4.3 Time-Dependent Representation
91
=
i
¯
h
∞
0
dt
g
ρ
(0)
g
g|α pq (t)μ r − μ r α pq (t)|g
exp (iω 2 t)
=
i
¯
h
∞
0
dt
α pq (t)μ r − μ r α pq (t)
exp (iω 2 t).
(4.17)
In the above derivation from the second line to third, we adopted the Heisenberg
picture to regard the Raman tensor α as a time dependent quantity,
g|α pq (t)|m
=
g|α pq |m
exp
i(−ω mg + ii mg )t
.
(4.18)
This relation is validated by the equivalence of Schrödinger and Heisenberg pictures, as discussed in the following. In general, time dependence of the expectation
value of an arbitrary physical quantity A is represented in two ways,
A(t) =
g,m
A gm ρ mg (t) =
g,m
A gm (t)ρ mg ,
(4.19)
by Schrödinger and Heisenberg pictures, respectively. The former regards the state
(density matrix ρ) is a function of time, while the latter regards the physical quantity
A as time dependent. In the Schrödinger picture, the time development of the density
matrix ρ is given by the Liouville equation (3.16),
i ¯
h
dρ mg (t)
dt
= [H 0 ρ − ρH 0 ] mg − i ¯
hh mg
ρ mg (t) − ρ
eq
mg
=
E m − E g
ρ mg (t) − i ¯
hh mg
ρ mg (t) − ρ
eq
mg
,
where the time development is driven by the Hamiltonian H = H 0 (with no
perturbation) and the states g and m are eigenstates of H 0 . Therefore, the offdiagonal element ρ mg (m = g) is given as a solution,
ρ mg (t) = ρ mg exp
i(−ω mg + ii mg )t
,
which satisfies the proper boundary condition ρ mg (t) → 0 at t → ∞ ( mg > 0).
Consequently, A(t) in Eq. (4.19) is presented by
A(t) =
g,m
A gm ρ mg (t)=
g,m
A gm ρ mg exp
i(−ω mg + ii mg )t
=
g,m
A gm (t)ρ mg .
This solution leads to the equivalent Heisenberg picture by
A gm (t) = g|A(t)|m = A gm exp
i(−ω mg + ii mg )t
.
(4.20)
91
=
i
¯
h
∞
0
dt
g
ρ
(0)
g
g|α pq (t)μ r − μ r α pq (t)|g
exp (iω 2 t)
=
i
¯
h
∞
0
dt
α pq (t)μ r − μ r α pq (t)
exp (iω 2 t).
(4.17)
In the above derivation from the second line to third, we adopted the Heisenberg
picture to regard the Raman tensor α as a time dependent quantity,
g|α pq (t)|m
=
g|α pq |m
exp
i(−ω mg + ii mg )t
.
(4.18)
This relation is validated by the equivalence of Schrödinger and Heisenberg pictures, as discussed in the following. In general, time dependence of the expectation
value of an arbitrary physical quantity A is represented in two ways,
A(t) =
g,m
A gm ρ mg (t) =
g,m
A gm (t)ρ mg ,
(4.19)
by Schrödinger and Heisenberg pictures, respectively. The former regards the state
(density matrix ρ) is a function of time, while the latter regards the physical quantity
A as time dependent. In the Schrödinger picture, the time development of the density
matrix ρ is given by the Liouville equation (3.16),
i ¯
h
dρ mg (t)
dt
= [H 0 ρ − ρH 0 ] mg − i ¯
hh mg
ρ mg (t) − ρ
eq
mg
=
E m − E g
ρ mg (t) − i ¯
hh mg
ρ mg (t) − ρ
eq
mg
,
where the time development is driven by the Hamiltonian H = H 0 (with no
perturbation) and the states g and m are eigenstates of H 0 . Therefore, the offdiagonal element ρ mg (m = g) is given as a solution,
ρ mg (t) = ρ mg exp
i(−ω mg + ii mg )t
,
which satisfies the proper boundary condition ρ mg (t) → 0 at t → ∞ ( mg > 0).
Consequently, A(t) in Eq. (4.19) is presented by
A(t) =
g,m
A gm ρ mg (t)=
g,m
A gm ρ mg exp
i(−ω mg + ii mg )t
=
g,m
A gm (t)ρ mg .
This solution leads to the equivalent Heisenberg picture by
A gm (t) = g|A(t)|m = A gm exp
i(−ω mg + ii mg )t
.
(4.20)