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Laboratory _1-Pipet teand Mi cropipette

14 pages
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
  • billionth 10-9 pico
  • mol micromol
  • volume of air from the piston
  • pipette bulb on the end of the pipette
  • pipette
  • tip
  • tube
  • liquid
  • volume
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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.
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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.
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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
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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.
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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
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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 7Dierentialequations(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.
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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
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Table2:OverviewofPhysics330 LaboratoriesTopics 1Dierentialequations(Maple)andstartlearningMatlab 2Dierentialequations(Maple),moreMatlab 3Drivendampedharmonicoscillator(Maple) Linearalgebraandpolynomialfitting(Matlab) 4Phasespace(Maple)andloopsandlogic(Matlab) 5Numericaldierentiationandintegration(Matlab) 6FastFouriertransform,uncertaintyprinciple(Matlab) 7Parametricinstability(Mathieusequation)usingMaple 8Pendulum(Maple)andsolvingnonlinearsystemsofequations(Matlab) 9Chaos:entrainment,intermittency,period-doubling(Matlab) 10Couplednon-linearoscillators(entrainment)usingMatlab 11Dynamicallystabilizedpendulum(two-timescaleanalysis)(Matlab) 12Gravitationalmotion,Keplerslaws(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
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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
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program.Keysectionsinthesetemplatesareoftenleftblanksothatthestudentcan programthemostimportantpartsofthealgorithmsbeingused,butdetailslikebuilding gridsandmakingplotsareoftensupplied.ThetopicscoveredineachlabarelistedinTable .3
Table3:OverviewofPhysics430 LaboratoriesTopics 1Spatialgridsinoneandtwodimensions 2Finite-dierenceapproximationstorstandsecondderivatives 3Convertingdierentialequationsintolinearalgebraproblems 4Resonanceandsteadystateforthewaveequation 5Vibrationsofahangingchainandquantumboundstates 6Time-stepping:thewaveequationandthestaggeredleapfrogalgorithm 7Two-dimensionalwaveequationviastaggeredleapfrog 8Thediusionequation,explicitmethod 9Diusionusinganimplicitalgorithm(Crank-Nicholson) 10AnimatingSchroedingersequationusingCrank-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
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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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