QMGreensite_merged

Hilbert vector space and vector algebra (2, 12). The orthogonality
condition is

Tr ByjBk

no

¼BjjBk

¼jkBkjBk

: ½ 2 : 202 Š

Unnormalized basis operators,Bk, can be normalized using

Bk^0 ¼Bk=hiBkjBk^1 =^2 : ½ 2 : 203 Š

The expectation value of an operatorAcan be written, by substitution of
[2.201] into [2.47],

hiðA tÞ¼TrfðÞtAg¼Tr

XK

k¼ 1

bkðÞtBkA

()

¼

XK

k¼ 1

bkðÞt TrfgBkA: ½ 2 : 204 Š

Note that Tr{AB} used in [2.204] and Tr{AyB} used in [2.202] in general
are not equal unlessAis a Hermitian operator. The time evolution of the
density operator can be expressed, by substitution of [2.201] into [2.53]
and [2.54], as

dðÞt

dt

¼
i½Š¼
H,ðÞt i

XK

k¼ 1

bkðÞt½ŠH,Bk, ½ 2 : 205 Š

ðÞ¼t expfg
iHtðÞ 0 expfg¼iHt

XK

k¼ 1

bkðÞ 0 expfg
iHtBkexpfgiHt:

½ 2 : 206 Š
The usefulness of [2.204]–[2.206] is that the evolution of the density
operator and expectation values can be calculated from a limited
number of trace operations Tr{BkBj} and transformation rules for
exp{–iHt}Bkexp{iHt}.
A transformation of the density operator is formally described as a
rotation of an initial density operator^1 to a new operator^2 under the
effect of a particular Hamiltonian,H. The notation to be employed has
the form

^1
!

Ht

^2 , ½ 2 : 207 Š

which represents the formal expression:

^2 ¼expfg
iHt^1 expfgiHt: ½ 2 : 208 Š

IfHandare expressed in terms of the angular momentum operators,
then the solutions to [2.208] are given by the expressions derived in

2.7 PRODUCTOPERATORFORMALISM 79