Quantum Mechanics

7/27/2015 IntroductiontoquantummechanicsWikipedia,thefreeencyclopedia

https://en.wikipedia.org/wiki/Introduction_to_quantum_mechanics 1/21

IntroductiontoquantummechanicsFromWikipedia,thefreeencyclopedia

Quantummechanicsisthescienceoftheverysmall:thebodyofscientificprinciplesthatexplainsthebehaviourofmatteranditsinteractionswithenergyonthescaleofatomsandsubatomicparticles.

Classicalphysicsexplainsmatterandenergyonascalefamiliartohumanexperience,includingthebehaviourofastronomicalbodies.Itremainsthekeytomeasurementformuchofmodernscienceandtechnology.However,towardstheendofthe19thcentury,scientistsdiscoveredphenomenainboththelarge(macro)andthesmall(micro)worldsthatclassicalphysicscouldnotexplain.[1]AsThomasKuhnexplainsinhisanalysisofthephilosophyofscience,TheStructureofScientificRevolutions,comingtotermswiththeselimitationsledtotwomajorrevolutionsinphysicswhichcreatedashiftintheoriginalscientificparadigm:thetheoryofrelativityandthedevelopmentofquantummechanics.[2]Thisarticledescribeshowphysicistsdiscoveredthelimitationsofclassicalphysicsanddevelopedthemainconceptsofthequantumtheorythatreplaceditintheearlydecadesofthe20thcentury.Theseconceptsaredescribedinroughlytheorderinwhichtheywerefirstdiscovered.Foramorecompletehistoryofthesubject,seeHistoryofquantummechanics.

Inthissense,thewordquantummeanstheminimumamountofanyphysicalentityinvolvedinaninteraction.Certaincharacteristicsofmattercantakeonlydiscretevalues.

Lightbehavesinsomerespectslikeparticlesandinotherrespectslikewaves.Matterparticlessuchaselectronsandatomsexhibitswavelikebehaviourtoo.Somelightsources,includingneonlights,giveoffonlycertaindiscretefrequenciesoflight.Quantummechanicsshowsthatlight,alongwithallotherformsofelectromagneticradiation,comesindiscreteunits,calledphotons,andpredictsitsenergies,colours,andspectralintensities.

Someaspectsofquantummechanicscanseemcounterintuitiveorevenparadoxical,becausetheydescribebehaviourquitedifferentfromthatseenatlargerlengthscales.InthewordsofRichardFeynman,quantummechanicsdealswith"natureasSheisabsurd".[3]Forexample,theuncertaintyprincipleofquantummechanicsmeansthatthemorecloselyonepinsdownonemeasurement(suchasthepositionofaparticle),thelesspreciseanothermeasurementpertainingtothesameparticle(suchasitsmomentum)mustbecome.

Contents

1Thefirstquantumtheory:MaxPlanckandblackbodyradiation2Photons:thequantisationoflight

2.1Thephotoelectriceffect2.2Consequencesofthelightbeingquantised

3Thequantisationofmatter:theBohrmodeloftheatom4Waveparticleduality

4.1Thedoubleslitexperiment4.2ApplicationtotheBohrmodel

5Spin6Developmentofmodernquantummechanics7Copenhageninterpretation

7.1Uncertaintyprinciple7.2Wavefunctioncollapse7.3Eigenstatesandeigenvalues7.4ThePauliexclusionprinciple7.5Applicationtothehydrogenatom

8Diracwaveequation9Quantumentanglement

7/27/2015 IntroductiontoquantummechanicsWikipedia,thefreeencyclopedia

https://en.wikipedia.org/wiki/Introduction_to_quantum_mechanics 2/21

Hotmetalwork.Theyelloworangeglowisthevisiblepartofthethermalradiationemittedduetothehightemperature.Everythingelseinthepictureisglowingwiththermalradiationaswell,butlessbrightlyandatlongerwavelengthsthanthehumaneyecandetect.Afarinfraredcameracanobservethisradiation.

10Quantumfieldtheory11Quantumelectrodynamics12StandardModel13Interpretations14Applications15Seealso16Notes17References18Bibliography19Furtherreading20Externallinks

Thefirstquantumtheory:MaxPlanckandblackbodyradiation

Thermalradiationiselectromagneticradiationemittedfromthesurfaceofanobjectduetotheobject'sinternalenergy.Ifanobjectisheatedsufficiently,itstartstoemitlightattheredendofthespectrum,asitbecomesredhot.

