TheoreticalTask2(T-2):Solutions
5of9
??
xn+ky0)??
(n0+ky0)2
??1/2??
0(n00=klnn+2?1
0n0
???????2
??1/2?
=30ln?221.5+1.5
?1???
=30ln4??7
??1/2??
3+
9
=30ln??
4
3
+0.88??=24.0cm
C.TheExtremumPrincipleandtheWaveNatureofMatter
WenowexplorebetweenthePLAandthewavenatureofamovingparticle.ForthisweassumethataparticlemovingfromOtoPcantakeallpossibletrajectoriesandwewillseekatrajectorythatdependsontheconstructiveinterferenceofdeBrogliewaves.
(C1)Astheparticlemovesalongitstrajectorybyanin?nitesimaldistance?s,relatethechange
?φinthephaseofitsdeBrogliewavetothechange?AintheactionandthePlanckconstant.
Solution:
FromthedeBrogliehypothesis
λ→λdB=h/mv
whereλisthedeBrogliewavelengthandtheothersymbolshavetheirusualmeaning
?φ=2πλ
?s=
2π
h
mv?s=
2π?Ah
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TheoreticalTask2(T-2):Solutions
(C2)
RecalltheproblemfrompartAwheretheparticletraversesfromOtoP(seeFig.4).LetanopaquepartitionbeplacedattheboundaryABbetweenthetworegions.ThereisasmallopeningCDofwidthdinABsuchthatd??(x0?x1)andd??x1.
ConsidertwoextremepathsOCPandODPsuchthatOCPliesontheclassicaltrajectorydiscussedinpartA.Obtainthephasedi?erence?φCDbetweenthetwopathsto?rstorder.Solution:6of9
[1.2]
Figure4
yIAIIPDCEθ1OFBθ2xx1ConsidertheextremetrajectoriesOCPandODPof(C1)Thegeometricalpathdi?erenceisEDinregionIandCFinregionII.Thisimplies(note:d??(x0?x1)andd??x1)?φCD=?φCD=2πdsinθ12πdsinθ2?λ1λ22πmv1dsinθ12πmv2dsinθ2?hhmd=2π(v1sinθ1?v2sinθ2)h=0(fromA2orB1)Thusneartheclasicalpaththereisinvariablyconstructiveinterference.TheoreticalTask2(T-2):Solutions
7of9
D.MatterWaveInterferenceConsideranelectrongunatOwhichdi-rectsacollimatedbeamofelectronstoanarrowslitatFintheopaquepartitionA1B1atx=x1suchthatOFPisastraightline.Pisapointonthescreenatx=x0(seeFig.5).ThespeedinIisv1=2.0000×107ms?1andθ=10.0000?.Thepoten-tialinregionIIissuchthatthespeedv2=1.9900×107ms?1.Thedistancex0?x1is250.00mm(1mm=10?3m).Ignoreelectron-electroninteraction.
Figure5
IftheelectronsatOhavebeenacceleratedfromrest,calculatetheacceleratingpotential
U1.
Solution:
qU1=
1
mv22
9.11×10?31×4×1014
=
2
J=2×9.11×10?17J2×9.11×10?17=1.6×10?19eV
=1.139×103eV(??1100eV)
U1=1.139×103V
AnotheridenticalslitGismadeinthepartitionA1B1atadistanceof215.00nm(1nm
=10?9m)belowslitF(Fig.5).Ifthephasedi?erencebetweendeBrogliewavesarivingatPfromFandGis2πβ,calculateβ.
Solution:Phasedi?erenceatPis
?φ2πdsinθ2πdsinθ
P=
λ?1λ2
=2π(vmd
1?v2)
h
sin10?=2πββ=5.13
(D1)[0.3]
(D2)[0.8]
TheoreticalTask2(T-2):Solutions
8of9
(D3)Whatisisthesmallestdistance?yfromPatwhichnull(zero)electrondetectionmaybe
expectedonthescreen?[Note:youmay?ndtheapproximationsin(θ+?θ)≈sinθ+?θcosθuseful]
Solution:yIF215 nm[1.2]
A1IIPGOB1x1xFrompreviouspartfornull(zero)electrondetection?φ=5.5×2π∴mv1dsinθmv2dsin(θ+?θ)?=5.5hhsin(θ+?θ)====mv1dsinθ?5.5hmv2dhh5.5v1sinθ?v2mv2d5.52sin10??1.991374.78×1.99×107××2.15×10?70.174521?0.000935Thisyields?θ=?0.0036?TheclosestdistancetoPis?y====(x0?x1)(tan(θ+?θ)?tanθ)250(tan9.9964?tan10)?0.0162mm?16.2μmThenegativesignmeansthattheclosestminimumisbelowP.ApproximateCalculationforθand?yUsingtheapproximationsin(θ+?θ)≈sinθ+?θcosθThephasedi?erenceof5.5×2πgivesdsin10?d(sin10?+?θcos10?)mv1?mv2=5.5hhFromsolutionofthepreviouspartdsin10?dsin10?mv1?mv2=5.13hhTheoreticalTask2(T-2):Solutions
9of9
Therefore
d?θcos10?
mv2
h
=0.3700Thisyields?θ≈0.0036?
?y=?0.0162mm=?16.2μmasbefore
(D4)Theelectronbeamhasasquarecrosssectionof500nm×500nmandthesetupis2m
long.Whatshouldbetheminimumbeam?uxdensityImin(numberofelectronsperunitnormalareaperunittime)if,onanaverage,thereisatleastoneelectroninthesetupatagiventime?
Solution:Theproductofthespeedoftheelectronsandnumberofelectronperunitvolumeonanaverageyieldstheintensity.
ThusN=1=Intensity×Area×Length/ElectronSpeed=Imin×0.25×10?12×2/2×107ThisgivesImin=4×1019m?2s?1
R.Bach,D.Pope,Sy-HLiouandH.Batelaan,NewJ.ofPhysicsVol.15,033018(2013).
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