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WignerDistributionfortheDoubleSlitExperiment

FrankRioux

Thequantummechanicalinterpretationofthedoubleslitexperimentisthatpositionismeasuredatthe

slitscreenandmomentumismeasuredatthedetectionscreen.Positionandmomentumareconjugate

observablesconnectedbyaFouriertransformandgovernedbytheuncertaintyprinciple.Knowingthe

slitscreengeometrymakesitpossibletocalculatethemomentumdistributionatthedetectionscreen.

Theslitscreengeometryandthereforethecoordinatewavefunctionismodeledasasuperpositionof

twoGaussianfunctions.

x( ) exp 4 x 3 2

exp 4 x 3 2

6 4 2 0 2 4 6

Position

x( )

x

ThecoordinatewavefunctionisFouriertransformedintomomentumspacetoyieldthediffraction

pattern.Notethatthiscalculationisinagreementwiththewellknowndoubleslitdiffractionpattern.

p( )1

2 6

6

xexp i p x x( )

d

6 4 2 0 2 4 6

p( ) 2

p

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TheWignerfunctionisaphasespacedistributionthatisobtainedbytheFouriertransformofeitherthe

coordinateormomentumwavefunction.Weusethecoordinatewavefunction.

W x p( )1

3

2

s xs

2

exp i s p xs

2

d

N 100 i 0 N xi 48 i

N j 0 N pj 6

12 j

N Wigneri j W xi pj

Wigner

TheWignerdistributionisfrequentlycalledaquasiprobabilitydistributionbecause,ascanbeseenin

thedisplayabove,itcanhavenegativevalues.IntegrationoftheWignerfunctionwithrespectto

momentumrecoversthecoordinatewavefunctionandintegrationwithrespecttopositionyieldsthe

momentumwavefunction.

w x( )6

6

pW x p( )

d x 5 4.98 5 w p( )5

5

xW x p( )

d p 6 5.98 6

w x( )

x

w p( )

p

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TheWignerdistributioncanbereconstructedfromexperimentalmeasurementsusingquantumstate

tomography. Reconstructivetomographyisawidelyusedtechniqueinmedicine,forexample,for

obtainingtheshapeofaninaccessibletwodimensioanlobjectfromasetofdifferentonedimensional

shadows castbythatobject.

Quantumstatereconstructionispossibleifasystemcanbepreparedrepeatedlyinthesamestate.

Subsequentmeasurementsonsuchasystemaretheneffectivelymutltiplemeasurementsonthesame

quantumstate.ThetheoreticalWignerdistributionshownaboveforthedoubleslitexperimenthasbeenreconstructedfortheheliumatom.[See,Nature,386,150(1997).

SeeFigure1of ShadowsandMirrors:ReconstructingQuantumStatesofAtomMotion, PhysicsToday,April1998,byLeibfried,Pfau,andMonroe.

Reference:DecoherenceandtheTransitionformQuantumtoClassical,WojciechJurek,PhysicsToday,October1991,pages3644.