Video Bonanza Assignment Solutions

BJES held the annual book bonanza this past week in the gym. Students in grades K-3 read a book and made a board with setting, summary, characters, problem and solution, author and illustrator. Students were judged by a rubric. Everybody did really great.

The winners were:

 

Kindergarten

1st Place: Nolan Ford

2nd Place: Hadley Lucas

3rd Place: Zona Stout

 

1st Grade

1st Place: Jonah Miller

2nd Place: Elijah Broadbent

3rd Place: Brady Stevenson

 

2nd Grade

1st Place: Chance Martin

2nd Place: Rita Carter

3rd Place: Emma Ford

 

3rd Grade

1st Place: Gracie Bishop

2nd Place: Rachel Schwartz (Not Pictured)

3rd Place: Zach Davis

 

Honorable Mentions were also named for each grade. They are listed below:

K-1

 

Keeylee Fraley, Isabella Brown, Jaylei Adams, Bree Daley, Kaleb Ford, Zoey Fogle, Ethan Wells, Clair Skaggs, Austin Hall, Ben Clark, David Warren, Rocker Fraley, Kaylee Morris, Allie Tucker, Gracie Stevenson, Isaac Miller, Katie Morgan, Karson Chandler, Mary Hilty, Landon Lawson, Jessie Blair, Abbi Davis, Cayden Kitchens, Kaitlin Lucas, Kelly May, Addison Pile, Kory May, Aubrey Vessels, Sarah Maier, Ellie Blanford, Noah McMahan, Jesyln Henning, Bailey Kessler

2-3

 

Landon Ford, Keilana Cargo, John Blair, Jason Riley, Austin Lawson, Adian Kiper, Izaiah Lucas, Mallory Crawford, Jesslyn Moorman, Elizabeth Ballman, Willis Henning, Erica Carman, Isabelle Pile, Gavin Cain, Jackson Henning, Natalie Hockenberry, Abigail Blair, Morgan Carwile, Braden Leslie

 

 

Written By: Kyle Saettel

 

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PROFESSOR: Hieveryone.I'minaverygoodmoodtoday.It'snothingtodowith theclass,butI'mhavingababy.

[APPLAUSE]

PROFESSOR: Sothat'skindofexciting.SoifIjuststartedgiggling,you'llknowwhy.AndinsixmonthsifIamjustweepingand ontheground,you'll alsoknow why.

Sotodaywe'regoingtodotwothings.ThefirstisI'mgoingtogiveyou--well,thefirstisreviewalittlebitofpracticefortheexamwe'regoingtohaveonThursday.So let metell you alittlebitmoreabouttheexam.Theexam, bytheway,hasbeenrescheduledtobeinthe6120,notinWalkerGym.Soit'sgoingtobetheusualplace,we'renotmoving.

AndthereasonisI'mchangingtheformatonthisexam,inparttomake italittlelessofaburden toeveryone.ButalsoinpartbecauseI'vebeenstrugglingwiththequestionofhowtomaketheexammostuseful.Thepurposeofanexamlikethisisnottogetgradesforyouguys,althoughthat'sanincidentalbyproduct.

Thepurposeistogiveyousomefeedbackonhowyou'redoing,howyourcommandofthematerialhasevolved.Andalsotohelpyoulearnsomeofthethingsthat youmightnothavemastered.Sothewaytheexamisgoingtobestructured isgoingabout15minutesofshortanswerquestions--acoupleofveryshortcomputationsbutmostlyshortanswerquestions--onpaper.You'llhandthoseback,andthenwe'llgooverthosequestionsinclassafterwards.

Soit'sarelativelylowpressureexamandit'smostlyconceptual.Itwillcovereverythingwe'vedonethroughthislastproblemsettothedegreethatwegettothelecturepartoftodayafterquestions.Today'slecturewillnot bedirectlycovered,howeverit willbefairgameforthenextmidterm--which willbemoreofatraditionalmidterm--andthat'scominginApril.

Sothestructureoftodayis,I'mgoingtogiveyouawholebunchofclickerquestions.Somakesureyou'vegotyourclickersout.Andthoseclickerquestionsaregoingtogiveyouasenseforthelevelandscopeoftheexam.

The examwillbealittleharderthantheclickerquestions,butnotawholelot.Andthedifferenceisjustthatit'sgoingtobeonpaperinfrontofyouinsteadofclickered.Andthus,thatgivesyoualittlemoreroomtodocalculationsonapieceofpaper,shortcalculations.

Andthenafterthatwe'llcomeonto--insomesenseareview,butalsoanintroductiontotheDirac Bra-Ketnotationthatmanyofourtextbooksuse,butthat wehaven'tintroducedinlecturessofar.Anyquestionsbeforewegetstarted?

