particles in three-dimensional boxes- …particles in three-dimensional boxes- demonstration of...

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?