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6 Quantum Dynamics and Random Matrix Theory
the probability, P ( ¯
H R )dd H R , of finding the matrix elements of ¯
H R in the interval
¯
H R → ¯
H R + d ¯
H R , where dd H R is the measure associated with the matrix elements
of ¯
H R . One way to obtain such a distribution is to require that the information
contained in the Hamiltonian matrix be minimum.
The information contained in the N×N Hamiltonian matrix ¯
H R can be defined
(Porter 1965) as
I =
dd H R P ( ¯
H R ) lnP ( ¯
H R ).
(6.23)
We extremize (minimize) the information subject to the condition that the probability is normalized to 1,
dd H R P ( ¯
H R ) = 1.
(6.24)
We also require that it be improbable to find extremely large matrix elements in the
Hamiltonian matrix. This can be accomplished by requiring that the variance of the
matrix elements be finite,
dd H R Tr[ ¯
H R · ¯
H R ] P ( ¯
H R ) =
N(N + 1)
2
δ
2 ,
(6.25)
where δ 2 is a constant. With this condition, the variance of the diagonal matrix
elements, h jj (i = 1, . . . , N), is h 2
jj = δ 2 and the variance of the off-diagonal
matrix elements, h ij (i =j ), is h 2
ij = δ 2 /2.
The information in Eq. (6.23) can be extremized subject to the constraints in
Eqs. (6.24) and (6.25) by using Lagrange multipliers. We define
I (a, b) =
dd H R [P ( ¯
H R ) lnP ( ¯
H R ) + aP ( ¯
H R ) + bTr[ ¯
H R · ¯
H R ] P ( ¯
H R )],
(6.26)
where a and b are the Lagrange multipliers associated with the constraints in
Eqs. (6.24) and (6.25), respectively. Information is extremized if the first variation
of I (a, b) (with respect to P ( ¯
H R )) is equal to zero,
δI =
dd H R δP ( ¯
H R ) {lnP ( ¯
H R ) + 1 + a + bTr[ ¯
H R · ¯
H R ]} = 0.
(6.27)
Since δP ( ¯
H R ) is arbitrary, we must have
P ( ¯
H R ) = exp[−(1 + a + bTr[ ¯
H R · ¯
H R ])].
(6.28)
The Lagrange multipliers can be determined from Eqs. (6.24) and (6.25). It is
straightforward to show that