QMGreensite_merged

Substituting [2.63] and [2.64] into [2.61] yields

dr

dt

¼iU½Š,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 Š

This system obeys [2.60] if the effective Hamiltonian,He, is written as

He¼UHU
^1 
iU

dU
^1

dt

: ½ 2 : 66 Š

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]:

rðtÞ¼expðÞ
iHetrðÞ 0 expðÞiHet: ½ 2 : 67 Š

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 expðÞ
iABexpðiAÞ, ½ 2 : 68 Š

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 Š

in which Afg¼[A, ] is a commutation superoperator andA^0 :¼E.
Thus, the propagator expression is evaluated as

BðÞ¼Bþi½ŠþA,B

ðiÞ^2

2

h

A,½ŠA,B

i

þ...: ½ 2 : 70 Š

2.2 THEDENSITYMATRIX 45