50
3 Microscopic Expressions of Nonlinear Polarization
We notice that A(t) involves two kinds of average manipulations, the quantum
mechanical average for the wavefunction ψ j and the statistical average over the
ensemble j . The definition of the density matrix ρ in Eq. (3.4) is accordingly
extended to take account of the statistical ensemble to
ρ mn (t) =
ensemble
j
P
j c
j
n (t)
∗ c
j
m (t) = c n (t) ∗ c m (t).
(3.10)
Such ensemble of states is called “mixed state”, whereas a state represented with a
single wavefunction is called “pure state”. The density matrix ρ in Eq. (3.4) account
for a pure state, while that in Eq. (3.10) for a mixed state.
Then we extend the formulas of ρ in the preceding subsection to the mixed states.
However, it is rather surprising to confirm that the fundamental formulas of ρ are
unchanged by the extension of ρ. The following formulas are valid for the mixed
states as well as the pure states,
• Eq.(3.5): expectation value A(t) or A(t),
• Eq.(3.6): normalization condition,
• Eq.(3.8): time development (Liouville equation),
once we replace ρ in Eq. (3.4) with that in Eq. (3.10).
[Problem 3.1] Confirm that Eqs. (3.5), (3.6) and (3.8) are valid for the mixed states
as well.
The density matrix allows us to treat pure states and mixed states with the common
formalism. This is a remarkable advantage of the density matrix in treating ensemble
of states. We also note that it is possible to distinguish whether a given density
matrix ρ refers to a mixed state or a pure state by the following criterion,
Tr [ρ 2 ] = 1 for pure state,
Tr [ρ 2 ] < 1 for mixed state.
(3.11)
[Problem 3.2] Prove the criterion Eq. (3.11) to distinguish the pure state and the
mixed state. Recall the normalization condition of states and the Schwarz inequality
for inner products.
On the other hand, the wavefunction is not as convenient as the density matrix
to treat the mixed state, since the mixed state is not represented with a single wavefunction. One cannot take the statistical average for the wavefunctions themselves,
since an “ensemble averaged state” ψ =
j P j ψ j would be a completely different
state! For example, the expectation value would be different as follows,
3 Microscopic Expressions of Nonlinear Polarization
We notice that A(t) involves two kinds of average manipulations, the quantum
mechanical average for the wavefunction ψ j and the statistical average over the
ensemble j . The definition of the density matrix ρ in Eq. (3.4) is accordingly
extended to take account of the statistical ensemble to
ρ mn (t) =
ensemble
j
P
j c
j
n (t)
∗ c
j
m (t) = c n (t) ∗ c m (t).
(3.10)
Such ensemble of states is called “mixed state”, whereas a state represented with a
single wavefunction is called “pure state”. The density matrix ρ in Eq. (3.4) account
for a pure state, while that in Eq. (3.10) for a mixed state.
Then we extend the formulas of ρ in the preceding subsection to the mixed states.
However, it is rather surprising to confirm that the fundamental formulas of ρ are
unchanged by the extension of ρ. The following formulas are valid for the mixed
states as well as the pure states,
• Eq.(3.5): expectation value A(t) or A(t),
• Eq.(3.6): normalization condition,
• Eq.(3.8): time development (Liouville equation),
once we replace ρ in Eq. (3.4) with that in Eq. (3.10).
[Problem 3.1] Confirm that Eqs. (3.5), (3.6) and (3.8) are valid for the mixed states
as well.
The density matrix allows us to treat pure states and mixed states with the common
formalism. This is a remarkable advantage of the density matrix in treating ensemble
of states. We also note that it is possible to distinguish whether a given density
matrix ρ refers to a mixed state or a pure state by the following criterion,
Tr [ρ 2 ] = 1 for pure state,
Tr [ρ 2 ] < 1 for mixed state.
(3.11)
[Problem 3.2] Prove the criterion Eq. (3.11) to distinguish the pure state and the
mixed state. Recall the normalization condition of states and the Schwarz inequality
for inner products.
On the other hand, the wavefunction is not as convenient as the density matrix
to treat the mixed state, since the mixed state is not represented with a single wavefunction. One cannot take the statistical average for the wavefunctions themselves,
since an “ensemble averaged state” ψ =
j P j ψ j would be a completely different
state! For example, the expectation value would be different as follows,