Lecture 34:The `Density Operator’

Phy851 Fall 2009

The QM `density operator’

• HAS NOTHING TO DO WITH MASS PERUNIT VOLUME

• The density operator formalism is ageneralization of the Pure State QM wehave used so far.

• New concept: Mixed state

• Used for:– Describing open quantum systems– Incorporating our ignorance into our

quantum theory

• Main idea:– We need to distinguish between a

`statistical mixture’ and a `coherentsuperposition’

– Statistical mixture: it is either a or b,but we don’t know which one

• No interference effects

– Coherent superposition: it is both aand b at the same time

• Quantum interference effects appear

Pure State quantum Mechanics

• The goal of quantum mechanics is tomake predictions regarding theoutcomes of measurements

• Using the formalism we have developedso far, the procedure is as follows:

– Take an initial state vector– Evolve it according to Schrödinger's

equation until the time the measurementtakes place

– Use the projector onto eigenstates of theobservable to predict the probabilities fordifferent results

– To confirm the prediction, one wouldprepare a system in a known initial state,make the measurement, then re-preparethe same initial state and make the samemeasurement after the same evolutiontime. With enough repetitions, the resultsshould show statistical agreement withthe results of quantum theory

Expectation Value

• The expectation value of an operator isdefined (with respect to state |ψ〉) as:

• The interpretation is the average of theresults of many measurements of theobservable A on a system prepared instate |ψ〉.– Proof:

ψψ AA ≡

ψψ AA ≡

ψψ Aaa nn

n∑=

ψψ nnn

n aaa∑=

nn

nn aaa∑= ψψ

nn

n aa∑=2

ψ

nn

n aap∑= )(

This is clearly the weighted average ofall possible outcomes

Statistical mixture of states

• What if we cannot know the exact initialquantum state of our system?– For example, suppose we only know the

temperature, T, of our system?

• Suppose I know that with probability P1,the system is in state |ψ1〉, while withprobability P2, the system is in state |ψ2〉.– This is called a statistical mixture of the

states |ψ1〉 and |ψ2〉.

• In this case, what would be theprobability of obtaining result an of ameasurement of observable A?– Clearly, the probability would be

〈ψ1|an〉〈an|ψ1〉 with probability p1, and〈ψ2|an〉〈an|ψ2〉 with probability p2.

• Thus the frequency with which an wouldbe obtained over many repetitions wouldbe

2

2

21

2

1)( PaPaap nnn ψψ +=

)()|()()|()( 2211 ψψψψ PaPPaPaP nnn +=

The Density `Operator’

• For the previous example, Let us definea `density operator’ for the system as:

• The probability to obtain result an couldthen obtained in the following manner:

• Proof:

222111 PP ψψψψρ +=

)}({)( nn aITraP ρ=nnn aaaI =)(

)}({)( nn aITraP ρ=

∑=m

nn maam ρ

{|m〉} is acomplete basis

( )∑ +=m

nn maaPPm 222111 ψψψψ

( )∑ +=m

nn aPPmma 222111 ψψψψ

( ) nn aPPa 222111 ψψψψ +=

nnnn aaPaaP 222111 ψψψψ +=

2

2

21

2

1 PaPa nn ψψ +=

This will describe thestate of the system, inplace of a wavefunction

Generic Density Operator

• For a ‘statistical mixture’ of the states{|ψj〉} with respective probabilities {Pj},the density operator is thus:

• The sum of the Pj’s is Unity:

• The |ψj〉’s are required to be normalizedto one, but are not necessarilyorthogonal– For example, we could say that with 50%

probability, an electron is in state |↑〉, andthe other 50% of the time it is in state(|↑〉+|↓〉)/√2

∑=j

jjjP ψψρ

∑ =j

jP 1

€

ρ =12↑ ↑ +

12↑ + ↓( )2

↑ + ↓( )2

=34↑ ↑ +

14↑ ↓ +

14↓ ↑ +

14↓ ↓

This state is only `partially mixed’,meaning interference effects are

reduced, but not eliminated

Density matrix of a pure state

• Every pure state has a density matrixdescription:

• Every density matrix does not have apure 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 |ψ〉 is the first basisvector

• In any such basis, we will have the matrixelements

ψψρ =

ψψρ =

1,1, nmnm δδρ =

=

OMMM

L

L

L

000

000

001

ρ

Testing for purity cont.

• Test #2:– In any basis, the pure state will satisfy for

every m,n:

– A partially mixed state will satisfy for atleast one pair of m,n values:

– And a totally mixed state will satisfy for atleast one pair of m, n values:

• Examples in spin-1/2 system:

nnmmnmmn ρρρρ =

nnmmnmmn ρρρρ <<0

00 ≠== nmmmnmmn and ρρρρ

↓↓+↑↓+↓↑+↑↑==4

1

4

1

4

1

4

3ρ

( )( )22

↓+↑↓+↑=ρ

↑↑=ρ

↓↓+↑↑==4

1

4

3ρ

00

01

21

21

21

21

41

41

41

43

41

43

0

0

Probabilities and`Coherence’

• In a given basis, the diagonal elementsare always the probabilities to be in thecorresponding states:

• The off diagonals are a measure of the‘coherence’ between any two of the basisstates.

