ch 6. commuting and noncommuting operators and the surprising consequence of entanglement ms310...

Ch 6. Commuting and Noncommuting Ch 6. Commuting and Noncommuting Operators and the Surprising Operators and the Surprising

Consequence of EntanglementConsequence of Entanglement

MS310 Quantum Physical Chemistry

- Applied simple quantum mechanical framework in Applied simple quantum mechanical framework in real experiment. (Stern-Gerlach experiment)real experiment. (Stern-Gerlach experiment)

- Noncommuting operators concerning position and - Noncommuting operators concerning position and monentum. ( Heisenberg uncertainty principle)monentum. ( Heisenberg uncertainty principle)

- Particle in a 3-D box and Quantum computersParticle in a 3-D box and Quantum computers


6.1 Commutation relations

There are 2 observables a and b, corresponding operatorWe can think two cases.1) Measurement A first, B after2) Measurement B first, A after

If ψn(x) is eigenfunction of operator A(no state change)

BA ˆ and ˆ

)x(AB))x(A(B :t measuremen second ,)x(A :t measuremenfirst 1) nnn

)(ˆ)(ˆ)(ˆˆ xBxBxAB nnnnn

Also, if ψn(x) is eigenfunction of operator B(no state change also)

)()(ˆ)(ˆˆ xxBxAB nnnnnn


If ψn(x) is eigenfunction of operator B

)(ˆ)(ˆ)(ˆˆ xAxAxBA nnnnn

)(ˆˆ))(ˆ(ˆ :t measuremen second ,)(ˆ :t measuremenfirst 2) xBAxBAxB nnn

Also, if ψn(x) is eigenfunction of operator B

)()(ˆ)(ˆˆ xxAxBA nnnnnn

ψn(x) is eigenfunction of operator A and B both → result is independent of the order of measurement

Otherwise, two results are different.Two operator A and B have a common set of eigenfunction → must satisfy the commutation relation(f(x) is arbitrary function) and if it satisfied, A and B commute.Notation :

0)](ˆ[ˆ)](ˆ[ˆ xfABxfBA

commutator:]ˆ,ˆ[),(]ˆ,ˆ[)](ˆ[ˆ)](ˆ[ˆ BAxfBAxfABxfBA


ψn(x) is eigenfunction of operator A : no change after the measurement the observable a

If satisfied : state ψn(x) is not change by the two measurement the observable a and b→ ‘can measure simultaneously and exactly two observable a and b’Ex) 6.1

Momentum and a) kinetic energy b) the total energy can be known simultaneously?

Sol)Use the commutator

a)

0)](ˆ[ˆ)](ˆ[ˆ xABxBA nn

)](ˆ[ˆ)](ˆ[ˆ xfABxfBA

0)())(2

()()2

()(]ˆ,ˆ[

2ˆ,ˆ

2

22

2

22

2

22

xfdx

di

dx

d

mxf

dx

d

mdx

dixfEp

dx

d

mE

dx

dip

kineticx

kineticx


Momentum and kinetic energy is commute.Therefore, momentum and kinetic energy can be known simultaneously.

b)

0)()()()()()()()(

)()())()((

)()))((2

()())(2

()(]ˆ,ˆ[

)(2

ˆ,ˆ

2

22

2

22

2

22

xVdx

dxfixf

dx

dxVixf

dx

dxVixV

dx

dxfi

xfdx

dxVixfxV

dx

di

xfdx

dixV

dx

d

mxfxV

dx

d

mdx

dixfHp

xVdx

d

mH

dx

dip

x

x

Therefore, we cannot be known the momentum and total energy simultaneously.

6.2 The Stern-Gerlach experiment


Consider the dipole in the inhomogeneous magnetic field.

In this situation, dipole orient and deflect to the magnetic field.(parallel and antiparallel to the magnetic field)


Stern and Gerlach did the experiment. condition : external magnetic field applied to the Ag beam

Result : Ag beam split two beams. → 2 eigenvalues of measure the z-component of the magnetic momentumWe write operator of measurement the z-component of the magnetic momentum as A, wavefunction of one spin as α, other spin as β.

