Laboratory _1-Pipet teand Mi cropipette

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Laboratory _1 – Pipette and Micropipette Reagents/Supplies: Micropipettes (P10, P20, P200, P1000) Pipettes (1ml, 2ml, 5ml) Micropipette tips Micro centrifuge tubes Sharpe pen 250ml Erlenmeyer flask Water Fine balance Balance Overview: The proper use and technique of using a pipette and a micropipette is essential in molecular biology. For this lab we will practice using a pipette by moving water from a flask to a culture tube.

prefixes for metric units
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mol micromol
volume of air from the piston
pipette bulb on the end of the pipette
pipette
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TEACHINGCOMPUTATIONALPHYSICSASA LABORATORYSEQUENCE

RossL.Spencer DepartmentofPhysicsandAstronomy BrighamYoungUniversity Provo,Utah84602

Abstract Theundergraduatecurriculuminmostphysicsdepartmentsissofullthataddingnewcourses isdiﬃcult,butthetrainingofanundergraduatephysicsstudentisgreatlyenhancedbylearning computationalmethods.Thestandardmethodofaddressingthisneedatmostuniversitiesisto oﬀerone3-credithourcomputationalphysicscourse,eitherearlyonasanintroduction,orlater, whenitcanaddressmoreadvancedproblems.IntheDepartmentofPhysicsandAstronomyat BrighamYoungUniversitywehavechosentousethestandard3credithoursallottedtocompu- tationalphysicsinadiﬀerentway.Computationalmethodsaretaughtas3separate1-credithour laboratories,oneforsophomores,oneforjuniors,andoneforseniors.(Theusualsemesterload ofaphysicsstudentis14-16credithours.)Studentsareintroducedtosymbolicmethodsusing Maplewhentheyaresophomores,andtonumericalmethodsusingMatlabbeginningintheirju- nioryear.Thishelpspreparethestudentsfortheirupperdivisioncourses,preparesthemforthe researchtheywilldofortheirseniorproject,andspreadscomputationalmethodsthroughoutthe undergraduatecurriculum.

I.INTRODUCTION

Physics,likeothermathematicallydevelopedsciences,hasaveryfullcurriculumbecause studentsneedtolearnchallengingmaterialdatingbackatleasttothe1700s.Theseold techniquesarestillimportant,soancientmaterialcan’tjustberemovedfromthecourse listtobereplacedbythemostrecentadvances.Soeventhoughcomputationalmethods havebeenimportantinphysicsformorethan30years,thephysicsdepartmentsthatteach them(abouthalfintheUnitedStates,accordingtoanInternetsearchofawidesample ofdepartments)onlyoﬀeronecourseonnumericalmethods.Aparticularlywell-developed exampleofsuchasinglecourseusingalaboratorysettingisthecoursedevelopedoveraspan ofnearly20yearsbyGouldandTobochnik 1,2 .(Twonotableexceptionsaretheinterdisci- plinarycomputationalphysicsdegreeatOregonStateUniversity,inwhichcomputational methodsareemployedthroughoutthecurriculum. 3,4 ,andtheCCLIProjectatLawrence University,whichintegratescomputationwithundergraduatecoursework 5–7 .) AtBrighamYoungUniversitywebeganteachingcomputationalphysicsthewaymost departmentsdo,byteachingcomputationalphysicslateinthecurriculumwhenthestu- dents’experienceallowedadvancedproblemstobetackled.Weoﬀereda3-credithour courseinnumericalmethodsatthesenior/ﬁrst-yeargraduatelevelwithanemphasison gridmethodsforsolvingpartialdiﬀerentialequations,usingtheexcellentbookbyAlejan- droGarciawhichtaughtus,amongotherthings,thevalueofMatlabasanintroductory programminglanguage. 8 Welearnedtwothingsbyteachingthiscourse.(i)Itcamesolate forourundergraduatesthatlittleconnectionbetweenthesemethodsandeithertheircourse workandortheirresearchprojectswaspossible.(ii)Lecturingandthensendingstudents oﬀtodotheirhomeworkmostlycreatedfrustration,mostlybecausemanyofourstudents havepoorprogrammingskillswhentheystartourprogram.Studentsstudyingthissubject simplyneedlotsofhelpfromtheinstructortobeabletodevelopagoodprogrammingstyle anddebuggingskills.Thelecture/homeworkstylealsomadeitdiﬃculttosurveyenough methodstogivethestudentsabroadbackground.