– Coherence is maximized when:

00

01

21

21

21

21

41

41

41

43

41

43

0

0

nnmmnmmn ρρρρ =

Rule 1: Normalization

• Consider the trace of the densityoperator

∑=j

jjjP ψψρ

jjj

jPTr ψψρ ∑=}{

∑=j

jP

1}{ =ρTr

Since the Pj’s are probabilities, theymust sum to unity

Rule 2: Expectation Values

• The expectation value of any operator Ais defined as:

• For a pure state this gives the usualresult:

• For a mixed state, it gives:€

A = Tr ψ ψ A{ }= ψ Aψ

}{ ATrA ρ=

jjj

j

jjjj

Ap

ApTrA

ψψ

ψψ

∑

∑

=

=

Rule 3: Equation of motion

• For a closed system:

– Pure state will remain pure underHamiltonian evolution

• For an open system, will have additionalterms:– Called ‘master equation’– Example: 2 –level atom interacting with

quantized electric field.

– Master equation describes state of systemonly, not the `environment’, but includeseffects of coupling to environment

– Pure state can evolve into mixed state

ψψρ =

+

= ψψψψρdt

d

dt

d

dt

d

Hi

Hi

ψψψψhh

+−=

[ ]ρρ ,Hih& −=

€

˙ ρ = −ihH,ρ[ ] − Γ

2e e ρ + ρ e e( ) + Γ g e ρ e g

Example: Interference fringes

• Consider a system which is in either acoherent, or incoherent (mixture)superposition of two momentum states k,and –k:– Coherent superposition:

– Incoherent mixture:

€

ψ =12

k + −k( )

kkkkkkkk −−+−+−+=2

1

2

1

2

1

2

1ρ

€

P(x) = Tr ρ x x{ } = x ρ x

)2cos(1)( kxxP += Fringes!

kkkk −−+=2

1

2

1ρ

NA=ψ

1)( =xP No fringes!

Entanglement Gives theIllusion of decoherence

• Consider a small system in a pure state. Itis initially decoupled from theenvironment:

• Then turn on coupling to the environment:

• Let the interaction be non-dissipative– System states do not decay to lower energy

states

• Strong interaction: assume that different|s〉 states drive |φ〉 into orthogonal states

)()(),( es

ss

essc φψ ⊗

= ∑

),(),(),()0(

esesesU ψψ =′

)()()()(),( e

s

seses ssU φφ ⊗=⊗

ss

e

ss ′′ = ,

)(δφφ

The `reduced system densityoperator’

• Suppose we want to make predictionsfor system observables only– Definition of ‘system observable’:

• Take expectation value:

• Define the `reduced system densityoperator’:

• Physical predictions regarding systemobservables depend only on ρ(s):

)()( ess IAA ⊗=

€

As = Tr ρ(s,e ) A(s) ⊗ I(e ){ })()()()(),()()(

,

eseseses

nm

nmIAnm ⊗⊗⊗=∑ ρ

)()()(),()()( ss

n

eeses

m

mAnnm ∑∑= ρ

€

ρ(s) = n (e )ρ(s,e ) n (e )

n∑ = Tre ρ

(s,e ){ }

)()()()( ssss

ms mAmA ρ∑=

€

As = Trs ρ(s)A(s){ }

Entanglement mimics`collapse’

• Return to our entangled state of thesystem + environment:

• Compute density matrix:

• Compute the reduced system densityoperator:

)()(),( e

s

s

ss

essc φψ ⊗=∑

),( esψψρ =

)()()()(

,

e

s

se

s

s

ssss sscc ′

′

∗′ ⊗′⊗=∑ φφ

€

= csc ′ s ∗

s, ′ s ∑ Tre s (s)

⊗ φs(e ) ′ s (s) ⊗ φ ′ s

(e ){ }

€

ρ(s) = Tre ψ ψ(s,e ){ }

ss

s

ssss sscc φφ ′

′

∗′ ′=∑ )(

,)(2 s

ss ssc∑=

‘Collapse’ of the state

• Conclusion: Any subsequent measurementon the system, will give results as if thesystem were in only one of the |s〉, chosen atrandom, with probability Ps = |cs|2

– This is also how we would describe the`collapse’ of the wavefunction

• Yet, the true state of the whole system is not`collapsed’:

• We see that the entanglement betweensystem and env. mimics `collapse’– Is collapse during measurement real or

illusion?

• Pointer States: for a measuring device towork properly, the assumption, 〈φs |φs’ 〉 = δs,s’will only be true if the system basis states,{|s 〉}, are the eigenstates of the observablebeing measured

)(2)( s

ss

s ssc∑=ρ

)()(),( e

s

s

ss

essc φψ ⊗=∑