One dimensional scattering of a two body interacting system by an infinite wall

The one-dimensional scattering of a two body interacting system by an infinite wall is studied in a quantum-mechanical framework. This problem contains some of the dynamical features present in the collision of atomic, molecular and nuclear systems. The scattering problem is solved exactly, for the case of a harmonic interaction between the fragments. The exact result is used to assess the validity of two different approximations to the scattering process. The adiabatic approximation, which considers that the relative co-ordinate is frozen during the scattering process, is found to be inadequate for this problem. The uncorrelated scattering approximation, which neglects the correlation between the fragments, gives results in accordance with the exact calculations when the scattering energy is high compared to the oscillator parameter.

💡 Research Summary

The paper investigates the quantum mechanical scattering of a two‑particle bound system from an infinitely hard wall in one dimension. The two particles interact via a harmonic oscillator potential, (v(r)=\frac{1}{2}\mu\omega^{2}r^{2}), where (r) is the relative coordinate and (\mu) the reduced mass. Each particle also feels an infinite‑wall potential, (v_i(r_i)=0) for (r_i>0) and (\infty) for (r_i\le 0). By transforming to the relative coordinate (r) and the center‑of‑mass coordinate (R), the Hamiltonian separates into an internal harmonic‑oscillator part (\hat H_r) and a free‑particle part (\hat H_R). The internal eigenstates are the usual harmonic‑oscillator wavefunctions (\phi_n(r)) with energies (\epsilon_n=(n+\tfrac12)\hbar\omega). Energy conservation links each internal state to a center‑of‑mass momentum (K_n) via (\frac{\hbar^2K_n^2}{2M}+\epsilon_n=E), where (E) is the total incident energy and (M) the total mass.

The total wavefunction is expanded as a superposition of incoming, reflected (open channels) and exponentially decaying (closed channels) components: \