Chapter 2
The Quantum Approach to the Two-Body
Problem
This chapter, after considering the drawbacks of a classical mechanics treatment
of the deflection angle in two-body collisions, tackles the problem of treating such
collisions using a quantum approach to both bound states and elastic scattering.
Some basic interaction models are then considered in order to provide support to
some fundamental relationships between the shape of the interaction and the formulation of bound states and scattering quantities. Basic techniques often adopted for
numerically integrating the Schrödinger equation are finally illustrated.
2.1 Quantum Mechanics and Bound States
2.1.1 The Limits of the Classical Mechanics Approach
So far, we have treated atomic and molecular processes as collisions of pointwise
particles using the concepts of classical mechanics. In classical mechanics, each
particle is assigned a well-defined position (say x in one dimension) and a momentum
(say p x = mv x = mdx/dt) at each value of time t. This assumption turns out not to
be valid in some cases (like in the case of light that in Newton’s view is treated
as made of particles, whereas in the Huygens’s view is treated as made of waves
propagating on an ether (like the waves generated by a stone thrown on water)).
While ether was proven not to exist by some sophisticated experiments wave-like
behaviors (such as diffraction patterns) were found to be typical of small particles
(like photons, electrons, etc.). Thanks to Born, the behaviour of elementary particles
was identified to be of probabilistic nature, that in one dimension can be associated
with a wave function, say ψ(x, t), representing the probability amplitude allowing
the evaluation of related physical observables. In the Dirac notation, the function ψ
can be written as a vector |ψ (more details about vectors, matrices, and vector spaces
© Springer International Publishing AG 2018
A. Laganà and G. A. Parker (eds.), Chemical Reactions, Theoretical Chemistry
and Computational Modelling, https://doi.org/10.1007/978-3-319-62356-6_2
39
The Quantum Approach to the Two-Body
Problem
This chapter, after considering the drawbacks of a classical mechanics treatment
of the deflection angle in two-body collisions, tackles the problem of treating such
collisions using a quantum approach to both bound states and elastic scattering.
Some basic interaction models are then considered in order to provide support to
some fundamental relationships between the shape of the interaction and the formulation of bound states and scattering quantities. Basic techniques often adopted for
numerically integrating the Schrödinger equation are finally illustrated.
2.1 Quantum Mechanics and Bound States
2.1.1 The Limits of the Classical Mechanics Approach
So far, we have treated atomic and molecular processes as collisions of pointwise
particles using the concepts of classical mechanics. In classical mechanics, each
particle is assigned a well-defined position (say x in one dimension) and a momentum
(say p x = mv x = mdx/dt) at each value of time t. This assumption turns out not to
be valid in some cases (like in the case of light that in Newton’s view is treated
as made of particles, whereas in the Huygens’s view is treated as made of waves
propagating on an ether (like the waves generated by a stone thrown on water)).
While ether was proven not to exist by some sophisticated experiments wave-like
behaviors (such as diffraction patterns) were found to be typical of small particles
(like photons, electrons, etc.). Thanks to Born, the behaviour of elementary particles
was identified to be of probabilistic nature, that in one dimension can be associated
with a wave function, say ψ(x, t), representing the probability amplitude allowing
the evaluation of related physical observables. In the Dirac notation, the function ψ
can be written as a vector |ψ (more details about vectors, matrices, and vector spaces
© Springer International Publishing AG 2018
A. Laganà and G. A. Parker (eds.), Chemical Reactions, Theoretical Chemistry
and Computational Modelling, https://doi.org/10.1007/978-3-319-62356-6_2
39