ParticlesinThree-DimensionalBoxes-DemonstrationofQuantumMechanics
Purpose:Thepurposeofthisexperimentistodemonstratethequantizednatureofenergywhenaparticleisconstrainedtoasmallregionofspace.
Background:
Quantumdotsareareal-worldparticleinabox.Thesearesmallsemiconductorparticlesthatcancontainoneelectronandone“hole”(theabsenceofanelectroninthevalencebandofasemiconductor.Justlikethesemiconductorsthatareusedtomakeflashdrivesandmicroprocessors,theseelectronsandholesactlikesmallparticlewhichcanmovefreelyinsidethesemiconductor,butcannotgetout(justlikeaparticleinabox).Bycarefullyobservingquantumdotswithdifferentsizes,wecanseetheeffectofchangingthesizeoftheboxontheenergylevelsofthesystem.
Itisimportanttomakesomeadjustmenttotheparticleintheboxequationsthatwerederivedinclasstoaccountfordifferencesbetweenour“real-world”boxandtheidealizedmodel.Forathree-dimensionalidealbox,thefollowingexpressionswerederived:
Ψ 𝑥,𝑦, 𝑧 =2𝐿
!!sin
𝑛!𝜋𝑥𝐿 sin
𝑛!𝜋𝑦𝐿 sin
𝑛!𝜋𝑧𝐿
and
𝐸!!,!!,!! = (𝑛!! + 𝑛!! + 𝑛!!)!!ℏ!
!!!!.
Inthecaseofthedots,theboxesarespherical,soinsteadofLtheradiusRisusedsothelowestenergyvalueis
𝐸!"!!"! =!!ℏ!
!!!!.
Sincethereareactuallytwoparticleswithineachquantumdotratherthanjustone,theminumumenergyisthesumoftheenergyoftheelectronandthehole:
𝐸!"!#$ !"#$%&'() =!!ℏ!
!!!!!+ !!ℏ!
!!!!!.
Inaddition,sincetheboxisnotemptybutcontainsasemiconductormaterial,thereisanenergythatmustbeovercometocreatetheelectron-holepairandthisiscalledtheenergygap,Eg.Asaresult,thetotalenergyis
𝐸!"!#$ =!!ℏ!
!!!!!+ !!ℏ!
!!!!!+ 𝐸!.
Foroursemiconductingmaterial,theenergygapisknowntobe2.15x10-19J,andtheeffectivemassesoftheelectronandholeare7.29x10-32kgand5.47x10-31kgrespectively.
Inthefollowingexperiment,youwillexcitetheelectron-holepairandthelightradiatedwilloccurasaresultofthedecayofthepairbacktoazeroenergystate.Therefore,thelightobservedwillhaveenergy:
𝐸!!!"!# =!!!= 𝐸!"!#$ =
!!ℏ!
!!!!!+ !!ℏ!
!!!!!+ 𝐸!.
Bymeasuringthewavelength,itispossibletofindtheradius,R,oftheparticlesinthesolutions.Itisimportanttounderstandthattheparticlesinthesolutionsareallthesamesemiconductingmaterialwiththeonlydifferencesbetweenthesolutionsbeingthesizeoftheparticles.
Procedure:
1. Youwillusetheblue/violetLEDlighttoexcitethequantumdotsineachvial.YouwillholdtheLEDundereachvialandcollectthespectraoflightemittedbythequantumdotsbyplacingthespectrometerfibertothesideofthesolutionvial.InthiswayyouavoidcollectedthelightfromtheLED.
2. CollectaspectraofthelightemittedbytheLEDwiththeusbspectrometertoconfirmitswavelength.Save,orprint,thisspectra.
3. Collectthespectrumofthelightemittedbythequantumdotsineachofthe4vials.4. Determinethepeakpositionineachspectraandprint5. Calculatetheradiioftheparticlesinthefoursolutions.6. Comparethesewiththeknownvaluesfortheradii:2.3674nm(green),2.5339nm(yellow),
2.7182nm(orange),and2.9249(red).
Handin:
DataandAnalysis:
1. Spectraoftheblue/violetLED2. Spectraofeachofthe4solutionsofquantumdots,withpeakwavelengthmarked.3. Calculation,withequationsoftheradiiforeachsolution4. Percenterror,discusserrorsources.
Questions:
1. Whydidwechoosetouseblue/violetlight(400nm)ratherthanredlightfortheLED2. Ifathree-dimensionalcubewasusedforthebox,andthesidesoftheboxwereeach2R,what
wouldhavebeenthefourwavelengthsemittedhadthetransitionoccurredbetweenthelowesttwoenergystates?