AUDIENCE: Whatchannelare we?

PROFESSOR: 41.Other questions?

AUDIENCE: So nopracticeexam?

PROFESSOR: Ithinktherewillprobablynotbeapracticeexambecauseoftheshiftinformat.Todaywillbasicallybe yourpracticeexam.IfIcangetmyworktogetherandgetyouguysapracticeexamoftherightformat,thenmaybe.Butitwouldn'tbeallthatusefulistheproblem.Sowatchwhathappenstoday.

AUDIENCE: Will youpost theclickerquestionsfromtoday andthelast timeon the website?

PROFESSOR: Iwillpostthequestionsfromtodayon the website,yes.

AUDIENCE: So it's50 minutes,short answer.Is thereacertainnumber of questionsthat's goingto be on it?

PROFESSOR: I'mnot going totellyouthat.Butit'snotgoingtobeatimetrial.You'renotgoing to beracingtogetthe,you know.

AUDIENCE: Will itstill beworth as muchasan exam?

PROFESSOR: Itwill.Becausetomymind,partofthereasontodesigntheexamthiswayisthatit'stestingyourconceptualunderstanding,whichistheimportantthing.Soitwillbe.Otherquestions?

Allright.Solet'sgetstartedwiththeclickerpartoftoday.Sowe're onchannel41.Soconsider thiseigenvalueequation,twoderivativesonf--it'sconstant--timesfofx.Howmanyoftheseareeigenfunctionswiththecorrespondingeigenvalue?

You'vegotabout10seconds.Sogo aheadandputinyourclicks.Andsoyour--whoops,ohsorry.Thatjustclearedyourresponses,unfortunately.Don'tworry,they'resavedinmemory.It justclearedthemoffmyscreen.

Sotheresponses--herewego.Oops,thatdidn'twork.Wow,itjusttotallydisappearedout of thatapp.Wow,that'ssoweird.

Ohno,thatdidn'twork.Let'strythis.Wherediditgo?Ah,thereitis.Whoa,thisisallveryconfusing.

Sothatwasyourresponse,BandC,63%and38%.Solet'sgobacktothis.Quicklydiscussthiswithyourneighbor.Andnowgoaheadand clickinyournewanswersifyou--you cankeeptalking,that'sfine.

Anotherfiveseconds.Awesome.OK,that'sit.SotheanswerisB.And99%ofyou allgotthat.

I suspectsomeonedidn'tclick.OKgood,nextquestion.PsiIandPsiIIaretwosolutionsoftheSchrodingerequation.

IsthesumofthetwoofthemwithcoefficientsA andB alsoasolutionoftheSchrodingerequation?Oh, and Iforgottostarttheclickythingy.Soclicknow.

AUDIENCE: Did you mean tosayB Psi 2on thatthing?

PROFESSOR: Ohyeah,it issupposedtosayB Psi 2.ItsaysB Psi2onthenext--sorry,it'sAPsi IplusBPsiI.ItshouldsayI PsiIplusB PsiII.Thankyou.

AUDIENCE: Does itmean thatthe sameSchrodingerequation withthe same potential?

PROFESSOR: Yeah,with thesamepotential.Yeah,Yeah.

[LAUGHTER]

PROFESSOR: Wow,Yeah.SoyouknowEinsteinsaid,Goddidn't playdice.Andletmeparaphrasethat as,Goddoesn'tmesswithyou inclickerquestions.AndyouguyshaveeffectivelyuniversallythattheanswerisA,yes,superpositionprinciple.

OKnextquestionisgoingtobeforanswer--herewe go.Consideraninfinitesquarewellwithwidtha.AndcomparedtotheinfinitesquarewellwithA,thegroundstateofafinitewellislower,higher,sameenergy,orundetermined.

You've got10seconds,socontinuethinkingthroughthis.5seconds.OK,andyoumostlygotit,buthaveaquickchatwiththepersonnextto you.

Allright,let'stryagain.Suchagoodtechnique.All right,anotherfivesecondstoputinyouranswer.4-3-2-1.Andyouvirtuallyallgotitright,alowerenergy.

Andlet'sjustthinkaboutthisintuitively.Intuitively,thegradientofthepotentialistheforce,right?Sointhesecondcase,you'vegotlessforcecrammingthearticleinsidetheboxsoit'sbeingsqueezedlesstightly.

Morephysically,youseethatthereisanevanescenttailon theoutside.Whatthattellsyou isthewavefunctiondidn'thavetogotozero attheends.Itjust had togetsmallandlatchontoadyingexponential.That'sfromthequalitativeanalysisof wavefunctions.

