Finite threshold value for two-body binding

Question

In his book Introduction to Superconductivity, Second Ed., Chapter 3.1, when discussing Cooper's 1956 paper and the formation of Cooper pairs, Tinkham states

[...] it is well known that binding does not ordinarily occur in the two-body problem in three dimensions until the strength of the potential exceeds a finite threshold value.

I'm having trouble trying to understand what the author meant.

Take the hydrogen atom for example. If we rescale the potential strength by a positive quantity $\lambda$: $U(r)\rightarrow\lambda U(r)$, then the energy levels will just be rescaled according to $E_n \rightarrow \lambda E_n$. This means there will still be bound levels for any positive $\lambda$, no matter how small.

• What did the author mean, then? Was he referring strictly to pairs of identical particles? Does it have something to do with Cooper's approximation of the attractive potential as a constant?

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@KevinKostlan 's answer below is very enlightening. However, as stated in the bounty, I'm still looking for a concrete source on the subject.

Answer 1

He is referring to short-ranged interactions that drop off much more steeply than inverse-square law.

The standard particle-in-a-box has length L and infinite depth. But for 3D boxes with finite "depth" $E_{box}$ we don't always have bound states. Why is this?

The strictly mathematical answer is "there is no eigensolutions with energy < $E_{box}$". But this can also be seen from a scaling argument:

It takes energy to confine a particle to mostly fit into a region. The order of magnitude energy cost is $E_{confine}$~$\frac {h^2} {2m\lambda^2}$ in any dimension, where $\lambda$ is the size in which most of the particle is found. This is balanced against how much of the particle is actually in the box and the depth of the box: $E_{bind}$~$E_{box}min(1,L/\lambda^{D})$, where D is the dimension. In three dimensions, doubling $\lambda$ amounts to spreading the wavefunction and corresponding probability distribution out in all three axes. In 1D, there will always be bound states. In 2D it is unclear under this crude scaling law if there are always bound states. However, in 3D if the box is too small or too shallow, no bound states can be found because $E_{bind}$ drops off too quickly as $\lambda$ increases.

For atoms, the potential well is inverse-linear (inverse square force) rather than a fixed box. This makes the scaling law $E_{bind}$~$Z/\lambda$ and guarantees bound states no matter how weak the binding forces are. However, for other systems we may have sharper drop-offs. For two helium atoms they are attracted by an extremely weak van der Waals interaction which also drops off very quickly. In this case bound states only exist if the two atoms are both Helium-4 atoms. The case of interactions between electrons in a lattice is similarly short-ranged.