18
1 From the Phenomenology of Chemical Reactions …
formulate dispersion van der Waals interaction) that will be discussed in more detail
later on. The trajectories were obtained by integrating Eqs. 1.38 and 1.39 using a
numerical method.
Before addressing the question of the methods used for numerical integration,
we discuss here the further simplification of the equations for the simple (though
very general and important) problem of the deflection from the central field. This
problem can in fact be further decomposed (thereby reducing from 4 to 2 the number
of differential equations to integrate), thanks to the use of conservation laws related
to total angular momentum L and the total energy E. For this purpose, use is made of
the magnitude of the impact parameter b defined as the distance between the center of
interaction and the initial velocity vector (or, equivalently, the perpendicular segment
drawn from the particle to the z axis, see Fig. 1.7). In many treatments of diatomic
molecules, one uses j as the angular momentum operator. For the interaction of two
spherically symmetric particles, L ≡ j. To be consistent with the scattering notation
used later on, we will use L as the total angular momentum when we consider
diatomic molecules. In fact we will always use L for systems where the position
vector r associated with with L ≡ r × p r is used to describe particles when are
infinitely separated. In the classical formulation of the conservation of total angular
momentum l as
6
|L| = μv 0 b = l = μr
2 ˙
θ
(1.40)
contrary to the quantum one
7 and combining Eqs. (1.33) and (1.40), one obtains the
two differential equations that characterize the system. Of these, the radial velocity
is
˙
r ≡
dr
dt
= ±
2
μ
E − V (r ) −
l
2
2μr 2
1/2
(1.41)
(in the following, we shall often compact the notation using the effective potential
V
l
(r ) which incorporates the centrifugal component l
2
/2μr
2 being V
l
(r ) = V (r ) +
l
2
/2μr
2 ) and then the angular velocity is
˙
θ ≡
dθ
dt
=
l
μr 2 =
v 0 b
r 2 .
(1.42)
6 By definition of angular momentum and vector product is in fact:
L = r × p = μ(r × ˙
r) = μ
z
dy
dt
− y
dz
dt
.
in this case, the motion is confined to a plane yz as in our case. Then, at time t = 0, since dy/dt = 0,
we have |L| = yμdz/dt = μbv, while at generic times t, you will have |L| = μr 2 ˙
θ (see next note).
7 In the quantum treatment in formulating the conservation of the diatomic total angular momentum
quantum number l, l 2 will be replaced (apart from a constant factor) by l(l + 1) which is the total
angular momentum eigenvalue of the quantum operator, as we will see more forward.
1 From the Phenomenology of Chemical Reactions …
formulate dispersion van der Waals interaction) that will be discussed in more detail
later on. The trajectories were obtained by integrating Eqs. 1.38 and 1.39 using a
numerical method.
Before addressing the question of the methods used for numerical integration,
we discuss here the further simplification of the equations for the simple (though
very general and important) problem of the deflection from the central field. This
problem can in fact be further decomposed (thereby reducing from 4 to 2 the number
of differential equations to integrate), thanks to the use of conservation laws related
to total angular momentum L and the total energy E. For this purpose, use is made of
the magnitude of the impact parameter b defined as the distance between the center of
interaction and the initial velocity vector (or, equivalently, the perpendicular segment
drawn from the particle to the z axis, see Fig. 1.7). In many treatments of diatomic
molecules, one uses j as the angular momentum operator. For the interaction of two
spherically symmetric particles, L ≡ j. To be consistent with the scattering notation
used later on, we will use L as the total angular momentum when we consider
diatomic molecules. In fact we will always use L for systems where the position
vector r associated with with L ≡ r × p r is used to describe particles when are
infinitely separated. In the classical formulation of the conservation of total angular
momentum l as
6
|L| = μv 0 b = l = μr
2 ˙
θ
(1.40)
contrary to the quantum one
7 and combining Eqs. (1.33) and (1.40), one obtains the
two differential equations that characterize the system. Of these, the radial velocity
is
˙
r ≡
dr
dt
= ±
2
μ
E − V (r ) −
l
2
2μr 2
1/2
(1.41)
(in the following, we shall often compact the notation using the effective potential
V
l
(r ) which incorporates the centrifugal component l
2
/2μr
2 being V
l
(r ) = V (r ) +
l
2
/2μr
2 ) and then the angular velocity is
˙
θ ≡
dθ
dt
=
l
μr 2 =
v 0 b
r 2 .
(1.42)
6 By definition of angular momentum and vector product is in fact:
L = r × p = μ(r × ˙
r) = μ
z
dy
dt
− y
dz
dt
.
in this case, the motion is confined to a plane yz as in our case. Then, at time t = 0, since dy/dt = 0,
we have |L| = yμdz/dt = μbv, while at generic times t, you will have |L| = μr 2 ˙
θ (see next note).
7 In the quantum treatment in formulating the conservation of the diatomic total angular momentum
quantum number l, l 2 will be replaced (apart from a constant factor) by l(l + 1) which is the total
angular momentum eigenvalue of the quantum operator, as we will see more forward.