Generalized Angular Momentum in Many-Body Collisions

Abstract

With short-range forces, initial and final states in a classical 3-body collision are straight-line trajectories into and out of a region where all three particles are close together at the same time. Using six coordinates, three describing the relative position of a pair of particles, and three the relative position of the third particle and the center of mass of the pair, the condition for simultaneous togetherness can be expressed with the help of the 6×6 grand angular momentum tensor, $\Lambda$, whose components are ${\Lambda }_{\mathrm{ij}}={\left(\frac{m_i}{m_j}\right)}^{\frac{1}{2}}{x}_i{p}_j-{\left(\frac{m_j}{m_i}\right)}^{\frac{1}{2}}{x}_j{p}_i$. For a close 3-body collision ${\Lambda }^2=\frac{1}{2}\Sigma {i,j}^{}{{\Lambda }_{\mathrm{ij}}}^2$ must be small. ${\Lambda }^2$ commutes with the ordinary angular momentum operators and with the kinetic energy; its eigenvalues are $\lambda (\lambda +4){\hslash }^2$, with integral $\lambda$, and its eigenfunctions hyperspherical harmonics. Initial and final 3-body states can be described quantally by the total energy $E$, ${\Lambda }^2$, and a commuting set of ordinary angular momenta; this description has the same relation to a momentum representation as the ordinary angular momentum analysis has for a 2-body collision. A collision of ($N+1\mathrm{}$) particles can be described by using a hierarchy of operators ${{\Lambda }_n}^2(2\mathrm{}<~n<~N)$; their eigenvalues are ${\lambda }_n({\lambda }_n+3n-2){\hslash }^2$.