Page 45 - STATISTICAL MECHANICS: From First Principles to Macroscopic Phenomena

Quantum mechanical density matrix 31

It is immediately evident that if the limit exists it is zero:
(a (τ)a ν (τ) − (a (0))a ν (0))
∗
∗
dρ νν ν ν
= lim = 0 (2.17)
dt τ→∞ τ
since the coefficients a ν must be finite if the wave functions (q, t) are to be nor-
malizable. Thus by this definition the density matrix is a constant. One can see
from this that the density matrix corresponds to an operator representing a con-
served quantity in the usual sense in quantum mechanics. We write the Schr¨odinger
equation in the representation of the states ν as

da ν
i¯h = H νν a ν (2.18)

dt
ν
which gives
d i
∗ ∗ ∗
(a (t)a ν (t)) = H ν ν a (t)a ν (t) − a (t)a ν (t)H νν (2.19)

ν
ν
ν
dt ¯ h
ν
1 τ
Then taking the time average lim τ→∞ (...)dt of both sides and assuming that
τ 0
H is not time dependent gives, with the same definition (2.16) of dρ νν /dt,

i
dρ νν
= [ρ νν H ν ν − H νν ρ ν ν ] (2.20)

dt ¯ h
ν
Thus with (2.17) we have
ρH − Hρ = 0 (2.21)
in matrix notation so that ρ can be regarded as an operator corresponding to a
constant of the motion in the quantum mechanical sense. This formulation will
also prove quite useful in describing time dependent phenomena in Part III. Now
consider a special basis in which the density matrix has a particularly simple form
which allows an unambiguous interpretation. Consider the complete set of com-
muting operators which includes the Hamiltonian. The operators represent all the
3N quantum mechanical constants of the motion of the system. In the basis ψ ν (q)
which are simultaneously eigenvalues of all these operators, ρ, which because it
is itself a constant of the motion must be a function of these 3N operators, must
also be diagonal. Because ρ is Hermitian, its diagonal matrix elements in this basis
must be real and, again from the definition, positive. Thus the quantities ρ νν can
be interpreted as the probabilities of finding the system with values of the 3N con-
stants of the motion designated by the 3N quantum numbers ν and there are no off
diagonal elements of ρ in this basis. Unfortunately, in a large interacting system,
the 3N operators associated with all the constants of the motion are never known.