Tosolvethesecondproblemwestartedusingalaboratoryformatsothatwhilethestu- dentswereworkingatcomputersdoingtheirhomeworktheinstructorandsometeaching assistantswerepresenttohelpthemwithdebuggingandtoanswerquestions.(Asurveyof computationalcoursesaroundtheUnitedStates,andtheworld,showsthatotherdepart- mentsusethissolutiontoo.)Inthisformattheinstructorgivesashortintroductiontothe subjectfortheday,afterwhichthestudentsgotowork.Occasionalmini-lecturesarealso givenduringthelabperiodtoenliventhecourse,toanticipatediﬃculties,andtoprovide breadth. Wewerepleasedtoﬁndthatteachingthissubjectasalabcreatesanactivelearning environmentinwhichstudentsareexposedtoproblems,helpeachothersolvethemmuchof thetime,andarereadytolistenwhentheinstructorhassomethingtosay 9 .Thisuseofan activelearningenvironmentworkedsowellthatwedecidedtouseittosolveproblem(i)as wellbyabandoningour3-credit-hourlecturecourseandputtinginitsplacethree1-credit hourlabcourses:Physics230,330,and430.(Theusualsemesterloadofaphysicsstudent is14-16credithours.) Physics230isdesignedforsophomorephysicsstudentsandteachesthembasiccomputa- tionalskills(withemphasisonsymbolicmethods)usingMaple.Intheprocessthestudents reviewthemathematicalmethodstheylearnedintheirﬁrstyearandapplythemtophysics problemsfromtheirﬁrstandsecondyearcourses.ThetextbookforthecourseisaMaple worksheettitled IntroductiontoMapleforPhysicsStudents .Thiscoursewillbediscussed inSec.II.(AsimilarelectronictutorialinMathematicaisincludedinDubin’stextbook 10 andaprintedMathematicatutorialwithsimilarphysicsexamplesisinDeJong’sbook 11 .) Physics330isdesignedforjuniorsandfocusesonordinarydiﬀerentialequationswith applicationsmostlyfromclassicalmechanics(andespeciallynonlineardynamics).Thestu- dentsstartbylearningtosolvediﬀerentialequationsinMaple,using IntroductiontoMaple forPhysicsStudents again(weskipthistopicinPhysics230becausethestudentstyp- icallyhaven’tstudieddiﬀerentialequationsyet.)WhiletheyareenhancingtheirMaple skills,theyarealsolearningthebasicsofMatlab,culminatinginlearninghowtouseits

diﬀerentialequationsolvers.Alongthewaythestudentsreviewclassicalmechanicsandare introducedtosomeofthebasicideasofnonlineardynamics,asdiscussedinSec.III.Other textbookswhichcoverthesamematerialasPhysics330,butnotinlaboratoryformat,are listedinthebibliography 10,12–21 . Physics430isdesignedforseniors,andisdiscussedinSec.IV.Theemphasisisonmethods forpartialdiﬀerentialequationsandweconcentrateongridmethods,usingMatlab.These methodsareillustratedwithexamplesinvolvingwavemotion,diﬀusion,Schroedinger’sequa- tion,andtwo-dimensionalelectrostatics.Othertextbookswhichcoverthesamematerialas Physics430,butnotinlaboratoryformat,arelistedinthebibliography 10,12–16,19,20,22–24 . Thesecoursesarerelativelynewandarestillbeingreﬁned,butourstudentsgivethem goodmarkswhentheyevaluatethecourses.ThestudentsareespeciallypleasedwithPhysics 230becauseofthepoweritgivesthemearlyintheirstudiestosolvediﬃcultproblemsintheir otherphysicscourses.Infact,mostofourstudentsnowuseMapleastheirstandardmath- ematicalhandbook.Whenstudentsbecomeinvolvedinresearch(nowagraduationrequire- mentinourdepartment)theyoftenusebothMapleandMatlabintheirwork.Wehavenot asyet,however,performedanysystematicstudyofstudentoutcomestoassessthequalityof thesecourses.Thetextsforthesecoursesarefreelyavailableathttp://www.physics.byu.edu undertheCourseslink. Tocomparetheapproachdescribedherewithwhatisbeingdoneelsewhereseethe extensivelistingsofarticlesandbooksoncomputationalphysicsatKalamazooCollege 25 andClarkUniversity 26 .

II.PHYSICS230:MAPLEANDSOPHOMOREPHYSICS

ThegoaloftheMapleworksheet IntroductiontoMapleforPhysicsStudents istohelp thestudentdeveloptheabilitytosolvediﬃcultproblemswithasymboliccomputingengine (Mapleinourcase),andtoreviewthebasicmathematicsoftheirﬁrsttwoyearsofstudy.