Heatingitfurthercausesthecolourtochangefromredtoyellow,white,andblue,aslightatshorterwavelengths(higherfrequencies)beginstobeemitted.Aperfectemitterisalsoaperfectabsorber:whenitiscold,suchanobjectlooksperfectlyblack,becauseitabsorbsallthelightthatfallsonitandemitsnone.Consequently,anidealthermalemitterisknownasablackbody,andtheradiationitemitsiscalledblackbodyradiation.

Inthelate19thcentury,thermalradiationhadbeenfairlywellcharacterizedexperimentally.[note1]However,classicalphysicswasunabletoexplaintherelationshipbetweentemperaturesandpredominantfrequenciesofradiation.Physicistssearchedforasingletheorythatexplainedalltheexperimentalresults.

ThefirstmodelthatwasabletoexplainthefullspectrumofthermalradiationwasputforwardbyMaxPlanckin1900.[4]Heproposedamathematicalmodelinwhichthethermalradiationwasinequilibriumwithasetofharmonicoscillators.Toreproducetheexperimentalresults,hehadtoassumethateachoscillatorproducedanintegernumberofunitsofenergyatitssinglecharacteristicfrequency,ratherthanbeingabletoemitanyarbitraryamountofenergy.Inotherwords,theenergyofeachoscillatorwasquantized.[note2]Thequantumofenergyforeachoscillator,accordingtoPlanck,wasproportionaltothefrequencyoftheoscillatortheconstantofproportionalityisnowknownasthePlanckconstant.ThePlanckconstant,usuallywrittenash,hasthevalueof6.63 1034Js.So,theenergyEofanoscillatoroffrequencyfisgivenby

[5]

Tochangethecolourofsucharadiatingbody,itisnecessarytochangeitstemperature.Planck'slawexplainswhy:increasingthetemperatureofabodyallowsittoemitmoreenergyoverall,andmeansthatalargerproportionoftheenergyistowardsthevioletendofthespectrum.

7/27/2015 IntroductiontoquantummechanicsWikipedia,thefreeencyclopedia

https://en.wikipedia.org/wiki/Introduction_to_quantum_mechanics 3/21

Predictionsoftheamountofthermalradiationofdifferentfrequenciesemittedbyabody.CorrectvaluespredictedbyPlanck'slaw(green)contrastedagainsttheclassicalvaluesofRayleighJeanslaw(red)andWienapproximation(blue).

AlbertEinsteininaround1905.

Light(redarrows,left)isshoneuponametal.Ifthelightisofsufficientfrequency(i.e.sufficientenergy),electronsareejected(bluearrows,right).

Planck'slawwasthefirstquantumtheoryinphysics,andPlanckwontheNobelPrizein1918"inrecognitionoftheservicesherenderedtotheadvancementofPhysicsbyhisdiscoveryofenergyquanta".[6]Atthetime,however,Planck'sviewwasthatquantizationwaspurelyamathematicalconstruct,ratherthan(asisnowbelieved)afundamentalchangeinourunderstandingoftheworld.[7]

Photons:thequantisationoflight

In1905,AlbertEinsteintookanextrastep.Hesuggestedthatquantisationwasnotjustamathematicalconstruct,butthattheenergyinabeamoflightactuallyoccursinindividualpackets,whicharenowcalledphotons.[8]TheenergyofasinglephotonisgivenbyitsfrequencymultipliedbyPlanck'sconstant:

Forcenturies,scientistshaddebatedbetweentwopossibletheoriesoflight:wasitawaveordiditinsteadcompriseastreamoftinyparticles?Bythe19thcentury,thedebatewasgenerallyconsideredtohavebeensettledinfavourofthewavetheory,asitwasabletoexplainobservedeffectssuchasrefraction,diffraction,interferenceandpolarization.JamesClerkMaxwellhadshownthatelectricity,magnetismandlightareallmanifestationsofthesamephenomenon:theelectromagneticfield.Maxwell'sequations,whicharethecompletesetoflawsofclassicalelectromagnetism,describelightaswaves:acombinationofoscillatingelectricandmagneticfields.Becauseofthepreponderanceofevidenceinfavourofthewavetheory,Einstein'sideasweremetinitiallywithgreatskepticism.Eventually,however,thephotonmodelbecamefavoured.Oneofthemostsignificantpiecesofevidenceinitsfavourwasitsabilitytoexplainseveralpuzzlingpropertiesofthephotoelectriceffect,describedinthefollowingsection.Nonetheless,thewaveanalogyremainedindispensableforhelpingtounderstandothercharacteristicsoflight:diffraction,refractionandinterference.

