Particles in Three-Dimensional Boxes- Demonstration of Quantum Mechanics Purpose: The purpose of this experiment is to demonstrate the quantized nature of energy when a particle is constrained to a small region of space. Background: Quantum dots are a real-world particle in a box. These are small semiconductor particles that can contain one electron and one “hole” (the absence of an electron in the valence band of a semiconductor. Just like the semiconductors that are used to make flash drives and microprocessors, these electrons and holes act like small particle which can move freely inside the semiconductor, but cannot get out (just like a particle in a box). By carefully observing quantum dots with different sizes, we can see the effect of changing the size of the box on the energy levels of the system. It is important to make some adjustment to the particle in the box equations that were derived in class to account for differences between our “real-world” box and the idealized model. For a three- dimensional ideal box, the following expressions were derived: Ψ , , = 2 ! ! sin ! sin ! sin ! and ! ! ,! ! ,! ! = ( ! ! + ! ! + ! ! ) ! ! ℏ ! !!! ! . In the case of the dots, the boxes are spherical, so instead of L the radius R is used so the lowest energy value is !"!!"! = ! ! ℏ ! !!! ! . Since there are actually two particles within each quantum dot rather than just one, the minumum energy is the sum of the energy of the electron and the hole: !"!#$ !"#$%&’() = ! ! ℏ ! !! ! ! ! + ! ! ℏ ! !! ! ! ! . In addition, since the box is not empty but contains a semiconductor material, there is an energy that must be overcome to create the electron-hole pair and this is called the energy gap, E g . As a result, the total energy is !"!#$ = ! ! ℏ ! !! ! ! ! + ! ! ℏ ! !! ! ! ! + ! .

Particles in Three-Dimensional Boxes- …Particles in Three-Dimensional Boxes- Demonstration of Quantum Mechanics Purpose: The purpose of this experiment is to demonstrate the quantized

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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!.

Page 2: Particles in Three-Dimensional Boxes- …Particles in Three-Dimensional Boxes- Demonstration of Quantum Mechanics Purpose: The purpose of this experiment is to demonstrate the quantized

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?