QMGreensite_merged

Equation [2.50] can be used to find a differential equation for the matrix
elements of the density operator,

dkhjjim

dt

¼

dckcm

dt

¼ck

dcm

dt

þ

dck

dt

cm

¼i

X

n

ckcnhjnHjim 
i

X

n

cncmhjkHjin

¼i

X

n

hjkjinhjnHjim 
i

X

n

hjkHjinhjnjim

¼ik½ŠhjHjim 
hjkHjim , ½ 2 : 51 Š

in whichHis assumed to be identical for all members of the ensemble
and the complex conjugate of [2.50] is written as

dckðÞt

dt

¼
i

X

n

cnðÞthjkHjin

"#

¼i

X

n

cnðÞthjkHjin

¼i

X

n

cnðÞthjnHjik: ½ 2 : 52 Š

The last line of [2.52] is obtained using the Hermitian property ofH
[2.13]. Equation [2.51] is written in operator form as

dðtÞ

dt

¼
i½ŠH,ðtÞ: ½ 2 : 53 Š

This is known as theLiouville–von Neumannequation and describes the
time evolution of the density operator.
The solution to [2.53] is straightforward if the Hamiltonian is time
independent:

ðÞ¼t expðÞ
iHtðÞ 0 expðÞiHt: ½ 2 : 54 Š

The exponential operator exp(A) used in [2.54] is defined by its Taylor
series expansion:

expðAÞ¼

X^1

k¼ 0

1

k!

Ak¼EþAþ

1

2

AAþ, ½ 2 : 55 Š

in which E is the identity operator. The operators A and exp(A)
necessarily commute. Using these results, [2.54] can be shown to be a

42 CHAPTER 2 THEORETICALDESCRIPTION OFNMR SPECTROSCOPY