Thephotoelectriceffect

In1887,HeinrichHertzobservedthatwhenlightwithsufficientfrequencyhitsametallicsurface,itemitselectrons.[9]In1902,PhilippLenarddiscoveredthatthemaximumpossibleenergyofanejectedelectronisrelatedtothefrequencyofthelight,nottoitsintensity:ifthefrequencyistoolow,noelectronsareejectedregardlessoftheintensity.Strongbeamsoflighttowardtheredendofthespectrummightproducenoelectricalpotentialatall,whileweakbeamsoflighttowardthevioletendofthespectrumwouldproducehigherandhighervoltages.Thelowestfrequencyoflightthatcancauseelectronstobeemitted,calledthethresholdfrequency,isdifferentfordifferentmetals.Thisobservationisatoddswithclassicalelectromagnetism,whichpredictsthattheelectron'senergyshouldbeproportionaltotheintensityoftheradiation.[10]:24Sowhenphysicistsfirstdiscovereddevicesexhibitingthephotoelectriceffect,theyinitiallyexpectedthatahigherintensityoflightwouldproduceahighervoltagefromthephotoelectricdevice.

Einsteinexplainedtheeffectbypostulatingthatabeamoflightisastreamofparticles("photons")andthat,ifthebeamisoffrequencyf,theneachphotonhasanenergyequaltohf.[9]Anelectronislikelytobestruckonlybyasinglephoton,whichimpartsatmostanenergyhftotheelectron.[9]Therefore,theintensityofthebeamhasnoeffect[note3]

7/27/2015 IntroductiontoquantummechanicsWikipedia,thefreeencyclopedia

https://en.wikipedia.org/wiki/Introduction_to_quantum_mechanics 4/21

andonlyitsfrequencydeterminesthemaximumenergythatcanbeimpartedtotheelectron.[9]

Toexplainthethresholdeffect,Einsteinarguedthatittakesacertainamountofenergy,calledtheworkfunctionanddenotedby,toremoveanelectronfromthemetal.[9]Thisamountofenergyisdifferentforeachmetal.Iftheenergyofthephotonislessthantheworkfunction,thenitdoesnotcarrysufficientenergytoremovetheelectronfromthemetal.Thethresholdfrequency,f0,isthefrequencyofaphotonwhoseenergyisequaltotheworkfunction:

Iffisgreaterthanf0,theenergyhfisenoughtoremoveanelectron.Theejectedelectronhasakineticenergy,EK,whichis,atmost,equaltothephoton'senergyminustheenergyneededtodislodgetheelectronfromthemetal:

Einstein'sdescriptionoflightasbeingcomposedofparticles,extendedPlanck'snotionofquantisedenergy,whichisthatasinglephotonofagivenfrequency,f,deliversaninvariantamountofenergy,hf.Inotherwords,individualphotonscandelivermoreorlessenergy,butonlydependingontheirfrequencies.Innature,singlephotonsarerarelyencountered.TheSunandemissionsourcesavailableinthe19thcenturyemitvastnumbersofphotonseverysecond,andsotheimportanceoftheenergycarriedbyeachindividualphotonwasnotobvious.Einstein'sideathattheenergycontainedinindividualunitsoflightdependsontheirfrequencymadeitpossibletoexplainexperimentalresultsthathadhithertoseemedquitecounterintuitive.However,althoughthephotonisaparticle,itwasstillbeingdescribedashavingthewavelikepropertyoffrequency.Onceagain,theparticleaccountoflightwasbeingcompromised[11][note4].

Consequencesofthelightbeingquantised

Therelationshipbetweenthefrequencyofelectromagneticradiationandtheenergyofeachindividualphotoniswhyultravioletlightcancausesunburn,butvisibleorinfraredlightcannot.Aphotonofultravioletlightwilldeliverahighamountofenergyenoughtocontributetocellulardamagesuchasoccursinasunburn.Aphotonofinfraredlightwilldeliveraloweramountofenergyonlyenoughtowarmone'sskin.So,aninfraredlampcanwarmalargesurface,perhapslargeenoughtokeeppeoplecomfortableinacoldroom,butitcannotgiveanyoneasunburn.

Allphotonsofthesamefrequencyhaveidenticalenergy,andallphotonsofdifferentfrequencieshaveproportionallydifferentenergies.However,althoughtheenergyimpartedbyphotonsisinvariantatanygivenfrequency,theinitialenergystateoftheelectronsinaphotoelectricdevicepriortoabsorptionoflightisnotnecessarilyuniform.Anomalousresultsmayoccurinthecaseofindividualelectrons.Forinstance,anelectronthatwasalreadyexcitedabovetheequilibriumlevelofthephotoelectricdevicemightbeejectedwhenitabsorbeduncharacteristicallylowfrequencyillumination.Statistically,however,thecharacteristicbehaviourofaphotoelectricdevicewillreflectthebehaviourofthevastmajorityofitselectrons,whichwillbeattheirequilibriumlevel.Thispointishelpfulincomprehendingthedistinctionbetweenthestudyofindividualparticlesinquantumdynamicsandthestudyofmassedparticlesinclassicalphysics.

