10.7 Density Matrix and Ensemble Averages
583
Interlude Time Dependence of the Density Matrix
The time evolution of the density matrix is given in general as
i ¯
h
∂ρ
∂t
= [H, ρ] −
in terms of the commutator [H, ρ] − ≡ Hρ − ρH of the system Hamiltonian and
the density matrix. For a time-independent Hamiltonian, the time evolution of ρ(t)
is given in terms of the initial density matrix ρ(0) as
ρ(t) = e
−iHt/¯ h ρ(0)e
iHt/¯ h ,
in which case, the matrix elements of ρ(t) may be written explicitly as
i|ρ(t)|j = e
i(E j −E i )t/¯ h
i|ρ(0)|j .
This last result shows, in particular, that for a time-independent Hamiltonian the
diagonal elements of ρ(t) are constant, while the off-diagonal elements of ρ(t)
oscillate in time with frequencies ν ij = (E j − E i )/ h. This expression is the source
of the so-called random phase approximation, whereby there is assumed to be no
phase correlation between the members of an ensemble.
The elements of the density matrix for an equilibrium system, given by
i|ρ|j = |c i | 2 δ ij
(10.7.7)
correspond to diagonal elements equal to the probabilities for finding the system in
a given energy eigenstate of H and all off-diagonal elements equal to zero. We have
seen that for a microcanonical ensemble, each of the accessible microstates
of a system having an energy that lies in an interval δE about energy E has the same
probability, namely 1//(E), so that the relevant density matrix elements are given
by
i|ρ|j =
1
(E)
δ ij .
(10.7.8)
The density matrix is thus simply a multiple of the identity matrix. Similarly, we
have seen that for the canonical ensemble thermal contact between the members
of the ensemble and a heat bath determines a reciprocal temperature parameter β ≡
(k B T ) −1 . The elements of the density matrix for a canonical ensemble are thus given
by
i|ρ|j =
1
Z(T )
e
−βE i δ ij ,
(10.7.9a)
583
Interlude Time Dependence of the Density Matrix
The time evolution of the density matrix is given in general as
i ¯
h
∂ρ
∂t
= [H, ρ] −
in terms of the commutator [H, ρ] − ≡ Hρ − ρH of the system Hamiltonian and
the density matrix. For a time-independent Hamiltonian, the time evolution of ρ(t)
is given in terms of the initial density matrix ρ(0) as
ρ(t) = e
−iHt/¯ h ρ(0)e
iHt/¯ h ,
in which case, the matrix elements of ρ(t) may be written explicitly as
i|ρ(t)|j = e
i(E j −E i )t/¯ h
i|ρ(0)|j .
This last result shows, in particular, that for a time-independent Hamiltonian the
diagonal elements of ρ(t) are constant, while the off-diagonal elements of ρ(t)
oscillate in time with frequencies ν ij = (E j − E i )/ h. This expression is the source
of the so-called random phase approximation, whereby there is assumed to be no
phase correlation between the members of an ensemble.
The elements of the density matrix for an equilibrium system, given by
i|ρ|j = |c i | 2 δ ij
(10.7.7)
correspond to diagonal elements equal to the probabilities for finding the system in
a given energy eigenstate of H and all off-diagonal elements equal to zero. We have
seen that for a microcanonical ensemble, each of the accessible microstates
of a system having an energy that lies in an interval δE about energy E has the same
probability, namely 1//(E), so that the relevant density matrix elements are given
by
i|ρ|j =
1
(E)
δ ij .
(10.7.8)
The density matrix is thus simply a multiple of the identity matrix. Similarly, we
have seen that for the canonical ensemble thermal contact between the members
of the ensemble and a heat bath determines a reciprocal temperature parameter β ≡
(k B T ) −1 . The elements of the density matrix for a canonical ensemble are thus given
by
i|ρ|j =
1
Z(T )
e
−βE i δ ij ,
(10.7.9a)