134
4 The Treatment of Few-Body Reactions
r
Fig. 4.8 A sketch of a simplified semiclassical model with the r being the translational degree
of freedom and α the angle of the bound motion whose action is represented as circular cuts of
the reaction channel. The distortions occurring during the advance inside the tube together with
the differences in the cuts generate a mismatch between the asymptotic fluxes and the one at the
junction between the reactant (LHS) and the product (RHS) half-channels
H =
1
2μ
p
2
+ H o (() + V ((, α, r )
(4.62)
where α is the angle of the action-angle ((, α) pair of conjugated variables for the
internal motion ( p, r ). Following Ref. [66], the wavefunction may be expressed in
the JWKB form as
ψ(r, α) = exp [i W (α, r/)]
(4.63)
and the Hamiltonian has the Taylor expansion
H o (() + V ((, α, r ) =
k
h k (r, α)I
k
.
(4.64)
Accordingly also W can be expanded in Taylor series W = W o +
k
k W k . In the
0 order, the wavefunction may be expressed as
ψ(r, α) = (2πv)
−1/2
(∂α 1 /∂α) n 1
1/2 exp [i W o (α, r )/]
(4.65)
or by transforming Eq. (4.65) in the asymptotic regions from (α, r ) to a new (α, t)
representation (α = α − ωt = α − ωμr/ p with ω being ∂ H o /∂ I and α a constant)
that by the comparison with the standard asymptotic form
ψ n 1 E (α, t)
t→∞
∼ (2π)
−1/2
n 2
S n 1 n 2 exp [in 2 α − i Et/]
(4.66)