Chapter 6
Quantum Dynamics and Random Matrix
Theory
Abstract The energy spectrum of a quantum system, whose classical counterpart
is chaotic, has statistical properties like those of random matrices that extremize
information. In the 1950s, Wigner surmised that classically chaotic quantum
systems have energy eigenvalue spacing distributions similar to those of a Hamiltonian matrix whose matrix elements are random numbers determined by Gaussian
distributions. This gives a distribution for nearest neighbor spacings (the Wigner
distribution) between eigenvalues that has been verified by experiment. The Wigner
distribution predicts a very low probability of finding small spacings between
nearest neighbor energy eigenvalues.
The random matrix theory of Hamiltonian systems is based on the assumption
that global symmetries alone determine the form of the Hamiltonian. These
symmetries impose restrictions on the form of the Hamiltonian matrix. Probability
distributions for matrix elements of orthogonal, complex Hermitian, and quaternion
real Hamiltonian matrices can be obtained. Eigenvalue cluster functions, nearest
neighbor spacing distributions, and two-body eigenvalue correlation functions can
then be derived.
A gas of particles, whose Hamiltonian belongs to the Gaussian orthogonal
ensemble, is a thermalized system and its single particle reduced probability density
is given by the Maxwell-Boltzmann distribution. This important example shows the
mechanism by which quantum systems become thermalized.
Keywords Random matrix theory · Nuclear scattering theory · Wigner surmise ·
Wigner distribution · Wigner semi-circle law · Staircase function · Delta-3
statistic · Invariant metric · Hamiltonian matrices · Gaussian orthogonal
ensemble · Circular ensembles · Cluster functions · Eigenvalue nearest neighbor
spacing · Eigenvector distribution · Thermalization of quantum systems
6.1 Introduction
Classical conservative systems that undergo a transition to chaos have very complex
dynamical behavior, as we have seen in previous chapters. How much of this
© 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_6
155
Quantum Dynamics and Random Matrix
Theory
Abstract The energy spectrum of a quantum system, whose classical counterpart
is chaotic, has statistical properties like those of random matrices that extremize
information. In the 1950s, Wigner surmised that classically chaotic quantum
systems have energy eigenvalue spacing distributions similar to those of a Hamiltonian matrix whose matrix elements are random numbers determined by Gaussian
distributions. This gives a distribution for nearest neighbor spacings (the Wigner
distribution) between eigenvalues that has been verified by experiment. The Wigner
distribution predicts a very low probability of finding small spacings between
nearest neighbor energy eigenvalues.
The random matrix theory of Hamiltonian systems is based on the assumption
that global symmetries alone determine the form of the Hamiltonian. These
symmetries impose restrictions on the form of the Hamiltonian matrix. Probability
distributions for matrix elements of orthogonal, complex Hermitian, and quaternion
real Hamiltonian matrices can be obtained. Eigenvalue cluster functions, nearest
neighbor spacing distributions, and two-body eigenvalue correlation functions can
then be derived.
A gas of particles, whose Hamiltonian belongs to the Gaussian orthogonal
ensemble, is a thermalized system and its single particle reduced probability density
is given by the Maxwell-Boltzmann distribution. This important example shows the
mechanism by which quantum systems become thermalized.
Keywords Random matrix theory · Nuclear scattering theory · Wigner surmise ·
Wigner distribution · Wigner semi-circle law · Staircase function · Delta-3
statistic · Invariant metric · Hamiltonian matrices · Gaussian orthogonal
ensemble · Circular ensembles · Cluster functions · Eigenvalue nearest neighbor
spacing · Eigenvector distribution · Thermalization of quantum systems
6.1 Introduction
Classical conservative systems that undergo a transition to chaos have very complex
dynamical behavior, as we have seen in previous chapters. How much of this
© 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_6
155