Thequantisationofmatter:theBohrmodeloftheatom

Bythedawnofthe20thcentury,evidencerequiredamodeloftheatomwithadiffusecloudofnegativelychargedelectronssurroundingasmall,dense,positivelychargednucleus.Thesepropertiessuggestedamodelinwhichtheelectronscirclearoundthenucleuslikeplanetsorbitingasun.[note5]However,itwasalsoknownthattheatominthismodelwouldbeunstable:accordingtoclassicaltheoryorbitingelectronsareundergoingcentripetalacceleration,andshouldthereforegiveoffelectromagneticradiation,thelossofenergyalsocausingthemtospiraltowardthenucleus,collidingwithitinafractionofasecond.

7/27/2015 IntroductiontoquantummechanicsWikipedia,thefreeencyclopedia

https://en.wikipedia.org/wiki/Introduction_to_quantum_mechanics 5/21

Asecond,related,puzzlewastheemissionspectrumofatoms.Whenagasisheated,itgivesofflightonlyatdiscretefrequencies.Forexample,thevisiblelightgivenoffbyhydrogenconsistsoffourdifferentcolours,asshowninthepicturebelow.Theintensityofthelightatdifferentfrequenciesisalsodifferent.Bycontrast,whitelightconsistsofacontinuousemissionacrossthewholerangeofvisiblefrequencies.Bytheendofthenineteenthcentury,asimpleruleknownasBalmer'sformulahadbeenfoundwhichshowedhowthefrequenciesofthedifferentlineswererelatedtoeachother,thoughwithoutexplainingwhythiswas,ormakinganypredictionabouttheintensities.Theformulaalsopredictedsomeadditionalspectrallinesinultravioletandinfraredlightwhichhadnotbeenobservedatthetime.Theselineswerelaterobservedexperimentally,raisingconfidenceinthevalueoftheformula.

Emissionspectrumofhydrogen.Whenexcited,hydrogengasgivesofflightinfourdistinctcolours(spectrallines)inthevisiblespectrum,aswellasanumberoflinesintheinfraredandultraviolet.

In1885theSwissmathematicianJohannBalmerdiscoveredthateachwavelength(lambda)inthevisiblespectrumofhydrogenisrelatedtosomeintegernbytheequation

whereBisaconstantwhichBalmerdeterminedtobeequalto364.56nm.

In1888JohannesRydberggeneralizedandgreatlyincreasedtheexplanatoryutilityofBalmer'sformula.HepredictedthatisrelatedtotwointegersnandmaccordingtowhatisnowknownastheRydbergformula:[13]

whereRistheRydbergconstant,equalto0.0110nm1,andnmustbegreaterthanm.

Rydberg'sformulaaccountsforthefourvisiblewavelengthsofhydrogenbysettingm=2andn=3,4,5,6.Italsopredictsadditionalwavelengthsintheemissionspectrum:form=1andforn>1,theemissionspectrumshouldcontaincertainultravioletwavelengths,andform=3andn>3,itshouldalsocontaincertaininfraredwavelengths.Experimentalobservationofthesewavelengthscametwodecadeslater:in1908LouisPaschenfoundsomeofthepredictedinfraredwavelengths,andin1914TheodoreLymanfoundsomeofthepredictedultravioletwavelengths.[13]

NotethatbothBalmerandRydberg'sformulasinvolveintegers:inmodernterms,theyimplythatsomepropertyoftheatomisquantised.Understandingexactlywhatthispropertywas,andwhyitwasquantised,wasamajorpartinthedevelopmentofquantummechanics,aswillbeshownintherestofthisarticle.

In1913NielsBohrproposedanewmodeloftheatomthatincludedquantizedelectronorbits:electronsstillorbitthenucleusmuchasplanetsorbitaroundthesun,buttheyareonlypermittedtoinhabitcertainorbits,nottoorbitatanydistance.[14]Whenanatomemitted(orabsorbed)energy,theelectrondidnotmoveinacontinuoustrajectoryfromoneorbitaroundthenucleustoanother,asmightbeexpectedclassically.Instead,theelectronwouldjumpinstantaneously

Themathematicalformuladescribinghydrogen'semissionspectrum.

