158
6 Quantum Dynamics and Random Matrix Theory
respectively, and the probability distributions are said to describe Gaussian unitary
ensembles (GUE) and Gaussian symplectic ensembles (GSE), respectively.)
When dealing with N × N-dimensional Hermitian matrices, where N is large,
we have far more information in the joint probability distribution than we can
possibly use. As we will see, we are often only interested in pair correlations
between energy eigenvalues. Therefore, in Sect. 6.4 we introduce reduced joint
probability distributions for only n of the N eigenvalues. These reduced probability
distributions can themselves be written in terms of n-body cluster functions that are
nonzero only when n-body (n-eigenvalue) correlations exist in the system.
There are a number of statistical properties of random matrices that are commonly used in analyzing the spectral properties of systems. These are the eigenvalue
density, the 3 -statistic, and the eigenvalue nearest neighbor spacing distribution.
All three of these quantities will be derived in Sect. 6.4 for the Gaussian orthogonal
ensemble.
In Sect. 6.5, we obtain the probability distribution for eigenvectors of Hamiltonian matrices that belong to the Gaussian orthogonal ensemble. More precisely we
find the distribution of components of a single eigenvector taken from the complete
set of eigenvectors. This eigenvector distribution will then allow us, in Sect. 6.6, to
obtain the reduced probability distribution of a “noninteracting” gas of particles,
where we assume that each particle is governed by a Hamiltonian that is a member
of the Gaussian orthogonal ensemble. We follow the procedure of Srednicki (1994)
and, like him, we find that the gas is thermalized. The single particle reduced
probability density is given by the Maxwell-Boltzmann distribution. The work
of Srednicki was important because it helped to establish how quantum systems
become thermalized. Finally, in Sect. 6.7, we make some concluding remarks.
6.2 Invariant Measure for the GOE
Hamiltonians that govern the dynamics of systems that are invariant under time
reversal and rotation have matrix representations that are real and symmetric. An
N ×N real symmetric matrix,
¯
H R =
⎛
⎜
⎜
⎜
⎝
h 1,1 h 1,2 . . . h 1,N
h 1,2 h 2,2 . . . h 2,N
. . .
. . .
. . .
. . .
h 1,N h 2,N . . . h N,N
⎞
⎟
⎟
⎟
⎠
,
(6.3)
has N +
1
2 (N 2 − N) =
1
2 N(N + 1) independent real elements.
An N ×N real symmetric matrix, ¯
H R , is diagonalized by an N ×N orthogonal
matrix, ¯
O. An orthogonal matrix has the property that its transpose is equal to its
inverse,
6 Quantum Dynamics and Random Matrix Theory
respectively, and the probability distributions are said to describe Gaussian unitary
ensembles (GUE) and Gaussian symplectic ensembles (GSE), respectively.)
When dealing with N × N-dimensional Hermitian matrices, where N is large,
we have far more information in the joint probability distribution than we can
possibly use. As we will see, we are often only interested in pair correlations
between energy eigenvalues. Therefore, in Sect. 6.4 we introduce reduced joint
probability distributions for only n of the N eigenvalues. These reduced probability
distributions can themselves be written in terms of n-body cluster functions that are
nonzero only when n-body (n-eigenvalue) correlations exist in the system.
There are a number of statistical properties of random matrices that are commonly used in analyzing the spectral properties of systems. These are the eigenvalue
density, the 3 -statistic, and the eigenvalue nearest neighbor spacing distribution.
All three of these quantities will be derived in Sect. 6.4 for the Gaussian orthogonal
ensemble.
In Sect. 6.5, we obtain the probability distribution for eigenvectors of Hamiltonian matrices that belong to the Gaussian orthogonal ensemble. More precisely we
find the distribution of components of a single eigenvector taken from the complete
set of eigenvectors. This eigenvector distribution will then allow us, in Sect. 6.6, to
obtain the reduced probability distribution of a “noninteracting” gas of particles,
where we assume that each particle is governed by a Hamiltonian that is a member
of the Gaussian orthogonal ensemble. We follow the procedure of Srednicki (1994)
and, like him, we find that the gas is thermalized. The single particle reduced
probability density is given by the Maxwell-Boltzmann distribution. The work
of Srednicki was important because it helped to establish how quantum systems
become thermalized. Finally, in Sect. 6.7, we make some concluding remarks.
6.2 Invariant Measure for the GOE
Hamiltonians that govern the dynamics of systems that are invariant under time
reversal and rotation have matrix representations that are real and symmetric. An
N ×N real symmetric matrix,
¯
H R =
⎛
⎜
⎜
⎜
⎝
h 1,1 h 1,2 . . . h 1,N
h 1,2 h 2,2 . . . h 2,N
. . .
. . .
. . .
. . .
h 1,N h 2,N . . . h N,N
⎞
⎟
⎟
⎟
⎠
,
(6.3)
has N +
1
2 (N 2 − N) =
1
2 N(N + 1) independent real elements.
An N ×N real symmetric matrix, ¯
H R , is diagonalized by an N ×N orthogonal
matrix, ¯
O. An orthogonal matrix has the property that its transpose is equal to its
inverse,