4.3 Beyond Full Quantum Calculations
133
on a given function of r and has built in, therefore, a particular behavior in r (the
wavefunction dies away in the classically forbidden region and oscillates in the classically allowed one). This allows one to handle in terms of special functions the
problem of isolated turning points, potential wells, potential barriers, potential singularities, etc., and in terms of a family of quantum numbers the bound states. In
particular, the Bohr–Sommerfield rule can be adopted in order to quantize the action
(see Eq. (1.52)) associated with vibrations between the two turning points (say a and
b) as follows
=
b
a
p(x)dx = (v +
1
2
)h.
(4.57)
The area covered by the integrand of Eq. (4.57) indicates the phase space associated with the bound motion on the potential considered. In order to formulate the
semiclassical S matrix
4 when considering all the degrees of freedom in non-separable
processes (like in the case either of the inclusion of the electronic structure of the
colliding partners or of the reactive or nonreactive atom oscillating diatom collision), one naturally turns, as already illustrated in the previous subsections, into the
S matrix relating the family of events linking the initial state to the desired final
ones (each bearing in the semiclassical approach an associated phase determined by
the classical action defined in Chap. 1 accumulated along its path). For the sake of
simplicity, it is assumed that the (single) translational variable (in the equation below
and in Fig. 4.8) is r and that the related motion is coupled to a single internal degree
of freedom α that is the angle to which is associated the action integral (defined
as in Eq. (4.57))
4 The semiclassical connection between the deflection angle θ and the JWKB approximation to the
phase shift δ l can be obtained from the semiclassical formulation of the wavefunction
ψ l (r )
r →∞
∼ sin (kr − lπ/2 + δ l )
(4.58)
that gives
δ l
r →∞
∼
r
a
k l (r )dr − kr + (l + 1/2)π/2
,
( 4.59)
where k l (r ) is the Langer-corrected wavenumber function (resulting from the transformation of r
into e x and to the scaling of the wavefunction by e x/2 to the end of taking into account the singularity
occurring at θ = 0) that reads
k l (r ) =
k
2 − U (r ) −
(l + 1/2) 2
r 2
1/2
(4.60)
from which a comparison with the quantum solution gives
θ l = 2(∂δ l /∂l).
(4.61)
133
on a given function of r and has built in, therefore, a particular behavior in r (the
wavefunction dies away in the classically forbidden region and oscillates in the classically allowed one). This allows one to handle in terms of special functions the
problem of isolated turning points, potential wells, potential barriers, potential singularities, etc., and in terms of a family of quantum numbers the bound states. In
particular, the Bohr–Sommerfield rule can be adopted in order to quantize the action
(see Eq. (1.52)) associated with vibrations between the two turning points (say a and
b) as follows
=
b
a
p(x)dx = (v +
1
2
)h.
(4.57)
The area covered by the integrand of Eq. (4.57) indicates the phase space associated with the bound motion on the potential considered. In order to formulate the
semiclassical S matrix
4 when considering all the degrees of freedom in non-separable
processes (like in the case either of the inclusion of the electronic structure of the
colliding partners or of the reactive or nonreactive atom oscillating diatom collision), one naturally turns, as already illustrated in the previous subsections, into the
S matrix relating the family of events linking the initial state to the desired final
ones (each bearing in the semiclassical approach an associated phase determined by
the classical action defined in Chap. 1 accumulated along its path). For the sake of
simplicity, it is assumed that the (single) translational variable (in the equation below
and in Fig. 4.8) is r and that the related motion is coupled to a single internal degree
of freedom α that is the angle to which is associated the action integral (defined
as in Eq. (4.57))
4 The semiclassical connection between the deflection angle θ and the JWKB approximation to the
phase shift δ l can be obtained from the semiclassical formulation of the wavefunction
ψ l (r )
r →∞
∼ sin (kr − lπ/2 + δ l )
(4.58)
that gives
δ l
r →∞
∼
r
a
k l (r )dr − kr + (l + 1/2)π/2
,
( 4.59)
where k l (r ) is the Langer-corrected wavenumber function (resulting from the transformation of r
into e x and to the scaling of the wavefunction by e x/2 to the end of taking into account the singularity
occurring at θ = 0) that reads
k l (r ) =
k
2 − U (r ) −
(l + 1/2) 2
r 2
1/2
(4.60)
from which a comparison with the quantum solution gives
θ l = 2(∂δ l /∂l).
(4.61)