In principle, differential and total cross-sections of processes can be calculated,
especially if this is done in the so-called semi-classical approximation. The author
will explain some features of the approximation and some results of this approach
briefly.
The principal idea of the semi-classical approximation is that one group of
degrees of freedom of the system under consideration, for example, the translational
one, is described in the framework of classical mechanics, and the rest in the
framework of quantum mechanics. They form the classical and quantum subsystems. It is believed that in the classical subsystem the motion occurs along a specific
trajectory, and the interaction between the classical and quantum subsystems
induces time-dependent transitions between the quantized states of the quantum
subsystem.
One of the conditions, the fulfillment of which is necessary for the applicability
of the semi-classical approximation, is that the de Broglie wavelength of the
classical subsystem:
k ¼ h Á p ¼
h
2
2E Á m
1=2
ð3:1:11Þ
(here, p and E are the momentum and energy of the species, m is its mass) has to be
much smaller than the characteristic size of the potential action, in which the
interaction of species occurs, which is about 10
–8 cm. This condition is equivalent
to the one when discussing the translational partition function: quantum numbers
corresponding to the classical motion must be much greater than 1; here, it is
equivalent to n = l/k, where l is the same size, *1 Å. Let us see if this condition is
fulfilled for thermal energies and the motion of an oxygen atom, for example,
p = 2:6 Á 10
À23
Á 10
5 g∙cm/s: k = 1:05 Á 10
À27
=2:6 Á 10
À18 erg s/s g = 3 Á 10
À10 cm
<< 10
–8 cm (see Sect. 2.2).
Another condition for the semi-classical approximation applicability is that the
quantum subsystem energy change should be small; more precisely, there should be
a lot less than the kinetic energy of the classical degrees of freedom. Such an
approximation is the first step to the adiabatic approximation, which will be discussed below.
The semi-classical approximation allows to formulate the scattering problem as a
problem to solve a time-dependent wave equation (and the solution to this problem
is more complicated than a stationary solution), but for a smaller number of degrees
of freedom, namely, only for the quantum ones.
Let the quantum subsystem be determined by a set of quantum numbers i,
j before a collision. Then the time-dependent wave function Ф(t ! - ∞) can be
represented as a product of the wave functions of non-interacting species u
A
i and
u
B
j
3.1 Cross-Sections, Rate Constants, and Probabilities of Elementary Processes …
41
especially if this is done in the so-called semi-classical approximation. The author
will explain some features of the approximation and some results of this approach
briefly.
The principal idea of the semi-classical approximation is that one group of
degrees of freedom of the system under consideration, for example, the translational
one, is described in the framework of classical mechanics, and the rest in the
framework of quantum mechanics. They form the classical and quantum subsystems. It is believed that in the classical subsystem the motion occurs along a specific
trajectory, and the interaction between the classical and quantum subsystems
induces time-dependent transitions between the quantized states of the quantum
subsystem.
One of the conditions, the fulfillment of which is necessary for the applicability
of the semi-classical approximation, is that the de Broglie wavelength of the
classical subsystem:
k ¼ h Á p ¼
h
2
2E Á m
1=2
ð3:1:11Þ
(here, p and E are the momentum and energy of the species, m is its mass) has to be
much smaller than the characteristic size of the potential action, in which the
interaction of species occurs, which is about 10
–8 cm. This condition is equivalent
to the one when discussing the translational partition function: quantum numbers
corresponding to the classical motion must be much greater than 1; here, it is
equivalent to n = l/k, where l is the same size, *1 Å. Let us see if this condition is
fulfilled for thermal energies and the motion of an oxygen atom, for example,
p = 2:6 Á 10
À23
Á 10
5 g∙cm/s: k = 1:05 Á 10
À27
=2:6 Á 10
À18 erg s/s g = 3 Á 10
À10 cm
<< 10
–8 cm (see Sect. 2.2).
Another condition for the semi-classical approximation applicability is that the
quantum subsystem energy change should be small; more precisely, there should be
a lot less than the kinetic energy of the classical degrees of freedom. Such an
approximation is the first step to the adiabatic approximation, which will be discussed below.
The semi-classical approximation allows to formulate the scattering problem as a
problem to solve a time-dependent wave equation (and the solution to this problem
is more complicated than a stationary solution), but for a smaller number of degrees
of freedom, namely, only for the quantum ones.
Let the quantum subsystem be determined by a set of quantum numbers i,
j before a collision. Then the time-dependent wave function Ф(t ! - ∞) can be
represented as a product of the wave functions of non-interacting species u
A
i and
u
B
j
3.1 Cross-Sections, Rate Constants, and Probabilities of Elementary Processes …
41