Butmeanwhile,whatthattellsyouis,ithastocurvelessinsideinorder--itdoesn'thavetogetto zero,itjusthastogettoasmallvaluewhere itmatchesto thedecayingexponential.Soif thecurvatureisless,thentheenergyisless.Cool?

OK,nextquestion.Soanyquestionson thatbefore I?Good.

Timezerowavefunctioninfinitewellwithaisthis,sinesquaredwithanormalization.What'sthewavefunctionat asubsequenttimet?I willremindyouthatyouhavesolvedtheproblemofthe infinitesquarewell,and youknowwhattheeigenenergiesandeigenfunctionsareoftheenergyoperatorintheinfinitesquarewell.Sorememberbacktowhatthoseare.

Allright,andyouhavefiveseconds.OK,weare atabout50-50correct.Sochat with thepersonnexttoyou.All right.Andnow,anymoment,goahead andvoteagain.

Good,fivemoresecondsandthenputinyourfinalvote.OK,that'sitfornow.Sowhat's theanswer?D isthe answer,butalotofpeoplestill had somedoubts.So whowantstogiveanexplanationforwhyit'sD?

AUDIENCE: Sosinesquaredis notan eigenfunction.

PROFESSOR: Fantastic.

AUDIENCE: Soin someway ithastto beasummationofeigenfunctions.Sonotevenhavingto knowwhat theeigenfunctionsare,there's onlyone summationin it.

PROFESSOR: Excellent,excellent.

AUDIENCE: And if you doknow what theeigenfunctionsare,now you know that[INAUDIBLE].

PROFESSOR: Brilliant,soI'mgoingtorestatethat.Thatwasexactlycorrectineverystep.Sothefirstthingis,thatwavefunctionattime0,sinesquaredofx,isnotaneigenfunctionoftheenergyoperatorforthesystem.Infact,we'vecomputedtheeigenfunctionsfortheenergyoperatorintheinfinitesquarewell,andthey'resineswheretheygeta zeroattheends.

Ontheotherhand,anyfunctionsatisfyingtheboundaryconditions--normalizable,hitszeroat theboundaries--isasuperpositionofenergyeigenfunctions,andwecanusethattodeterminethetimeevolutionwetakethatsuperpositionandaddonaphase--etotheminusietuponhbarforeachoftheenergyeigenfunctions.AndinpartDweexpressthatwavefunctionasasuperpositionwiththecoefficientcndeterminedfromtheoverlapofouroriginalfunctionandtheenergyeigenfunctions.

Everyonecool withthat?Sothisisliterallyjustatranscriptionofoneofourpostulates.OK,questions?Ifyouhaveanyquestionsatall,askthem.Thisisthetimeto askthem.Yeah.

AUDIENCE: So the waygetcn[INAUDIBLE]?

PROFESSOR: Exactly.Sothewayyougetcnisbysaying,lookIhavemywavefunction,psiofxisequaltosumoverncnphinof z,wherethesephi n'senergyeigenfunctions.Ephi nisequaltoenphimofx.Andwealsoknowthattheintegralofphin'scomplexconjugatephimisequaltodeltamn.Thisisthestatement thatthey'reorthogonalandproperlynormalized.

Wealsowritethisasequaltophinphim.Andwecanusethistodeterminethecnisequaltotheinnerproductofphimwithourwavefunction.Thisisequaltotheintegraldxphistarphinstarpsi,whichisequaltotheintegraldxphi nstarsumovermofcmphim.

Butthissumovermofcmcanbepulledoutbecausethis isjustanintegraloverasumofterms,whichisthesameasthesumovermcmintegralphim--complexconjugate--phim.Andthat'sdeltamn,whichis0unlessnisequaltom,becausetheseguysareorthogonalandproperlynormalized.Sothisiszerounlessnisequaltom.Sointhesum,theonlytermthatcontributesiswhenmisequalton,thisisequaltocn.

Socnisgivenbytheoverlapofourwavefunctionwiththecorrespondingeigenfunction.Thisallowsustotakeourfunction,aknownfunction--forexample,sinesquared--andexpressthecoefficientsintermsofanoverlapofourwavefunctionsinesquaredwiththewavefunctionssine.Andthat'sexactlytheexpressionyouseebelow,cnisequaltotheintegralofsinesquared--ourwavefunction--times sine,whichistheenergyeigenfunction.

Otherquestions?OK,great.Nextproblem.Cometomecomputer.

