6.6 Thermalization of Quantum Systems
189
=
V
. . .
V
dr 1 . . . dr N m 1 , .., m N |r 1 , .., r N r 1 , .., r N |ψ
α
E
=
V
. . .
V
dr 1 . . . dr N
1
L 3N/2 e
−i
2π
L m 1 ·r 1 . . . e
−i
2π
L m N ·r N
×
n 1
. . .
n N
A
α
n 1 ,...,n N
α ({n i })
1
L 3N/2 e
i
2π
L n 1 ·r 1 . . . e
i
2π
L n N ·r N
=
n 1
. . .
n N
A
α
n 1 ,...,n N
α ({n i })δ n 1 ,m 1 . . . δ n N ,m N . (6.144)
The energy eigenstates states are normalized to one so
ψ
α
E |ψ
α
E =
n 1
. . .
n N
|A
α
n 1 ,...,n N
|
2
α ({n i }) = 1.
(6.145)
For the case when the system is chaotic, the GOE tells us that the collection of
amplitudes {A α
n 1 ,...,n N
} for the eigenstate |ψ α
E forms an ensemble of independent
random variables with average value |A α
n 1 ,...,n N
|≈
1
N 3N
E
. Following Eq. (6.134), if
we average over the ensemble of independent random variables we obtain
A
α ∗
m 1 ,...,m N
A
β
n 1 ,...,n N EE =
1
N
3N
E
δ α,β δ n 1 ,−m 1 . . . δ n N ,−m N
α ({n i }) (6.146)
Now consider the average of the off-diagonal product of states
ψ
E,α ∗
m 1 ,...,m N
ψ
E,β
n 1 ,...,n N EE =
m o
1
. . .
m o
N
n o
1
. . .
n o
N
A
α ∗
m o
1 ,...,m o
N
A
β
n o
1 ,...,n o
N
EE
α ({m
o
i }))
α ({n
o
i })
×δ m o
1 ,m 1 . . . δ m o
N ,m N δ n o
1 ,n 1 . . . δ n o
N ,n N
=
1
N
3N
E
δ α,β
m o
1
. . .
m o
N
n o
1
. . .
n o
N
α ({m
o
i }))
α ({n
o
i })
δ n o
1 ,−m o
1
. . . δ n o
N ,−m o
N
δ m o
1 ,m 1 . . . δ m o
N ,m N δ n o
1 ,n 1 . . . δ n o
N ,n N
=
1
N
3N
E
δ α,β
m o
1
. . .
m o
N
α ({m
o
i })
δ m 1 ,m o
1
. . . δ m N ,m o
N
δ n 1 ,−m o
1
. . . δ n N ,−m o
N
=
1
N
3N
E
δ α,β δ n 1 ,−m 1 . . . δ n N ,−m N
α ({n i }).
(6.147)