of the numerical calcuiations of Bonch-Bruevich and Glasko. The Bargmann and Lieb limits and the Sobolev inequality are substantially easier to evaluate than the Schwinger limit. Among the three limits, the Schwinger limit gives the most restrictive limit for the existence of only one bound state and, therefore, is the best one to use for the approach to no binding, i.e., $1.65 < (C/\alpha) < 1.98$. The Lieb limit is the best among the three when $(C/\alpha) > 1.98$. The Bargmann limit is the least restrictive.

(Emphasis mine.)

For more details, this may also be useful:

Brau, Fabian and Calogero, Francesco. "Upper and lower limits on the number of bound states in a central potential". Journal of Physics A: Mathematical and General, Volume 36, Number 48, 19 November 2003.


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