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Review of Density Matrix
•
A more general description of a quantum system
is via a density operator,
!
.
–
The density operator description incorporates
ignorance into the theory
•
Answers the question: What equation do we use if we
don’t know the exact state of our system?
•
The general form of a density operator is:
–
Here
p
j
is the (classical) probability to be in state

"
j
#
.
–
The normalization condition is:
Tr
{
}=1
•
For a pure state we have a single state with
probability 1, so that:
•
When the state is not pure, the form of the
density operator depends on the situation
•
The expectation value of any operator is given
by:
j
j
j
j
p
#
=
=
{
}
m
A
m
A
Tr
A
m
"
=
=
Quantum Description of a System in
Thermodynamic Equilibrium
•
Having the most information about a system
means knowing the exact state vector

j
#
.
•
Having no knowledge of a state means that it is
equally likely to be in any state in the
N
dimensional Hilbert space:
•
It should be clear that in this sense ignorance is
equivalent to microscopic ‘disorder’ in classical
statistical mechanics
•
The measure of ignorance is called the Entropy
–
The entropy is a measure of the size of the
available statespace, subject to the constraint of
fixed mean energy
N
I
n
n
N
N
n
=
=
!
=
1
1
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View Full DocumentVonNeumann Entropy
•
The entropy should be maximum for the state
!
=I/N
and minimum for the pure state
•
There are many such functions of
.
The one
which gives a result which agrees with
experiment is the ‘VonNeumann Entropy’:
•
Method to evaluate the matrix logarithm
ln(
A
)
1.
First find the eigenvectors and eigenvalues of
$
.
Using these as a basis gives:
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This note was uploaded on 10/25/2010 for the course PHYSICS PHYS 851 taught by Professor Michaelmoore during the Fall '08 term at Michigan State University.
 Fall '08
 MichaelMoore
 mechanics

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