156
6 Quantum Dynamics and Random Matrix Theory
complex behavior remains in the corresponding quantum systems? We will show,
in much of the remainder of this book, that quantum systems, whose classical
counterpart is chaotic, have spectra whose statistical properties are similar to those
of random matrices that extremize information. Thus, any study of the quantum
manifestations of chaos requires an analysis of information content of quantum
systems, using concepts from random matrix theory (RMT). We have attempted
to give a complete grounding on random matrix theory in this book. Much of our
discussion of random matrix theory is in the appendices, but we give an overview of
some key results in this chapter. Our analysis of quantum dynamics and the behavior
of solutions of the Schrödinger equation will actually begin in Chap. 7.
The use of random matrix theory as a tool to study the statistical properties of
quantum systems was introduced by Wigner (1951, 1955, 1957a,b, 1958) in an
attempt to understand nuclear scattering data. Wigner used it to analyze complex
nuclear energy-level sequences (Wigner 1959). At that time there was a shortage
of close spacings in experimentally obtained data on energy levels. This lack of
close spacings was generally thought to result from the inability of experimental
apparatus to resolve them. Wigner was able to give an explanation using statistical
arguments. Wigner surmised (Wigner 1959) a possible energy eigenvalue spacing
distribution assuming that matrix elements of the Hamiltonian matrix were random
numbers with Gaussian distributions. He obtained a distribution for nearest neighbor
spacings, s, between eigenvalues,
P W (s) =
πs
2D 2 exp
−πs 2
4D 2
,
(6.1)
where D is the average spacing between nearest neighbor eigenvalues in the
eigenvalue sequence being considered. Equation (6.1) is now called the Wigner
distribution. The Wigner distribution predicts a very low probability of finding small
spacings between nearest neighbor energy eigenvalues. This is very different from
the case where the eigenvalues are randomly distributed. For random eigenvalue
sequences, the nearest neighbor spacing satisfies a Poisson distribution,
P P (s) =
1
D
exp
−s
D
,
(6.2)
where D is again the average spacing between nearest neighbor eigenvalues. In
Fig. 6.1, we compare the Wigner distribution, P W (s), with the Poisson distribution,
P P (s), for the case D = 1. For systems whose eigenvalues are distributed at random, there is a large probability of finding very small spacing between eigenvalues.
For systems whose Hamiltonian matrix elements are distributed at random, there is
a very small probability of finding close spacings between eigenvalues.
Random matrix theory, as applied to Hamiltonian systems, is based on the
assumption that we know very little about the Hamiltonian matrix except for certain
symmetry properties. These symmetry properties impose restrictions on the form of
the Hamiltonian matrix, as described in Appendix C. In this chapter, we develop
6 Quantum Dynamics and Random Matrix Theory
complex behavior remains in the corresponding quantum systems? We will show,
in much of the remainder of this book, that quantum systems, whose classical
counterpart is chaotic, have spectra whose statistical properties are similar to those
of random matrices that extremize information. Thus, any study of the quantum
manifestations of chaos requires an analysis of information content of quantum
systems, using concepts from random matrix theory (RMT). We have attempted
to give a complete grounding on random matrix theory in this book. Much of our
discussion of random matrix theory is in the appendices, but we give an overview of
some key results in this chapter. Our analysis of quantum dynamics and the behavior
of solutions of the Schrödinger equation will actually begin in Chap. 7.
The use of random matrix theory as a tool to study the statistical properties of
quantum systems was introduced by Wigner (1951, 1955, 1957a,b, 1958) in an
attempt to understand nuclear scattering data. Wigner used it to analyze complex
nuclear energy-level sequences (Wigner 1959). At that time there was a shortage
of close spacings in experimentally obtained data on energy levels. This lack of
close spacings was generally thought to result from the inability of experimental
apparatus to resolve them. Wigner was able to give an explanation using statistical
arguments. Wigner surmised (Wigner 1959) a possible energy eigenvalue spacing
distribution assuming that matrix elements of the Hamiltonian matrix were random
numbers with Gaussian distributions. He obtained a distribution for nearest neighbor
spacings, s, between eigenvalues,
P W (s) =
πs
2D 2 exp
−πs 2
4D 2
,
(6.1)
where D is the average spacing between nearest neighbor eigenvalues in the
eigenvalue sequence being considered. Equation (6.1) is now called the Wigner
distribution. The Wigner distribution predicts a very low probability of finding small
spacings between nearest neighbor energy eigenvalues. This is very different from
the case where the eigenvalues are randomly distributed. For random eigenvalue
sequences, the nearest neighbor spacing satisfies a Poisson distribution,
P P (s) =
1
D
exp
−s
D
,
(6.2)
where D is again the average spacing between nearest neighbor eigenvalues. In
Fig. 6.1, we compare the Wigner distribution, P W (s), with the Poisson distribution,
P P (s), for the case D = 1. For systems whose eigenvalues are distributed at random, there is a large probability of finding very small spacing between eigenvalues.
For systems whose Hamiltonian matrix elements are distributed at random, there is
a very small probability of finding close spacings between eigenvalues.
Random matrix theory, as applied to Hamiltonian systems, is based on the
assumption that we know very little about the Hamiltonian matrix except for certain
symmetry properties. These symmetry properties impose restrictions on the form of
the Hamiltonian matrix, as described in Appendix C. In this chapter, we develop