QMGreensite_merged

Htime independent is the quantum mechanical equivalent of the
rotating frame transformation in the Bloch picture.
Asimilarity transformationapplied to the laboratory frame density
operatorgenerates a transformed density operatorr, such that

r¼UU
^1 , ½ 2 : 59 Š

in which U is an operator. A unitary operator is defined by the
relationshipU–1¼Uy.IfUis a unitary operator, then [2.59] is called a
unitary transformation. The equation of motion forris described by

drðtÞ

dt

¼
i½ŠHe,rðtÞ, ½ 2 : 60 Š

in whichHeis a transformed Hamiltonian. The form of He can be
established as follows:

dr

dt

¼

dUU
^1



dt

¼U

d

dt

U
^1 þ

dU

dt

U
^1 þU

dU
^1

dt

¼iU½Š,HU
^1 þ

dU

dt

U
^1 UU
^1 þUU
^1 U

dU
^1

dt

¼iU½Š,HU
^1 þ

dU

dt

U
^1 rþrU

dU
^1

dt

: ½ 2 : 61 Š

The common technique of insertingE¼U–1Uhas been utilized. To
proceed, the following identities are established:

dE

dt

¼

dUU
^1



dt

¼

dU

dt

U
^1 þU

dU
^1

dt

¼0, ½ 2 : 62 Š

which yields

dU

dt

U
^1 ¼
U

dU
^1

dt

, ½ 2 : 63 Š

and

U½Š,HU
^1 ¼UðÞH
HU
^1 ¼UU
^1 UHU
^1 
UHU
^1 UU
^1

¼ r,UHU
^1

   

:

½ 2 : 64 Š

44 CHAPTER 2 THEORETICALDESCRIPTION OFNMR SPECTROSCOPY