Thetextisawkwardtouseinprintedform;itisdesignedtobereadbysomeonesittingin frontofacomputeronwhichMapleisinstalled. Forexample,whenthestudentslearntointegrateinMapletheyaregivenproblems involvingintegrationoveraknownchargedistributiontoﬁndtheelectrostaticpotential andtheelectricﬁeld,andtheyalsointegrateovermassdistributionstoﬁndmomentsof inertia.WhenthestudentslearntouseMapletodosumstheyreviewthebasicideasof waveinterference,including2-slitinterferenceandbeats,usingMaple’spowerfulgraphic capabilities.Thereisalsoaratherlongexerciseonwavepacketsinwhichthestudents produceMapleanimationsofpacketsindispersivemedia,illustratingtheideasofphaseand groupvelocity.Inthisunittheyalsoworkthroughthebasicsoftheuncertaintyprinciple, allowingthemtoseethatthelimitationontheproductΔ ω Δ t isnotaquantumprinciple butawaveprinciple. Wealsointroducethemtosomeofthemorediﬃcultmathematicalideasandfunctions thattheywillencounterintheirthirdandfourthyears.TheyplotBesselandLegendre functionsandareintroducedtotheideaoforthogonalfunctionsandFourieranalysisas theylearntomakeplotsandperformintegrationsinMaple. Theclassroomenvironmentconsistsofaroomﬁlledwithcomputersandstaﬀedbyan instructorandoneormoreteachingassistants.Thelabperiodlastsfor3hours.Typically eachstudentsitsinfrontoftheirowncomputer,butweencouragethemtoworkaslab partnersbecausewiththishelpsbothstudentstolearnmoreeﬀectively.Theinstructorand thestudentteachingassistantsroamtheroomansweringquestionsandhelpingthestudents overcomeprogrammingandconceptualerrors.Theinstructoralsogivesmini-lecturesabout thephysicsthatwillbecoveredthatdayandanticipatesdiﬃculties.Whenthestudents discusswhattheyaredoingwithapartner,readthetextaloudtoeachother,andfrequently calltheinstructororanassistantoverforhelp,theclassroombecomesanactivelearning environmentinwhichdoingdiﬃcultmathematicsforthreehoursisalotmorefunthan itsoundslikeitwouldbe.Thestudentsdon’tprinttheirresultsbecauseprintingwastes paperandslowsthemdown.Instead,whentheyﬁnishanassignedproblem,theinstructor

orateachingassistantsitswiththemforafewminuteslisteningtothemexplainwhatthey did,askingquestions,andcorrectingerrorsormisconceptions.Duringthesemesterthree examinationsareusuallygivenandthestudentsalsotakeaﬁnalexamination,ofteninoral format.AbrieflistofthetopicsintheninechaptersoftheMapletextaregiveninTable .1

Table1:OverviewofPhysics230 ChapterTopics 1IntroductionandusingtheMapleinterface 2Plotting(andreviewofelementaryfunctions) 3Calculus:limits,derivatives,integration,sums 4Complexarithmeticandfunctions,includingbranchcuts 5Linearalgebra:linearsystems,eigenvalues,vectorcalculus 6Solvingequations,linearandnonlinear 7Diﬀerentialequations(postponedtoPhysics330) 8Logic,loops,andprocedures 9ReviewofsymbolicalgebracommandsinMaple Inadditiontohelpingthestudentshandlediﬃcultproblems,Physics230isalsoanexcel- lentpreparationforourstandardmathematicalmethodscourseonsolvingpartialdiﬀerential equationsusingseparationofvariablesandorthogonalfunctions.Becauseourstudentsal- readyknowhowtouseMaple,themathematicalmethodscourseusesitextensively.Note, however,thatPhysics230isnotamathematicalmethodscourse.Theemphasisisonlearn- ingtowriteanddebugMaplecode,andthemathematicsandphysicsideasinthecourse aremostlyareviewofmaterialpreviouslycoveredinothercourses. TheMapletext IntroductiontoMapleforPhysicsStudents ispubliclypostedatthe Physics230linkunderCoursesathttp://www.physics.byu.edu.Solutionstotheproblems inthetextareavailablefromtheauthoruponrequest.