7/27/2015 IntroductiontoquantummechanicsWikipedia,thefreeencyclopedia

https://en.wikipedia.org/wiki/Introduction_to_quantum_mechanics 6/21

TheBohrmodeloftheatom,showinganelectrontransitioningfromoneorbittoanotherbyemittingaphoton.

NielsBohrasayoungman

fromoneorbittoanother,givingofftheemittedlightintheformofaphoton.[15]Thepossibleenergiesofphotonsgivenoffbyeachelementweredeterminedbythedifferencesinenergybetweentheorbits,andsotheemissionspectrumforeachelementwouldcontainanumberoflines.[16]

Startingfromonlyonesimpleassumptionabouttherulethattheorbitsmustobey,theBohrmodelwasabletorelatetheobservedspectrallinesintheemissionspectrumofhydrogentopreviouslyknownconstants.InBohr'smodelthe

electronsimplywasn'tallowedtoemitenergycontinuouslyandcrashintothenucleus:onceitwasintheclosestpermittedorbit,itwasstableforever.Bohr'smodeldidn'texplainwhytheorbitsshouldbequantisedinthatway,norwasitabletomakeaccuratepredictionsforatomswithmorethanoneelectron,ortoexplainwhysomespectrallinesarebrighterthanothers.

AlthoughsomeofthefundamentalassumptionsoftheBohrmodelweresoonfoundtobewrong,thekeyresultthatthediscretelinesinemissionspectraareduetosomepropertyoftheelectronsinatomsbeingquantisediscorrect.Thewaythattheelectrons

actuallybehaveisstrikinglydifferentfromBohr'satom,andfromwhatweseeintheworldofoureverydayexperiencethismodernquantummechanicalmodeloftheatomisdiscussedbelow.

Bohrtheorisedthattheangularmomentum,L,ofanelectronisquantised:

wherenisanintegerandhisthePlanckconstant.Startingfromthisassumption,Coulomb'slawandtheequationsofcircularmotionshowthatanelectronwithnunitsofangularmomentumwillorbitaprotonatadistancergivenby

,

wherekeistheCoulombconstant,misthemassofanelectron,andeisthechargeonanelectron.Forsimplicitythisiswrittenas

wherea0,calledtheBohrradius,isequalto0.0529nm.TheBohrradiusistheradiusofthesmallestallowedorbit.

Theenergyoftheelectron[note6]canalsobecalculated,andisgivenby

.

AmoredetailedexplanationoftheBohrmodel.

7/27/2015 IntroductiontoquantummechanicsWikipedia,thefreeencyclopedia

https://en.wikipedia.org/wiki/Introduction_to_quantum_mechanics 7/21

LouisdeBrogliein1929.DeBrogliewontheNobelPrizeinPhysicsforhispredictionthatmatteractsasawave,madeinhis1924PhDthesis.

ThusBohr'sassumptionthatangularmomentumisquantisedmeansthatanelectroncanonlyinhabitcertainorbitsaroundthenucleus,andthatitcanhaveonlycertainenergies.Aconsequenceoftheseconstraintsisthattheelectronwillnotcrashintothenucleus:itcannotcontinuouslyemitenergy,anditcannotcomeclosertothenucleusthana0(theBohrradius).

Anelectronlosesenergybyjumpinginstantaneouslyfromitsoriginalorbittoalowerorbittheextraenergyisemittedintheformofaphoton.Conversely,anelectronthatabsorbsaphotongainsenergy,henceitjumpstoanorbitthatisfartherfromthenucleus.

Eachphotonfromglowingatomichydrogenisduetoanelectronmovingfromahigherorbit,withradiusrn,toalowerorbit,rm.TheenergyEofthisphotonisthedifferenceintheenergiesEnandEmoftheelectron:

SincePlanck'sequationshowsthatthephoton'senergyisrelatedtoitswavelengthbyE=hc/,thewavelengthsoflightthatcanbeemittedaregivenby

ThisequationhasthesameformastheRydbergformula,andpredictsthattheconstantRshouldbegivenby

Therefore,theBohrmodeloftheatomcanpredicttheemissionspectrumofhydrogenintermsoffundamentalconstants.[note7]However,itwasnotabletomakeaccuratepredictionsformultielectronatoms,ortoexplainwhysomespectrallinesarebrighterthanothers.

Waveparticleduality

Justaslighthasbothwavelikeandparticlelikeproperties,matteralsohaswavelikeproperties.[17]

Matterbehavingasawavewasfirstdemonstratedexperimentallyforelectrons:abeamofelectronscanexhibitdiffraction,justlikeabeamoflightorawaterwave.[note8]Similarwavelikephenomenawerelatershownforatomsandevensmallmolecules.

