Chapter 8
Manifestations of Chaos in Quantum
Scattering Processes
Abstract A statistical theory of nuclear energy-level structure, using random
matrix theory, was developed in the 1950s by Wigner to explain a shortage of close
spacings in experimentally obtained nuclear energy levels. Random matrix theory
and chaos theory merged in the 1970s when numerical studies of quantized chaotic
billiards showed agreement with some nuclear scattering data.
The theory best suited to study the statistical properties of scattering processes,
and their relation to the underlying Hamiltonian, is the R-matrix theory developed
by Wigner and Eisenbud. It allows the scattering matrix (S-matrix) to be written
explicitly in terms of a Hamiltonian for the reaction region and as a function of the
incident particle energy. The slopes of eigenphases of the S-matrix, when plotted
as a function of energy, are the partial delay times. The distribution of partial delay
times can show signatures of chaos.
Under special circumstances, one can use the ensemble of Hamiltonians,
obtained from the Gaussian Orthogonal Ensemble for the reaction region, to build
an ensemble of S-matrices that belong to the Circular Orthogonal Ensemble, thus
providing a direct link between scattering data and chaos in the reaction region.
This link has been shown explicitly for scattering in electron waveguides, and for
nuclear level sequences and molecular level sequences obtained from scattering
experiments.
Keywords Nuclear scattering · Wigner distribution · Wigner–Eisenbud
scattering theory · Scattering matrix · Wigner–Smith delay time · R-matrix
theory · Circular orthogonal ensemble · Gaussian orthogonal ensemble ·
Lorentzian ensemble · Electron waveguides
8.1 Introduction
Before 1955, there was no systematic statistical theory of nuclear energy-level
structure. There was, however, a shortage of close spacings in experimentally
obtained energy levels that was generally dismissed as being due to instrumental
resolution failings. Wigner was able to explain this shortage of close spacings
© Springer Nature Switzerland AG 2021
L. Reichl, The Transition to Chaos, Fundamental Theories of Physics 200,
https://doi.org/10.1007/978-3-030-63534-3_8
239
Manifestations of Chaos in Quantum
Scattering Processes
Abstract A statistical theory of nuclear energy-level structure, using random
matrix theory, was developed in the 1950s by Wigner to explain a shortage of close
spacings in experimentally obtained nuclear energy levels. Random matrix theory
and chaos theory merged in the 1970s when numerical studies of quantized chaotic
billiards showed agreement with some nuclear scattering data.
The theory best suited to study the statistical properties of scattering processes,
and their relation to the underlying Hamiltonian, is the R-matrix theory developed
by Wigner and Eisenbud. It allows the scattering matrix (S-matrix) to be written
explicitly in terms of a Hamiltonian for the reaction region and as a function of the
incident particle energy. The slopes of eigenphases of the S-matrix, when plotted
as a function of energy, are the partial delay times. The distribution of partial delay
times can show signatures of chaos.
Under special circumstances, one can use the ensemble of Hamiltonians,
obtained from the Gaussian Orthogonal Ensemble for the reaction region, to build
an ensemble of S-matrices that belong to the Circular Orthogonal Ensemble, thus
providing a direct link between scattering data and chaos in the reaction region.
This link has been shown explicitly for scattering in electron waveguides, and for
nuclear level sequences and molecular level sequences obtained from scattering
experiments.
Keywords Nuclear scattering · Wigner distribution · Wigner–Eisenbud
scattering theory · Scattering matrix · Wigner–Smith delay time · R-matrix
theory · Circular orthogonal ensemble · Gaussian orthogonal ensemble ·
Lorentzian ensemble · Electron waveguides
8.1 Introduction
Before 1955, there was no systematic statistical theory of nuclear energy-level
structure. There was, however, a shortage of close spacings in experimentally
obtained energy levels that was generally dismissed as being due to instrumental
resolution failings. Wigner was able to explain this shortage of close spacings
© Springer Nature Switzerland AG 2021
L. Reichl, The Transition to Chaos, Fundamental Theories of Physics 200,
https://doi.org/10.1007/978-3-030-63534-3_8
239