III.PHYSICS330:DIFFERENTIALEQUATIONSANDDYNAMICS

Therearethreetextsforthiscourse: IntroductiontoMapleforPhysicsStudents , Intro- ductiontoMatlab (88pages),and ComputationalPhysics330 (49pages),thelasttwoposted inPDFformatatthePhysics330linkunderCoursesathttp://www.physics.byu.edu.The ﬁrstisthesameMapleworksheetusedforPhysics230,butinthiscoursethematerialon diﬀerentialequationsthatwasskippedinPhysics230iscovered.Thesecondtextisablend ofaMatlabtutorialandabriefreferencemanual,andthethirdisalaboratorymanual whichgivesthestudentstheassignedtasksforeachlabperiod,andexplainssomeofthe physicsideasthatwillbeencountered. TheclassroomenvironmentisthesameasthatdiscussedinSec.IIforPhysics230, exceptthattheinstructorhastolecturealittlemorebecausethephysicalandmathematical contentofthecourseismoreadvanced.Mostofthelabsbeginwitha15-minutelectureby theinstructor.Perhapsthebestwaytoshowhowthecourseisorganizedistolisteachlab withashortdescription

Table2:OverviewofPhysics330 LaboratoriesTopics 1Diﬀerentialequations(Maple)andstartlearningMatlab 2Diﬀerentialequations(Maple),moreMatlab 3Drivendampedharmonicoscillator(Maple) Linearalgebraandpolynomialﬁtting(Matlab) 4Phasespace(Maple)andloopsandlogic(Matlab) 5Numericaldiﬀerentiationandintegration(Matlab) 6FastFouriertransform,uncertaintyprinciple(Matlab) 7Parametricinstability(Mathieu’sequation)usingMaple 8Pendulum(Maple)andsolvingnonlinearsystemsofequations(Matlab) 9Chaos:entrainment,intermittency,period-doubling(Matlab) 10Couplednon-linearoscillators(entrainment)usingMatlab 11Dynamicallystabilizedpendulum(two-timescaleanalysis)(Matlab) 12Gravitationalmotion,Kepler’slaws(Matlab) 13Hysteresisandnonlinearresonance(MapleandMatlab) 14Hysteresisandnonlinearresonancecontinued

AstheyworkthroughtheseprojectsthestudentsusethepowerofMatlabgraphicsto seetheself-similarityoffractalsandtobeintroducedtothebasicideasofdynamicalchaos. Eachofthelast5labsisalongsingleproject,includingthenon-linearsynchronizationof twoweaklycoupledoscillators(Huygen’sfamousneighboring-clocksonthewallproblem) andthestabilizationofaverticalpendulumbyrapidlyvibratingitspointofsupport.The studentsalsousegraphicsandthenumericalsolutionofdiﬀerentialequationstoillustrate andreviewallofthestandardanalyticmachineryof2-bodygravitation,includinganu- mericalveriﬁcationofKepler’slaws.Finally,theyspendtwoweeksworkingthroughthe hysteresisthatoccurswhennonlineardampedoscillatorsareexternallydriven. Thislistoftopicsmayseemoverlyambitiousforaclassthatonlymeetsforthreehours

eachweek,butthegoalofthecourseisforthestudentstolearntowriteanddebugcode, nottomastertheintricaciesofdynamics.Weassumethatthestudentseitherhavealready taken,orarecurrentlyenrolled,inourjunior-levelclassicalmechanicscourse,sothephysics topicsineachlabareeitherareviewofmaterialalreadycovered,orasupplementtoit. Thosewhohavetaughtthiscoursehavefoundthatitisespeciallyimportantforthe instructortocarefullymonitortheprogressofthestudentsduringtheﬁrst4labs,tohelp studentsnottobecomediscouraged.Studentswhohavenoprogrammingbackgroundstrug- gleatthebeginningandmanyjustgiveupunlesstheyreceivespecialattention.Ifitbecomes clearthatsomestudentsarenotgoingtobeabletoﬁnishalaboratory,theinstructoroften helpsthestudentscatchupbydoingpartofalaboratoryasaclassprojectbylecturing attheboardandhavingthestudentswritethecodeasitisbeingexplained.Lotsofhelp fromtheinstructorandfromteachingassistants(almostexclusivelyalumniofthecourse) isessentialforallofthestudentsinthecoursetohaveasuccessfulexperience. Thelasttwoweeksofthecourseinvolvealonganddiﬃcultprojectinnonlineardynamics. Alternativestothisﬁnalprojectmightbetoeitherstretchouttheﬁrst4labsinto5or6,to helpstudentswithaweakprogrammingbackgroundcomeuptospeed,orperhapstohave thelasttwolabsinvolveprojectsofthestudentschoice. SolutionstothelabproblemsforPhysics330areavailablefromtheauthoruponrequest.

