Scattering of a weakly bound dimer from a hard wall in one dimension

We consider a dimer formed by two particles with an attractive contact interaction in one dimension, colliding with a hard wall. We compute the scattering phase shifts and the reflection coefficients for various collision energies and various mass ratios of the two particles. For low-energy collisions (with dimer kinetic energies much smaller than the binding energy) our results are consistent with those of D. Lee and M. Pine, The European Physical Journal A 47, 41 (2011). For mass ratios much greater than 1 we use the Born-Oppenheimer approximation to show that the scattering length and the effective range of the dimer-wall collision both depend logarithmically on the mass ratio. For collision energies much greater than the binding energy, the dissociation probability is inversely proportional to the square of the incident momentum of the dimer and we find the constant of proportionality analytically, and we use a semiclassical analysis to approximately derive the angular distribution" of the dissociated pair, where the angle" $θ$ depends on the ratio of the velocities of the two outgoing unbound particles.

💡 Research Summary

The paper investigates the quantum scattering of a weakly bound dimer—two distinguishable particles interacting via an attractive delta‑function potential—in one dimension when it collides with an impenetrable hard wall. The authors formulate the problem using center‑of‑mass (X) and relative (r) coordinates, with the total energy expressed as E = K²/(2M) − 1/(2µa²), where K is the incident center‑of‑mass momentum, M the total mass, µ the reduced mass, and a the 1D scattering length (binding length). The hard wall imposes Dirichlet boundary conditions Ψ(0,x₂)=Ψ(x₁,0)=0.

Two analytically solvable (integrable) cases are treated exactly by Bethe Ansatz: equal masses (m₁/m₂ = 1) and the ratio 3 (m₁/m₂ = 3). In both cases the reflection coefficient R = 1 for all incident momenta, indicating perfect elastic reflection even above the dissociation threshold. The corresponding dimer‑wall scattering lengths are a_R = a/2 (mass‑balanced) and a_R = 3a/4 (ratio 3), with zero effective range.

For arbitrary mass ratios the authors recast the Schrödinger equation into a Lippmann‑Schwinger integral equation for the diagonal wavefunction ψ(x)=Ψ(x,x) using a Green’s function constructed by the method of images. Numerical solution of this equation yields the reflection amplitude f(K) and the coefficient R = |f|². The results show that for K < K_th (the dissociation threshold) scattering is purely elastic (R≈1). For K > K_th the reflection coefficient drops, reaching a minimum that becomes deeper as the mass imbalance grows; for m₁/m₂≈75 the minimum essentially vanishes, indicating almost certain breakup. The phase shift δ, extracted from the asymptotic form of ψ, displays a rapid initial decline with K for large mass ratios and approaches –π/2 at high K for all ratios.

In the limit of a heavy particle (m₁≫m₂) the Born‑Oppenheimer (BO) approximation is employed. The light particle experiences an effective potential V_eff(x)=½ m₂⁻¹