3.4 Solutions to Problems
69
2. Eq. (3.6): Normalization condition.
Tr [ρ] =
n
ρ nn = 1
( 3.6)
=
n
⎛
⎝
j
P
j c
j
n (t)
∗ c
j
n (t)
⎞
⎠ =
j
P
j
n
|c
j
n (t)|
2
= 1,
because the probability in the ensemble P j and each state ψ j are both normalized to unity,
j
P
j
= 1
and
|ψ
j (t)|
2
=
n
|c
j
n (t)|
2
= 1.
3. Eq. (3.8): Time development (Liouville equation).
i ¯
h
dρ mn (t)
dt
= [Hρ − ρH ] mn
(3.8)
= i ¯
h
j
P
j
dc
j
n (t) ∗
dt
c
j
m (t) + c
j
n (t)
∗ dc
j
m (t)
dt
=
j
P
j
−
l
H
∗
nl c
j
l (t)
∗ c
j
m (t) +
l
c
j
n (t)
∗ H ml c
j
l (t)
= −
l
H ln
j
P
j c
j
l (t)
∗ c
j
m (t) +
l
H ml
j
P
j c
j
n (t)
∗ c
j
l (t)
= −
l
H ln ρ ml +
l
H ml ρ ln = [Hρ − ρH ] mn .
3.4.2 Pure and Mixed States
[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.
First we note the normalization condition for the density matrix ρ,
Tr [ρ] =
n
j
P
j c
j
n
∗
c
j
n =
j
P
j
n
c
j
n
2
=
j
P
j
= 1
(P
j
≥ 0).