Substituting [2.63] and [2.64] into [2.61] yields
dr
dt
¼iU½,HU^1 þ
dU
dt
U^1 rþrU
dU^1
dt
¼ir,UHU^1
U
dU^1
dt
rþrU
dU^1
dt
¼ir,UHU^1
þ r,U
dU^1
dt
¼i r,UHU^1 iU
dU^1
dt
: ½ 2 : 65
This system obeys [2.60] if the effective Hamiltonian,He, is written as
He¼UHU^1 iU
dU^1
dt
: ½ 2 : 66
If a unitary transformation can be found that renders He time
independent, then the solution to [2.60] can be obtained by straight-
forward adaptation of [2.54]:
rðtÞ¼expðÞiHetrðÞ 0 expðÞiHet: ½ 2 : 67
The general procedure for solving [2.53] is as follows: find a unitary
transformation that rendersHtime independent; transform(0) andH
tor(0) andHe; solve [2.60] forr(t); and, finally, transformr(t) back to
(t).
Spin operator calculations involving unitary transformations
frequently involve propagator expressions of the general form
BðÞ¼ expðÞiABexpðiAÞ, ½ 2 : 68
in whichAandBare Hermitian operators andis a real parameter.
A series representation ofB() is given by one form of the BCH formula:
BðÞ¼
X^1
k¼ 0
ðiÞk
k!
AkfgB, ½ 2 : 69
in which Afg¼[A, ] is a commutation superoperator andA^0 :¼E.
Thus, the propagator expression is evaluated as
BðÞ¼Bþi½þA,B
ðiÞ^2
2
h
A,½A,B
i
þ...: ½ 2 : 70
2.2 THEDENSITYMATRIX 45