1||||,22

22

21

21 cccc


Cannot specify the value of c1 and c2.However, ratio of two beam is 1 by the individual measurement.

2

1|||| 2

22

1 averageaverage cc

Measure the direction of x-component of magnetic momentum of the beam of state α : ‘split’ 2 beams!(in this case, write the operator : B and wavefunction : γ, δ)

1||||,1||||,22

24

23

24

23

43 averageaverage cccccc


Then, operator A and B commute? No.

If two operator commute → eigenfunctions of 2 operators same→ result of second measurement is only 1 state. Why?

‘after’ the first measurement, wave function collapse to only 1 measured state. Second measurement measures the ‘collapsed’ state, one of the eigenfunctions of the first measurement. If two operator commute, second measurement measures the eigenfunction of operator B, and result must be one state.

However, result of second measurement also split to 2 beams.Therefore, measurement of z-component of magnetic moment and measurement of x-component of magnetic moment do not commute.

Result : Ag atom doesn’t have well-defined values for both μz and μx simultaneously.

6.2.1 The history of the Stern-Gerlach experiment


Experiment did in 1921Ag beam generation : oven in a vacuum chamber was collimated by 2 narrow slits of 0.03mm widthBeam passed into inhomogeneous magnetic field 3.5cm and impinged on a glass plate.1 hr operation in this experiment.How can see the Ag?

→ ‘sulfur’

Sulfur reacts to Ag and makes Ag2S.Ag2S : black, and it can see less than 10-7 mol of Ag → reason of successful experiment


6.3 The Heisenberg uncertainty principle

Heisenberg uncertainty principle : ‘cannot know simultaneously position and momentum of particle’

It starts that position and momentum do not commute.Wavefunction of free particle : Ψ(x,t)=Aexp[i(kx – ωt – φ)]Set φ=0 and t=0 : focus on spatial variation of ψ(x)We normalized wavefunction into finite interval [-L,L]

L

L

ikxikxL

L LAdxeeAAdxxAxA

2

1||,1,1)()( ***

Probability of x=x0 : P(x0)dx=ψ*(x0)ψ(x0)dx

L → ∞ : probability approaches to 0! → no data of positionIt gives this result : if we know momentum exactly, position is completely unknownSimilarly, if we know position exactly, momentum is completely unknown


Consider the superposition of plane waves of very similar wave vectors

See the case of m=10(21 waves superposition)

0)( ,

2

1

2

1)( 00 kkeAAex

mn

mn

xknkixik


Wave vector k0 : 7.00 x 10-10 mCase of 21 waves, peak of the probability : 0 , 3.14 x 10-10 m→ range of probability exist decrease(wave packet) : probability localized into finite interval → uncertainty of position increase.Superposition of a lot of plane wave : cannot know exactly the wave vector of particle → ‘uncertainty’ of momentum

More wave superposition occurs, uncertainty of particle decrease, but uncertainty of momentum increase!

Consider the ∆k << k0, momentum of wavefunction is given by

It means, range of momentum increase when m increase.Finally, we can obtain Heisenberg uncertainty principle

)()( 00 kmkpkmk

2

xp


Text p.88


Text p.89

6.4 The Heisenberg uncertainty principle expressed in terms of standard deviation


Heisenberg uncertainty principle can be written in the form

σx,σp : standard deviation of position and momentum

2

px

222222 , xxpp xp

This 4 values are defined by postulate 4.