Why can theyjust bewritteninPython.OK,good.theeigenstatesphin--whichweusuallycallpsi n,butthereitis--formanorthonormalset.Meaningintegralof phimstarphinis deltamn.Whatisthevalueofintegralpsimagainstthesumcnphin?

You've gotfiveseconds.I'll giveyou alittleextratime,becausepeopleareclickingaway.OKyounowhavesevenseconds,becausetimeisnon-linear.OK,soquicklydiscuss,becausethere'sstillsomeambiguityhere.

Allright,youhave10secondstomodifyyourclicks.Click.Allright,yesandtheansweris,C,excellent.Right?Becauseinfact,wejustdidthat.

Thatwon'thappenontheactualexam.Sohere'sthenextquestion.LettheHamiltonianonadaggerUnequalenplushomega--thatshouldbean h bar--adaggerUn.WhatcanyousayaboutadaggerUn?Hereweshouldprobablysay,whatcanonesay,becauseit'spossible--OK,whatcanonesay?

Heretheassumption--justtosay itout loud--theassumptionisthatUnisaneigenfunctionoftheenergyoperatorhwitheigenvalueen.You'vegotfiveseconds.Allright,weareat50/50.Sodiscussamongstyourselves.

Allright.OK,goahead andenteragain.Enteryourmodifiedguesses.Andyouhave10secondstodoso.Thisisalotbetter.

OK,thisisgreat.Sothisisoneofthosereallysatisfyingmoments.It'simproved,butthere'sstillsomerealdoubt here.SoIwouldliketogetonepersontoargueforbandonepersontoargueforc.So whowantstovolunteerfor each ofthose?

AUDIENCE: I'llargue forc.

PROFESSOR: Who'sgoingtoarguefor b?Someone'sgot to argueforb,comeon.

AUDIENCE: You canargue.

PROFESSOR: I'm notgoing toarguefor b.I'mnot goingtoargueforceither.Thatdefeatsthepurpose.

I'm the professor.I havetosaythisallthetime.Sowho'sgoingto argueforb?

OK,youargueforb.Who'sgoing toargueforc?Allright,yeah,thatworkedout well.Great.Arguefor b.You candoit.

AUDIENCE: SoIoriginallyaccidentally mis-clickedb, so IguessIcando this.SooriginallyIdidn'treadthequestion,andIthoughtsinceyou wereactingthe letteroperatorontheeigenstatethatyou'dget aproportionalityconstanttimessomeeigenstate.Sothat'swhyit couldpotentiallybe b.ButIdisbelieve.

PROFESSOR: It'snotthebestargumentimaginableforb,butwe'lltakeit.Sothankyou.WhenIputsomeone in animpossibleposition.Soc.

AUDIENCE: All right.I argueforc.Sowecansee justfrom thefirstline herethat this isclearly goingto be astationary[INAUDIBLE].However,wecanalso seethat it isadistinctlydifferentenergyfromthestatethatyou get aneigenfunction[INAUDIBLE].SoIsaid c,because itwasnotthe same state.

PROFESSOR: Fantastic.That'sexactlyright.Soletmewalkthroughthat.Let mesaythatallowed.

Sofromthefirstline,itisclearthattheobjectaUnisaneigenfunctionoftheenergyoperator.Andit'sgoteigenvalueenplushomega.Soitisastationarystate.It's an eigenfunctionof theenergy[INAUDIBLE].

However,itisnotthesameone,becauseithasadifferenteigenvalue.Sowhatitmeanstobethesameenergyeigenstateisithasthesameeigenvalue.If ithasadifferentenergyeigenvalue,it isadifferentstateandthey'reorthogonal.Weprovedthisbefore.

Twostateswithdifferentenergy areorthogonal.Sonotonlyare theynotthesame,theydon'tevenhaveanyoverlap.Soitis astationarystate,butit'snotproportionalwiththestateUn.It'sanimportantone.Questionsaboutthat onebeforewemoveontothenext?

AUDIENCE: Howdid you tellthat it'sproportionalto the stateUn?

PROFESSOR: If itwereproportionaltothestateUn,thenitwouldbesomeconstanttimesUn.That'swhat we meanbysaying it'sproportional.Butthenifweactedonitwiththeenergyoperator,whatwouldtheeigenvaluebe?

Letmesaythisagain.SupposeIhavea statetheUn,whichIknowthatif I actwiththeenergyoperatoronit--orthisissometimescalledhUnisequaltoEnUn.IfIactwithEonalphaUn,wherealpha'saconstant,whatisthisequalto?AlphaEnUn.

Sotheeigenvalueisthesame.It'sEn,becauseitcandividethroughby[INAUDIBLE].Soifyouhavethesamestate--meaningproportionaltoit--thenwe'll

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