Hilbert vector space and vector algebra (2, 12). The orthogonality
condition is
Tr ByjBk
no
¼BjjBk
¼jkBkjBk
: ½ 2 : 202
Unnormalized basis operators,Bk, can be normalized using
Bk^0 ¼Bk=hiBkjBk^1 =^2 : ½ 2 : 203
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
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
¼i½¼H,ðÞt i
XK
k¼ 1
bkðÞt½H,Bk, ½ 2 : 205
ðÞ¼t expfgiHtðÞ 0 expfg¼iHt
XK
k¼ 1
bkðÞ 0 expfgiHtBkexpfgiHt:
½ 2 : 206
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
which represents the formal expression:
^2 ¼expfgiHt^1 expfgiHt: ½ 2 : 208
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