I was reading a lecture note, where it talks about the two-body bound state in 1D, 2D and 3D. It says that, in 1D or 2D situation, two particles with any arbitrary attractive interaction, can form a bound state, while in 3D, there appears a threshold for this attraction to form a bound state.
I did not find a very direct and understandable document for this.. if anyone can explain to me briefly or has any good materials on this, please help! Thank you very much!
I would guess the notes mean that in 3D we expect a central force to vary with distance as:
$$ F = \frac{k}{r^2} $$
for some value of the constant $k$ that depends on the details of the force. To get the energy needed to remove the bound object to infinity we simply integrate this to get:
$$ V(R) = \int_R^\infty \frac{k}{r^2} = \frac{k}{R} $$
So any object with a kinetic energy greater than $k/R$ is not bound. This is the sense in which there is a threshold.
In 2D we expect the force to vary as:
$$ F = \frac{k}{r} $$
and if we attempt to integrate this to calculate the energy to remove the particle to infinity we get:
$$ V(R) = \int_R^\infty \frac{k}{r} = \left[k \ln(r) \right]_R^\infty $$
and this is infinite. So regardless of the value of $k$ in our equation no object can ever have enough kinetic energy to escape to infinity - there is no binding theshold. In 1D the force is just a constant and does not depend on $r$ so again we get an infinite binding energy.