IV.PHYSICS430:PARTIALDIFFERENTIALEQUATIONS

Thetextforthiscourseisalabmanualtitled ComputationalPhysics430 (88pages)which containsbothphysicalexplanationsanddescriptionsofcomputingtechniques.Theemphasis isonﬁnite-diﬀerencemethodsusinguniformgridsandtimestepping.Allprogrammingis doneinMatlab,andthesameactivelearningenvironmentdescribedintheprevioustwo sectionsisusedinthiscourse.Becauseextensiveprogrammingisrequiredinthiscourse severalMatlabscriptsareprovidedtothestudentsforthemtouseasatemplateasthey

program.Keysectionsinthesetemplatesareoftenleftblanksothatthestudentcan programthemostimportantpartsofthealgorithmsbeingused,butdetailslikebuilding gridsandmakingplotsareoftensupplied.ThetopicscoveredineachlabarelistedinTable .3

Table3:OverviewofPhysics430 LaboratoriesTopics 1Spatialgridsinoneandtwodimensions 2Finite-diﬀerenceapproximationstoﬁrstandsecondderivatives 3Convertingdiﬀerentialequationsintolinearalgebraproblems 4Resonanceandsteadystateforthewaveequation 5Vibrationsofahangingchainandquantumboundstates 6Time-stepping:thewaveequationandthestaggeredleapfrogalgorithm 7Two-dimensionalwaveequationviastaggeredleapfrog 8Thediﬀusionequation,explicitmethod 9Diﬀusionusinganimplicitalgorithm(Crank-Nicholson) 10AnimatingSchroedinger’sequationusingCrank-Nicholson 11Two-dimensionalelectrostaticsusingSuccessiveOver-Relaxation 12Two-dimensionalelectrostaticsusingSuccessiveOver-Relaxation 13Advectionandintroductiontoone-dimensionalgasdynamics 14Advectionalgorithmsappliedtosolitons(Korteweg-deVries)

Thiscourseisacompaniontoelectrodynamics,quantummechanics,andthermalphysics. Weassumethatstudentseitherhavehadcoursesinthesesubjects,orarecurrentlyenrolled, sothatthephysicsideascanbereviewedandillustratedratherthanfullydeveloped.The unifyingideainthiscourseisthepoweroflinearalgebratoeﬀectnumericalsolutionsof diﬃcultproblems.Thisturnsouttobeachallengetomostofourstudentsbecauselinear algebraseemedratherremoteandabstractwhentheysawitfortheﬁrsttime.Butitcomes aliveforthemwhentheyuseittoﬁndthevibrationfrequenciesofahangingchainand

tocomputequantumboundstateenergiesinpotentialwellsforwhichnoanalyticsolution isavailable.Theythenuselinearalgebratodevelopimplicitalgorithmsforthediﬀusion andtime-dependentSchroedingerequations,culminatinginananimationofawavepacket tunnelingthroughabarrierandananimationofsolitonsformingand”colliding”witheach other. AswithPhysics330,theinvolvementoftheinstructorineachlabiscrucial,especially atthebeginningwheresomestudentsbecomediscouragedanddropout.Eachlabbegins withalecture,andextensivecoachingbybothinstructorandteachingassistantsisrequired throughoutthelabperiod.Becauseofthemoreadvancednatureoftheproblemsinthis course,theinstructorusuallyrequiresthestudentstohavereadthroughtheassignedlabo- ratorybeforeclass,anactivitythatisencouragedbygivingashortquizonthereadingat thebeginningofeachclassperiod. Thelasttwolabsareinterestingapplicationsofthegridmethodsdevelopedinthecourse, butarenotcrucialtoﬂowofthecourse.Someinstructorsmightwanttostretchtheﬁrst fewlabsoutandeliminatetheselasttwo,orperhapsreplacethemwithlabsthatintroduce othercomputationaltopicssuchasMonte-Carlotechniquesandparticlesimulations,orallow theselasttwolabperiodstoinvolvestudent-designedprojects. Adrawbackofthislabformatisthatitisdiﬃculttofullycoverthetheorybehindthe algorithmsthatarepresented.Studentswhoareinterestedinamorecompletetreatmentare encouragedtotakeacourseonnumericalanalysisfromtheDepartmentofMathematics. Anotherconcernisthatthiscoursefocusesexclusivelyonﬁnite-diﬀerencemethodsusing uniformgrids,ignoringﬁnite-elementmethods,non-uniformgrids,Monte-Carlomethods, particlesimulationmethods,etc..Wehope,however,thatwiththisintroductionbehind themstudentswillbeabletobetterlearnothermethodsontheirownastheyencounter them.Finally,thereisalsothepotentialproblemthatprogrammingintheMatlabenviron- mentisdiﬀerentfromprogramminginacompiledlanguage.Wefeel,however,thatMatlab isanexcellentpreparationforlearningacompiledlanguage,andﬁndthatknowingMatlab isagoodintroductiontoothercomputerlanguages.

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