Thewavelength,,associatedwithanyobjectisrelatedtoitsmomentum,p,throughthePlanckconstant,h:[18][19]

Therelationship,calledthedeBrogliehypothesis,holdsforalltypesofmatter:allmatterexhibitspropertiesofbothparticlesandwaves.

7/27/2015 IntroductiontoquantummechanicsWikipedia,thefreeencyclopedia

https://en.wikipedia.org/wiki/Introduction_to_quantum_mechanics 8/21

Thediffractionpatternproducedwhenlightisshonethroughoneslit(top)andtheinterferencepatternproducedbytwoslits(bottom).Themuchmorecomplexpatternfromtwoslits,withitssmallscaleinterferencefringes,demonstratesthewavelikepropagationoflight.

Thedoubleslitexperimentforaclassicalparticle,awave,andaquantumparticledemonstratingwaveparticleduality

Theconceptofwaveparticledualitysaysthatneithertheclassicalconceptof"particle"norof"wave"canfullydescribethebehaviourofquantumscaleobjects,eitherphotonsormatter.Waveparticledualityisanexampleoftheprincipleofcomplementarityinquantumphysics.[20][21][22][23][24]Anelegantexampleofwaveparticleduality,thedoubleslitexperiment,isdiscussedinthesectionbelow.

Thedoubleslitexperiment

Inthedoubleslitexperiment,asoriginallyperformedbyThomasYoungandAugustinFresnelin1827,abeamoflightisdirectedthroughtwonarrow,closelyspacedslits,producinganinterferencepatternoflightanddarkbandsonascreen.Ifoneoftheslitsiscoveredup,onemightnaivelyexpectthattheintensityofthefringesduetointerferencewouldbehalvedeverywhere.Infact,amuchsimplerpatternisseen,asimplediffractionpattern.Closingoneslitresultsinamuchsimplerpatterndiametricallyoppositetheopenslit.Exactlythesamebehaviourcanbedemonstratedinwaterwaves,andsothedoubleslitexperimentwasseenasademonstrationofthewavenatureoflight.

Thedoubleslitexperimenthasalsobeenperformedusingelectrons,atoms,andevenmolecules,andthesametypeofinterferencepatternisseen.Thusithasbeendemonstratedthatallmatterpossessesbothparticleandwavecharacteristics.

Evenifthesourceintensityisturneddown,sothatonlyoneparticle(e.g.photonorelectron)ispassingthroughtheapparatusatatime,thesame

interferencepatterndevelopsovertime.Thequantumparticleactsasawavewhenpassingthroughthedoubleslits,butasaparticlewhenitisdetected.Thisisatypicalfeatureofquantumcomplementarity:aquantumparticlewillactasawaveinanexperimenttomeasureitswavelikeproperties,andlikeaparticleinanexperimenttomeasureitsparticlelikeproperties.Thepointonthedetectorscreenwhereanyindividualparticleshowsupwillbetheresultofarandomprocess.However,thedistributionpatternofmanyindividualparticleswillmimicthediffractionpatternproducedbywaves.

ApplicationtotheBohrmodel

DeBroglieexpandedtheBohrmodeloftheatombyshowingthatanelectroninorbitaroundanucleuscouldbethoughtofashavingwavelikeproperties.Inparticular,anelectronwillbeobservedonlyinsituationsthatpermitastandingwavearoundanucleus.Anexampleofastandingwaveisaviolinstring,whichisfixedatbothendsandcanbemadetovibrate.Thewavescreatedbyastringedinstrumentappeartooscillateinplace,movingfromcresttotroughinanupanddownmotion.Thewavelengthofastandingwaveisrelatedtothelengthofthevibratingobjectandtheboundaryconditions.Forexample,becausetheviolinstringisfixedatbothends,itcancarrystandingwavesofwavelengths2l/n,wherelisthelengthandnisapositiveinteger.DeBrogliesuggestedthattheallowedelectronorbitswerethoseforwhichthecircumferenceoftheorbitwouldbeanintegernumberofwavelengths.Theelectron'swavelengththereforedeterminesthatonlyBohrorbitsofcertaindistancesfromthenucleusarepossible.Inturn,atanydistancefromthenucleussmallerthanacertainvalueitwouldbeimpossibletoestablishanorbit.TheminimumpossibledistancefromthenucleusiscalledtheBohrradius.[25]

7/27/2015 IntroductiontoquantummechanicsWikipedia,thefreeencyclopedia

https://en.wikipedia.org/wiki/Introduction_to_quantum_mechanics 9/21

QuantumspinversusclassicalmagnetintheSternGerlachexperiment.

DeBroglie'streatmentofquantumeventsservedasastartingpointforSchrdingerwhenhesetouttoconstructawaveequationtodescribequantumtheoreticalevents.

