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2.3 The density matrix
There
are several ways of introducing the notion of density matrix. Often, one
considers the expectation value of an operator when the system is in a state
expressed as a linear combination of states. One then takes an ensemble average
of the coefficients. See e.g. Slichter or Ernst. The significance of the density
matrix in statistical physics is described in detail in Laudau and Lifshitz,
vol. of Statistical Physics. L. Ballentines poses as a postulate that the
mathematical expression of the notion of state is the state operator or density
matrix, which is
defined by the expectation value of an operator. Here we follow F. Reuse.
Statistical
average over an ensemble of systems of the expectation values of operators
Consider
an ensemble of equivalent systems with a probability pi of being in a normalized state
The
expectation value of a statistical ensemble of equivalent systems is
We now
express this average using projections in an orthonormal basis
We can
then write
The
operator r thus
defined, is called the density matrix. When using
the density matrix, I will generally omit the bar above the expectation value. Remarks
: 1) nothing
imposes that the states
2) Often,
we will use for the basis set
3) the
density matrix is hermitian :
4) in the
basis which diagonalize
Equation
of evolution of the density matrix
From
Schroedinger's equation,
In short :
If H is a
time-independent hamiltonian Ho, the
solution for the density matrix is
as can be
verified by operating the time derivative on this solution. Consider
for the basis
Then
Thermal
equilibrium
If the
system is at thermal equilibrium at the temperature T,
where Z
insures that the trace of the density matrix is one, that is :
In the
so-called high-temperature
approximation, one can consider
then
Consider a
single spin in a magnetic field. The Zeeman coupling gives
There are
2I+1 possible states. In this approximation
In the
case of many spins, it can be shown that the same approximation can be used,
provided the energy of a single spin is much less than kT (see CPS appendix E or
chapter 3.2). With this
approximation to the density matrix, one finds Curie's law for independent spins
(see exercices). The
quantum mechanical equivalent to the rotating coordinate system for the density
matrix
Consider a
spin
The
hamiltionian is then :
The
evolution of the density matrix is given by :
Consider
The
similtarity of this expression with the one for the rotating field suggest that
Using
first the definition of
Multiplying
to the left by
with the
same definition of
Note 1 :
further on we will be looking at coupled spins. So long as the couplings commute
with
Note 2 :
This type of transformation is not to be confused with the so-called interaction
representation which will be used later on. In the interaction representation we
will introduce :
where
it is the
frequency of the rf field which is involved. Description
of the spin echo with the density matrix
We
consider again the pulse sequence
with the
usual assumption that during the pulses the rf field dominates the evolution,
and in between pulses the effective field is
The system
is prepared in a state which corresponds to equilibrium in the applied magnetif
field Ho :
The first
term corresponds to an infinite temperature, where no magnetization occurs,
therefore no contribution from this term to the magnetization is expected. This
can be verified readily in the developpement below. Omitting the first term and
the coefficients, we write
Then we
have for the evolution of the density matrix
Then,
using the rotation properties of the angular momentum seen under chapter 2.1
we get successively :
Put
together, this leads to the type of "sandwich" structure which we have
already encountered :
We add the
necessary operators to this expression which bring out that the action of the
180x pulse is to modify the hamiltonian :
In the
last steps, we have introduced the identity in the form of products such as
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