LINEAR RESPONSE THEORY - Rare Earth Magnetism: Structures and Excitations - page 138

corresponding to the equation of motion

dt B ( t )= i

d

, B ( t )]

h [

H (3 . 2 . 2)

(assuming that B does not depend explicitly on time). Because the
wave functions are independent of time, in the Heisenberg picture, the
corresponding density operator ρ H must also be. Hence we may write

(3.1.3)

B ( t ) =Tr ρ ( t ) B =Tr ρ H B ( t ) .

(3 . 2 . 3)

Introducing (3.2.1) into this expression, and recalling that the trace is
invariant under a cyclic permutation of the operators within it, we obtain

ρ ( t )= e −iHt/h ρ H e iHt/h ,

or

d
dt ρ ( t )=

i
h [

− H ,ρ ( t )] . (3 . 2 . 4)

The equation of motion derived for the density operator, in the Schro-
dinger picture, is similar to the Heisenberg equation of motion above,
except for the change of sign in front of the commutator.
The density operator may be written as the sum of two terms:

ρ ( t )= ρ 0 + ρ 1 ( t ) ith[ H 0 ,ρ 0 ]=0 , (3 . 2 . 5)

where ρ 0 is the density operator (3.1.1) of the thermal-equilibrium state
which, by definition, must commute with

H 0 , and the additional contri-
bution due to f ( t ) is assumed to vanish at t

. In order to derive
ρ 1 ( t ) to leading order in f ( t ), we shall first consider the following density
operator, in the interaction picture ,

→−∞

≡ e iH 0 t/h ρ ( t ) e −iH 0 t/h ,

ρ I ( t ) (3 . 2 . 6)

for which

dt ρ I ( t )= e iH 0 t/h i dt ρ ( t ) e −iH 0 t/h

d

H 0 ,ρ ( t )]+ d

h [

i
h e iH 0 t/h [

H 1 ,ρ ( t )] e −iH 0 t/h .

= −

Because H 1 is linear in f ( t ), we may replace ρ ( t )by ρ 0 in calculating

the linear response, giving

h e iH 0 t/h

H 1 e −iH 0 t/h ,ρ 0 =

d
dt ρ I ( t )

i i

h [ A 0 ( t ) ,ρ 0 ] f ( t ) ,

−