2*2

2*2

2*2

2*2

))(ˆ)((

)(ˆ)(

))(ˆ)((

)(ˆ)(

dxxxxx

dxxxxx

dxxpxp

dxxpxp


Consider the particle in a box

257.0

2

1

122

1

12:1

2

1

12,

2

1

12

1,

sin2

))sin2

()(sin2

(

0cossin2

))sin2

()(sin2

(

)2

1

3

1(sin

2)sin

2()sin

2(

2

1sin

2)sin

2()sin

2(

sin2

)(,ˆ,ˆ

222

22

222

222

2

222

0

23

222

02

222

02

0

222

0

22

0

22

0

2

0

nn

n

na

a

n

a

n

a

ndx

a

xn

a

ndx

a

xn

axa

xn

ap

dxa

xn

a

xn

a

nidx

a

xn

axi

a

xn

ap

nadx

a

xnx

adx

a

xn

ax

a

xn

ax

adxa

xnx

adx

a

xn

ax

a

xn

ax

a

xn

ax

xipxx

xp

xpxp

aa

aa

aa

aa

n


n=1 : minimum → uncertainty principle satisfied for all nRelative uncertainty in x and p

0) because of instead , use(

1/

/,

2

3

1

2/2

1121

2

222

22

ppp

an

an

pnan

a

xpx

when n→∞, uncertainty of position increases. → related to probability of finding particle is equal everywhere case of large n.

However, uncertainty of momentum is independent to n. → uncertainty of momentum can be negligible. But, it is not enough : there are 2 p values when p2 determined.

Solution : change the wavefunction as eigenfunctions of momentum operator


k

k

ikxkn

n

n

eAx

x

xana

xn

ax

)(

otherwise,0)(

and 0...,4,3,2,1,sin2

)(

relative probability density of wave vector

Momentum approaches to classical value when n increase! → relative uncertainty of momentum ‘decrease’ as n increase!

6.5 A thought experiment using a particle in a 3-dimensional box


We do thought experiment by these steps.

1) A particle in a box → know the wavefunction of particle

2) Insert barrier → tunneling probability decrease in middle region

3) Move apart : separate to 2 boxes → wavefunction represent by(each function satisfies the particle in a box and can assume a=b) : superposition state

4) Look in box(measurement one of the boxes) → we can see only particle is in the left or right : ‘rapidly decay of superposition state when measurement occurs’

1||||, 22 baba rightleft

6.6 Entangled states, teleportation, and quantum computer


Consider the case in 6.5, the particle(single particle) is in the superposition state.

Also, this wavefunction is not an eigenfunction of position.

If two quantum particles are strongly coupled : entangled state

Beam of photons is incident on transparent cystalline BaTiO3. → Only 2 direction of electric field vector of photon : Horizontal(H) and Vertical(V) : polarization state

Probability of horizontal and vertical is same by the measurement, and ‘if polarization of first photon measured, the other polarization will be measured exactly!’

1||||, 22 baba rightleft


Wavefunction can be described by

This wavefunction is not an eigenfunction of single particle operator, and measure the single particle have no meaning because this system is ‘entangled’ state.

))()()()((2

1212112 HVVH


How can use this result : ‘teleportation’

There are a pair of entangled photons, Alice has photon A and Bob has photon B.

Consider the photon A is entangled to photon X. It means photon A and photon X is orthogonal.

Photon B is entangled to photon A and photon B must be orthogonal to photon A. Therefore, state of photon B is same as photon X, the message of Alice : teleportation!


More interesting application : quantum computer

Classical computer(our PC) : bit n bit memory : 000…0 to 111…1 : 2n state

Quantum computer : qubitUse the superposition of different quantum state.In photon system, H and V can be correspond to 0 and 1Superposition state → qubit n-qubit system : entangled 2n state

In a bit, 2M state stored in length M. However, in a qubit, 2M state stored in M-qubit, only one superposition state!Therefore, 2M simultaneous calculation can be parallel in M-qubit quantum computer.If M=30, 1030 calculation can be parallel, and it expects the speed of calculation improve surprisingly.


- The Heisenberg uncertainty principle limits the degree to which observables of noncommuting operators can be known simultaneously.

- The Stern-Gerlach experiment clearly demonstrates that the prediction of quantum mechanics is obeyed at the atomic level.

- Entanglement is the basis of both teleportation and quantum computing

Summary Summary

ch 6. commuting and noncommuting operators and the surprising consequence of entanglement ms310...

Documents

measurement b

operator of measurement

b commute

state n x

kinetic energy b

corresponding operator

magnetic field slide

order of measurement