8.3 Scattering Theory
243
The theory that has been most widely used to study the statistical properties of
scattering processes, and their relation to the underlying Hamiltonian, is the reaction
matrix theory of scattering developed by Wigner and Eisenbud (1947). The original
version of this theory was created to describe nuclear scattering processes. The basic
idea was to separate the spatial region occupied by the nucleus (the reaction region
in which the dynamics is largely unknown) from the asymptotic scattering region,
for which the dynamics is completely known. This formalism then allows one to
write the scattering matrix (S-matrix) explicitly in terms of a Hamiltonian for the
reaction region and as a function of the incident particle energy. In the next section,
we derive the relation between the Hamiltonian governing the quantum dynamics of
an open quantum system, and the scattering dynamics of the system.
8.3 Scattering Theory
In the late 1930s, Kapur and Peierls (1938) formulated a nonperturbative approach
to nuclear scattering theory in which the compound nucleus was viewed as a stable
object that was made unstable by weak coupling to the continuum. In the late
1940s, Wigner and Eisenbud (1947) built on this picture and developed the reaction
matrix (R-matrix) approach to scattering theory (Lane and Thomas 1958). The
idea is to decompose configuration space into an internal reaction region and an
external asymptotic scattering region. The reaction region can be modeled in terms
of a complete set of basis states, with fixed boundary conditions on the surface
of the reaction region. The reaction region basis states can then be coupled to the
external asymptotic states. However, as Bloch (1957) and Feshbach (1962) showed,
a consistent theory requires a singular coupling between the reaction region and the
asymptotic region.
Verbaarschot et al. (1985) used this formulation of scattering theory to provide
a framework with which to compare the predictions of random matrix theory to
experimental nuclear scattering data. As we show in this section, this approach to
scattering theory can also be used to obtain the scattering matrix for a quantum.
particle in a two-dimensional waveguide. The expressions we obtain will form a
basis for understanding how chaos manifests itself in open quantum systems in the
remainder of this chapter. Although we focus on 2D waveguide scattering in this
section, the method can be applied to other 2D and 3D scattering systems (Barr and
Reichl 2010; Reichl and Porter 2018; Porter et al. 2019; Barr et al. 2020).
We will derive the scattering matrix for a quantum particle (matter wave) with
mass m in the two dimensional waveguide shown in Fig. 8.3. A particle with energy
E enters the cavity region from the left (right) after traveling along a straight lead,
which we will assume has infinitely hard walls. The assumption that the leads have
hard walls is not essential, but it does simplify our discussion. The matter wave is
partly transmitted to the right (left) or reflected back to the left (right) after having
interacted with the cavity.
243
The theory that has been most widely used to study the statistical properties of
scattering processes, and their relation to the underlying Hamiltonian, is the reaction
matrix theory of scattering developed by Wigner and Eisenbud (1947). The original
version of this theory was created to describe nuclear scattering processes. The basic
idea was to separate the spatial region occupied by the nucleus (the reaction region
in which the dynamics is largely unknown) from the asymptotic scattering region,
for which the dynamics is completely known. This formalism then allows one to
write the scattering matrix (S-matrix) explicitly in terms of a Hamiltonian for the
reaction region and as a function of the incident particle energy. In the next section,
we derive the relation between the Hamiltonian governing the quantum dynamics of
an open quantum system, and the scattering dynamics of the system.
8.3 Scattering Theory
In the late 1930s, Kapur and Peierls (1938) formulated a nonperturbative approach
to nuclear scattering theory in which the compound nucleus was viewed as a stable
object that was made unstable by weak coupling to the continuum. In the late
1940s, Wigner and Eisenbud (1947) built on this picture and developed the reaction
matrix (R-matrix) approach to scattering theory (Lane and Thomas 1958). The
idea is to decompose configuration space into an internal reaction region and an
external asymptotic scattering region. The reaction region can be modeled in terms
of a complete set of basis states, with fixed boundary conditions on the surface
of the reaction region. The reaction region basis states can then be coupled to the
external asymptotic states. However, as Bloch (1957) and Feshbach (1962) showed,
a consistent theory requires a singular coupling between the reaction region and the
asymptotic region.
Verbaarschot et al. (1985) used this formulation of scattering theory to provide
a framework with which to compare the predictions of random matrix theory to
experimental nuclear scattering data. As we show in this section, this approach to
scattering theory can also be used to obtain the scattering matrix for a quantum.
particle in a two-dimensional waveguide. The expressions we obtain will form a
basis for understanding how chaos manifests itself in open quantum systems in the
remainder of this chapter. Although we focus on 2D waveguide scattering in this
section, the method can be applied to other 2D and 3D scattering systems (Barr and
Reichl 2010; Reichl and Porter 2018; Porter et al. 2019; Barr et al. 2020).
We will derive the scattering matrix for a quantum particle (matter wave) with
mass m in the two dimensional waveguide shown in Fig. 8.3. A particle with energy
E enters the cavity region from the left (right) after traveling along a straight lead,
which we will assume has infinitely hard walls. The assumption that the leads have
hard walls is not essential, but it does simplify our discussion. The matter wave is
partly transmitted to the right (left) or reflected back to the left (right) after having
interacted with the cavity.