Spinless particles scattering through states of
J=0,1, or 2 (from the problem set)
For spinless particles A and B in two to two scattering A+B→C→A+B through the state
C with J=0, 1, or 2, the general result for the scattering amplitude specializes to
€
T
θ
,
ϕ
( )
=TJP
Jcos
θ
( )
J
∑
. (This was given in a handout. The energy dependence in T and
TJ is suppressed.) Thus if a single J channel dominates, we have
€
d
σ
dΩ∝P
Jcos
θ
( )
2
, and
the angular distribution is 1, cos2θ, and [(3/2)cos2θ−1/2]2 for J=0, 1, and 2, respectively.
For J ≠ 0, this is not spherically symmetric because not all the J3 states are available. In
particular, spinless particles approaching along the 3-axis can have J3=0 only.