2.1 Quantum Mechanics and Bound States
41
The different operators do not necessarily commute. Given two operators (say ˆ
A
and ˆ
B), their property of commuting is checked through the commutator
ˆ
A, ˆ
B
= ˆ
A ˆ
B − ˆ
B ˆ
A
(2.9)
When two operators do not have common eigenvectors and their associated observables are not exactly determined at the same time their commutator is not a zero
operator, ˆ
O, and they do not commute. It can be shown that this is the case of the
position and momentum operators (ˆ x and ˆ
p x ) for which
ˆ
x, ˆ
p x
= i = 0 (similarly
for ˆ t and ˆ
E one obtains
ˆ t, ˆ
E
= −i ). This means that these pairs of variables
cannot be simultaneously determined. Formally, this is expressed by the uncertainty
principle (Heisenberg relationship)
1 :
xp x ≥
2
.
(2.10)
To relate the uncertainty of b and θ in our choice of coordinates, one has to recall
that the uncertainty relationship that has to be applied is bp z . Because the
z-component of the momentum p z transferred during the collision is p z = μv sin θ
μvθ, we get immediately
bθ /μv
(2.11)
or
μvvbθ = Lθ .
(2.12)
Let us consider now the singularities associated with the classical formulation
(see Eq. 1.54) of the differential cross section at both θ = 0 and dθ/db = 0 as is
the case shown in Fig. 2.1 for the Lennard-Jones, LJ, potential (see the end of the
previous chapter). In the figure, the value of the deflection angle θ corresponding to
a minimum (angle of rainbow) leads to a singularity corresponding in the quantum
treatment to a broad maximum with superimposed a highly oscillating structure.
The fact that the singularity of the classical result is smoothed in the quantum
solution (see Eq. 2.79) can be traced back to the uncertainty principle of Eq. 2.12 that
does not allow θ and b to be simultaneously defined. This inability of a pure classical
mechanics approach to embody such condition can be regained in semiclassical
treatments by taking into account the interference effects between waves associated
with different classical paths (trajectories) though leading to the same value of θ.
This result provides the theoretical basis for defining the abovementioned wave function ψ as a scattering physical distribution in space (obviously with different probabilities in different regions) of the system. Given a precise value L (or equivalently of
1 The original heuristic argument was given by Heisenberg in 1927. The formal statement was
proved by Earl Hesse Kennard and Hermann Weyl.
41
The different operators do not necessarily commute. Given two operators (say ˆ
A
and ˆ
B), their property of commuting is checked through the commutator
ˆ
A, ˆ
B
= ˆ
A ˆ
B − ˆ
B ˆ
A
(2.9)
When two operators do not have common eigenvectors and their associated observables are not exactly determined at the same time their commutator is not a zero
operator, ˆ
O, and they do not commute. It can be shown that this is the case of the
position and momentum operators (ˆ x and ˆ
p x ) for which
ˆ
x, ˆ
p x
= i = 0 (similarly
for ˆ t and ˆ
E one obtains
ˆ t, ˆ
E
= −i ). This means that these pairs of variables
cannot be simultaneously determined. Formally, this is expressed by the uncertainty
principle (Heisenberg relationship)
1 :
xp x ≥
2
.
(2.10)
To relate the uncertainty of b and θ in our choice of coordinates, one has to recall
that the uncertainty relationship that has to be applied is bp z . Because the
z-component of the momentum p z transferred during the collision is p z = μv sin θ
μvθ, we get immediately
bθ /μv
(2.11)
or
μvvbθ = Lθ .
(2.12)
Let us consider now the singularities associated with the classical formulation
(see Eq. 1.54) of the differential cross section at both θ = 0 and dθ/db = 0 as is
the case shown in Fig. 2.1 for the Lennard-Jones, LJ, potential (see the end of the
previous chapter). In the figure, the value of the deflection angle θ corresponding to
a minimum (angle of rainbow) leads to a singularity corresponding in the quantum
treatment to a broad maximum with superimposed a highly oscillating structure.
The fact that the singularity of the classical result is smoothed in the quantum
solution (see Eq. 2.79) can be traced back to the uncertainty principle of Eq. 2.12 that
does not allow θ and b to be simultaneously defined. This inability of a pure classical
mechanics approach to embody such condition can be regained in semiclassical
treatments by taking into account the interference effects between waves associated
with different classical paths (trajectories) though leading to the same value of θ.
This result provides the theoretical basis for defining the abovementioned wave function ψ as a scattering physical distribution in space (obviously with different probabilities in different regions) of the system. Given a precise value L (or equivalently of
1 The original heuristic argument was given by Heisenberg in 1927. The formal statement was
proved by Earl Hesse Kennard and Hermann Weyl.