sin2kRj

sin2kRj is the shorthand notation for the mathematical function sin(2 k R_j), where k is a wave number and R_j is a length scale associated with the j-th channel or region in a physical system. The subscript j indicates that multiple such radii or scales may be present in a problem with several channels or interaction regions.

In non-relativistic quantum mechanics, k is typically k = sqrt(2 m E)/ħ, with m the particle mass and E its energy. R_j may represent a channel radius in scattering theory, a boundary radius in a finite-range potential, or a characteristic distance in a multilayer model. The quantity sin(2 k R_j) is dimensionless and oscillates as energy changes, with period Δk determined by the relation Δk = π/(R_j) and ΔE through the dispersion between k and E.

Zeros occur when 2 k R_j = n π, i.e., k = n π / (2 R_j). The function is bounded between -1 and 1 and can be written as sin(2 k R_j) = 2 sin(k R_j) cos(k R_j).

In practice, sin(2 k R_j) can appear as a factor in approximate expressions for scattering amplitudes, phase shifts, or transition probabilities in models with finite-range interactions or channel decompositions (for example, in simplified treatments within R-matrix theory or optical models). Its exact role is context-dependent and depends on how R_j is defined in the chosen theoretical framework.