Spin

In1922,OttoSternandWaltherGerlachshotsilveratomsthroughan(inhomogeneous)magneticfield.Inclassicalmechanics,amagnetthrownthroughamagneticfieldmaybe,dependingonitsorientation(ifitispointingwithitsnorthernpoleupwardsordown,orsomewhereinbetween),deflectedasmallorlargedistanceupwardsordownwards.TheatomsthatSternandGerlachshotthroughthemagneticfieldactedinasimilarway.However,whilethemagnetscouldbedeflectedvariabledistances,theatomswouldalwaysbedeflectedaconstantdistanceeitherupordown.Thisimpliedthatthepropertyoftheatomwhichcorrespondstothemagnet'sorientationmustbequantised,takingoneoftwovalues(eitherupordown),asopposedtobeingchosenfreelyfromanyangle.

RalphKronigoriginatedtheideathatparticlessuchasatomsorelectronsbehaveasiftheyrotate,or"spin",aboutanaxis.Spinwouldaccountforthemissingmagneticmoment,andallowtwoelectronsinthesameorbitaltooccupydistinctquantumstatesifthey"spun"inoppositedirections,thussatisfyingtheexclusionprinciple.Thequantumnumberrepresentedthesense(positiveornegative)ofspin.

ThechoiceoforientationofthemagneticfieldusedintheSternGerlachexperimentisarbitrary.Intheanimationshownhere,thefieldisverticalandsotheatomsaredeflectedeitherupordown.Ifthemagnetisrotatedaquarterturn,theatomswillbedeflectedeitherleftorright.Usingaverticalfieldshowsthatthespinalongtheverticalaxisisquantised,andusingahorizontalfieldshowsthatthespinalongthehorizontalaxisisquantised.

If,insteadofhittingadetectorscreen,oneofthebeamsofatomscomingoutoftheSternGerlachapparatusispassedintoanother(inhomogeneous)magneticfieldorientedinthesamedirection,alloftheatomswillbedeflectedthesamewayinthissecondfield.However,ifthesecondfieldisorientedat90tothefirst,thenhalfoftheatomswillbedeflectedonewayandhalftheother,sothattheatom'sspinaboutthehorizontalandverticalaxesareindependentofeachother.However,ifoneofthesebeams(e.g.theatomsthatweredeflectedupthenleft)ispassedintoathirdmagneticfield,orientedthesamewayasthefirst,halfoftheatomswillgoonewayandhalftheother.Theactionofmeasuringtheatoms'spinwithrespecttoahorizontalfieldhaschangedtheirspinwithrespecttoaverticalfield.

TheSternGerlachexperimentdemonstratesanumberofimportantfeaturesofquantummechanics:

afeatureofthenaturalworldhasbeendemonstratedtobequantised,andonlyabletotakecertaindiscretevaluesparticlespossessanintrinsicangularmomentumthatiscloselyanalogoustotheangularmomentumofaclassicallyspinningobjectmeasurementchangesthesystembeingmeasuredinquantummechanics.Onlythespinofanobjectinonedirectioncanbeknown,andobservingthespininanotherdirectionwilldestroytheoriginalinformationaboutthespin.quantummechanicsisprobabilistic:whetherthespinofanyindividualatomsentintotheapparatusispositiveornegativeisrandom.

Developmentofmodernquantummechanics

7/27/2015 IntroductiontoquantummechanicsWikipedia,thefreeencyclopedia

https://en.wikipedia.org/wiki/Introduction_to_quantum_mechanics 10/21

TheNielsBohrInstituteinCopenhagen,whichwasafocalpointforresearchersinquantummechanicsandrelatedsubjectsinthe1920sand1930s.Mostoftheworld'sbestknowntheoreticalphysicistsspenttimethere.

In1925,WernerHeisenbergattemptedtosolveoneoftheproblemsthattheBohrmodelleftunanswered,explainingtheintensitiesofthedifferentlinesinthehydrogenemissionspectrum.Throughaseriesofmathematicalanalogies,hewroteoutthequantummechanicalanaloguefortheclassicalcomputationofintensities.[26]Shortlyafterwards,Heisenberg'scolleagueMaxBornrealisedthatHeisenberg'smethodofcalculatingtheprobabilitiesfortransitionsbetweenthedifferentenergylevelscouldbestbeexpressedbyusingthemathematicalconceptofmatrices.[note9]

