QMGreensite_merged

The ensemble average of coefficients,cncm, forms a matrix that is
referred to as the density matrix. The density matrix is the matrix
representation of an operator, referred to as thedensity operator,
such that

cncm¼hjnPjim ¼hjnjim ¼nm: ½ 2 : 46 Š

BecausePis a Hermitian operator, so is. An expression similar to
[2.41] for the expectation value of the propertyAin an ensemble of spins
in a mixed state can be written as

hAi¼Trfg¼A TrfgA: ½ 2 : 47 Š

The overbar will now be dropped for convenience, but an ensemble
average is implied. To evaluate the expectation value of an observable,
the matrix representation of the appropriate operator and, most
importantly, the form of the density operator must be known. The
time evolution of the system, say as it passes through a particular
sequence of rf pulses and delays, is described by the time evolution of the
density operator.

2.2.3 THELIOUVILLE–VONNEUMANNEQUATION
A differential equation that describes the evolution in time of the
density operator must be derived. Using the Dirac notation, the time-
dependent Schro ̈dinger equation [2.2] is written as

X

n

dcnðÞt

dt

jin ¼
i

X

n

cnðÞtHjin: ½ 2 : 48 Š

Multiplying both sides byhjk yields

X

n

dcnðÞt

dt

hikjn ¼
i

X

n

cnðÞthjkHjin: ½ 2 : 49 Š

The set of basis kets is orthonormal; therefore,hikjn¼0 unlessn¼k,
and [2.49] reduces to

dckðÞt

dt

¼
i

X

n

cnðÞthjkHjin: ½ 2 : 50 Š

2.2 THEDENSITYMATRIX 41