Chapter 1
Essentials of Non-relativistic Two-Body
Scattering Theory
1.1 Introduction
In this section, we retrace the basic steps required to formulate scattering theory in
non-relativistic quantum mechanics with two-body potential [23]. We start with a
time-dependent Schrodinger equation for the scattering state which is to be developed
from a wave packet that is free in the infinite past. The requirement that the scattered
wave packet coincides with the free packet demands that the norm of the difference
between the two be zero in the strong limit. This leads us to introduce and define
Moller operator. Although the domain of Moller operator is the Hilbert space of
square-integrable free states, the work by Faddeev shows that this domain can be
extended to include states of sharp momentum. This helps us to replace the strong
limit by an Euler limit. With the introduction of the Euler limit, we are able to get
two-particle scattering theory which is time independent.
We then introduce Green’s function or the resolvent through which the scattering state and the free particle states can be related. The next step is to derive the
integral equation for the scattering states incorporating proper boundary condition
which consists of an incident plane wave plus an outgoing scattered wave. Next,
we examine whether the kernel of the integral equation so obtained is a compact
operator, or, in simple terms, whether the Schmidt norm is finite or not. We find that
Schmidt norm can only be finite if (i) the integral over the absolute square of the
potential exists thereby ruling out the possibility of long infinite range potentials and
(ii) the imaginary part of
√
2μ(E + iε) is zero. The latter condition excludes the
scattering energies in which we are interested. However, a finite Schmidt norm is
only a sufficient but not the necessary condition. Lovelace in his paper shows that the
kernel of the integral equation is compact in the Banach space of continuous bounded
functions with continuous derivatives. This ensures that the Lippmann–Schwinger
equation has a unique solution.
To get the link between scattering states and the experimentally measured data,
we introduce and define the S-matrix and obtain its relationship with the T-matrix
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
V. S. Bhasin and I. Mazumdar, Few Body Dynamics, Efimov Effect and Halo Nuclei,
SpringerBriefs in Physics,
https://doi.org/10.1007/978-3-030-56171-0_1
1
Essentials of Non-relativistic Two-Body
Scattering Theory
1.1 Introduction
In this section, we retrace the basic steps required to formulate scattering theory in
non-relativistic quantum mechanics with two-body potential [23]. We start with a
time-dependent Schrodinger equation for the scattering state which is to be developed
from a wave packet that is free in the infinite past. The requirement that the scattered
wave packet coincides with the free packet demands that the norm of the difference
between the two be zero in the strong limit. This leads us to introduce and define
Moller operator. Although the domain of Moller operator is the Hilbert space of
square-integrable free states, the work by Faddeev shows that this domain can be
extended to include states of sharp momentum. This helps us to replace the strong
limit by an Euler limit. With the introduction of the Euler limit, we are able to get
two-particle scattering theory which is time independent.
We then introduce Green’s function or the resolvent through which the scattering state and the free particle states can be related. The next step is to derive the
integral equation for the scattering states incorporating proper boundary condition
which consists of an incident plane wave plus an outgoing scattered wave. Next,
we examine whether the kernel of the integral equation so obtained is a compact
operator, or, in simple terms, whether the Schmidt norm is finite or not. We find that
Schmidt norm can only be finite if (i) the integral over the absolute square of the
potential exists thereby ruling out the possibility of long infinite range potentials and
(ii) the imaginary part of
√
2μ(E + iε) is zero. The latter condition excludes the
scattering energies in which we are interested. However, a finite Schmidt norm is
only a sufficient but not the necessary condition. Lovelace in his paper shows that the
kernel of the integral equation is compact in the Banach space of continuous bounded
functions with continuous derivatives. This ensures that the Lippmann–Schwinger
equation has a unique solution.
To get the link between scattering states and the experimentally measured data,
we introduce and define the S-matrix and obtain its relationship with the T-matrix
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
V. S. Bhasin and I. Mazumdar, Few Body Dynamics, Efimov Effect and Halo Nuclei,
SpringerBriefs in Physics,
https://doi.org/10.1007/978-3-030-56171-0_1
1