The 'Density Operator': Phy851 Fall 2009
The 'Density Operator': Phy851 Fall 2009
The 'Density Operator': Phy851 Fall 2009
Main idea:
We need to distinguish between a `statistical mixture and a `coherent superposition Statistical mixture: it is either a or b, but we dont know which one
No interference effects
Expectation Value
The expectation value of an operator is defined (with respect to state |) as:
A A
The interpretation is the average of the results of many measurements of the observable A on a system prepared in state |.
Proof:
A A
= an an A
n
= an an an
n
= an an an
n
= an an
n
= p ( an ) an
n
Suppose I know that with probability P1, the system is in state |1, while with probability P2, the system is in state |2.
This is called a statistical mixture of the states |1 and |2.
In this case, what would be the probability of obtaining result an of a measurement of observable A?
Clearly, the probability would be 1|anan|1 with probability p1, and 2|anan|2 with probability p2.
p ( an ) = an 1
P 1 + an 2
P2
= 1 1 P 1 + 2 2 P 2
The probability to obtain result an could then obtained in the following manner:
I ( an ) = an an
{|m} is a complete basis
= m (P 1 1 1 + P 2 2 2 )an an m = an m m (P 1 1 1 + P 2 2 2 )an
m
= an (P 1 1 1 + P 2 2 2 )an =P 1 an 1 1 an + P 2 an 2 2 an
2 2
= an 1
P 1 + an 2
P2
= Pj j j
j
P
j
=1
The |js are required to be normalized to one, but are not necessarily orthogonal
For example, we could say that with 50% probability, an electron is in state |, and the other 50% of the time it is in state (|+|)/2
1 1 ( + ) ( + ) = + 2 2 2 2 3 1 1 1 = + + + 4 4 4 4
This state is only `partially mixed, meaning interference effects are reduced, but not eliminated
=
Every density matrix does not have a pure state description
Any density matrix can be tested to see if it corresponds to a pure state or not:
Test #1:
If it is a pure state, it will have exactly one non-zero eigenvalue equal to unity Proof:
Start from: Pick any orthonormal basis that spans the Hilbert space, for which | is the first basis vector In any such basis, we will have the matrix elements
m n = m ,1 n ,1
1 0 = 0 M
0 0 0 M
0 0 0 M
L L L O
mn nm = mm nn
A partially mixed state will satisfy for at least one pair of m,n values:
0 < mn nm < mm nn
And a totally mixed state will satisfy for at least one pair of m, n values:
mn = nm = 0 and
mm nm 0
1 2 1 2
1 2 1 2
( + )( + )
2 2
0 3 4 0 1 4
3 1 1 1 == + + + 4 4 4 4
1 0 0 0
3 4 1 4
1 4 1 4
3 1 == + 4 4
Probabilities and`Coherence
In a given basis, the diagonal elements are always the probabilities to be in the corresponding states: The off diagonals are a measure of the coherence between any two of the basis states.
1 2 1 2
1 2 1 2
1 0 0 0
3 4 1 4
1 4 1 4
0 3 4 0 1 4
mn nm = mm nn
Rule 1: Normalization
Consider the trace of the density operator
= Pj j j
j
Tr{} = Pj j j
j
= Pj
j
Tr{} = 1
Since the Pjs are probabilities, they must sum to unity
A = Tr{A}
For a pure state this gives the usual result:
A = Tr{ A} = A
A = Tr p j j j A j = p j j A j
j
=
d d d = + dt dt dt i i = H + H h h & = ih[H , ]
Pure state will remain pure under Hamiltonian evolution
i [ H, ] ( e e + e e ) + g e e g h 2
Master equation describes state of system only, not the `environment, but includes effects of coupling to environment Pure state can evolve into mixed state
=
=
1 1 1 1 k k + k k + k k + k k 2 2 2 2
1 k + k ( 2
P ( x ) = Tr{ x x } = x x
P( x) = 1 + cos(2kx)
Incoherent mixture:
Fringes!
= NA
1 1 = k k + k k 2 2
P( x) = 1
No fringes!
( s ,e )
= cs s s
( s ,e )
(s)
( s ,e )
(e)
( s ,e )
=U
(0)
( s ,e )
(s)
(e)
= s
(s)
(e)
Strong interaction: assume that different |s states drive | into orthogonal states
s s
(e)
= s , s
As = A( s ) I ( e )
Take expectation value:
As = Tr{ ( s,e ) A( s) I ( e )}
= m
m,n (s)
(e)
( s ,e )
A I
(s)
(e)
(s)
(e)
= m
m
(s)
(e)
( s ,e )
(e)
(s)
(s)
( s) = n
n
(e )
( s,e ) n
(e )
= Tre { ( s,e )}
As = m
m
(s)
(s)
(s)
(s)
As = Trs{ ( s) A( s)}
= cs s
s
(s)
(e)
=
s s s , s
( s ,e )
= c c s
(s)
(e)
(s)
(e)
= Tre
s s s, s
( s)
( s, e )
}
s
(e)
= c c Tre s
= c c s s
s s s , s
( s)
(s)
( s)
(e)
s s
= cs s s
s
(s)
(s)
= cs s s
s
(s)
Conclusion: Any subsequent measurement on the system, will give results as if the system were in only one of the |s, chosen at random, with probability Ps = |cs|2
This is also how we would describe the `collapse of the wavefunction
( s ,e )
= cs s
s
(s)
(e)
We see that the entanglement between system and env. mimics `collapse
Is collapse during measurement real or illusion?
Pointer States: for a measuring device to work properly, the assumption, s |s = s,s will only be true if the system basis states, {|s }, are the eigenstates of the observable being measured