Inthesameyear,buildingondeBroglie'shypothesis,ErwinSchrdingerdevelopedtheequationthatdescribesthebehaviourofaquantummechanicalwave.[27]Themathematicalmodel,calledtheSchrdingerequationafteritscreator,iscentraltoquantummechanics,definesthepermittedstationarystatesofaquantumsystem,anddescribeshowthequantumstateofaphysicalsystemchangesintime.[28]Thewaveitselfisdescribedbyamathematicalfunctionknownasa"wavefunction".Schrdingersaidthatthewavefunctionprovidesthe"meansforpredictingprobabilityofmeasurementresults".[29]

Schrdingerwasabletocalculatetheenergylevelsofhydrogenbytreatingahydrogenatom'selectronasaclassicalwave,movinginawellofelectricalpotentialcreatedbytheproton.ThiscalculationaccuratelyreproducedtheenergylevelsoftheBohrmodel.

InMay1926,SchrdingerprovedthatHeisenberg'smatrixmechanicsandhisownwavemechanicsmadethesamepredictionsaboutthepropertiesandbehaviouroftheelectronmathematically,thetwotheorieswereidentical.Yetthetwomendisagreedontheinterpretationoftheirmutualtheory.Forinstance,Heisenbergsawnoprobleminthetheoreticalpredictionofinstantaneoustransitionsofelectronsbetweenorbitsinanatom,butSchrdingerhopedthatatheorybasedoncontinuouswavelikepropertiescouldavoidwhathecalled(asparaphrasedbyWilhelmWien)"thisnonsenseaboutquantumjumps."[30]

Copenhageninterpretation

Bohr,Heisenbergandotherstriedtoexplainwhattheseexperimentalresultsandmathematicalmodelsreallymean.Theirdescription,knownastheCopenhageninterpretationofquantummechanics,aimedtodescribethenatureofrealitythatwasbeingprobedbythemeasurementsanddescribedbythemathematicalformulationsofquantummechanics.

ThemainprinciplesoftheCopenhageninterpretationare:

1. Asystemiscompletelydescribedbyawavefunction,usuallyrepresentedbytheGreekletter ("psi").(Heisenberg)

2. How changesovertimeisgivenbytheSchrdingerequation.3. Thedescriptionofnatureisessentiallyprobabilistic.Theprobabilityof

aneventforexample,whereonthescreenaparticlewillshowupinthetwoslitexperimentisrelatedtothesquareoftheabsolutevalueoftheamplitudeofitswavefunction.(Bornrule,duetoMaxBorn,whichgivesaphysicalmeaningtothewavefunctionintheCopenhageninterpretation:theprobabilityamplitude)

4. Itisnotpossibletoknowthevaluesofallofthepropertiesofthesystematthesametimethosepropertiesthatarenotknownwithprecisionmustbedescribedbyprobabilities.(Heisenberg'suncertaintyprinciple)

5. Matter,likeenergy,exhibitsawaveparticleduality.Anexperimentcandemonstratetheparticlelikepropertiesofmatter,oritswavelikepropertiesbutnotbothatthesametime.(ComplementarityprincipleduetoBohr)

6. Measuringdevicesareessentiallyclassicaldevices,andmeasureclassicalpropertiessuchaspositionandmomentum.

7. Thequantummechanicaldescriptionoflargesystemsshouldcloselyapproximatetheclassicaldescription.(CorrespondenceprincipleofBohrandHeisenberg)

7/27/2015 IntroductiontoquantummechanicsWikipedia,thefreeencyclopedia

https://en.wikipedia.org/wiki/Introduction_to_quantum_mechanics 11/21

WernerHeisenbergattheageof26.HeisenbergwontheNobelPrizeinPhysicsin1932fortheworkthathedidataroundthistime.[31]

Variousconsequencesoftheseprinciplesarediscussedinmoredetailinthefollowingsubsections.

Uncertaintyprinciple

Supposeitisdesiredtomeasurethepositionandspeedofanobjectforexampleacargoingthrougharadarspeedtrap.Itcanbeassumedthatthecarhasadefinitepositionandspeedataparticularmomentintime.Howaccuratelythesevaluescanbemeasureddependsonthequalityofthemeasuringequipmentiftheprecisionofthemeasuringequipmentisimproved,itwillprovidearesultthatisclosertothetruevalue.Inparticular,itwouldbeassumedthattheprecisionofmeasuringthespeedofthecardoesnotaffectitsposition,andviceversa.

In1927,Heisenbergprovedthattheseassumptionsarenotcorrect.[32]Quantummechanicsshowsthatcertainpairsofphysicalproperties,likepositionandspeed,cannotbothbeknowntoarbitraryprecision:themorepreciselyonepropertyisknown,thelesspreciselytheothercanbeknown.Thisstatementisknownastheuncertaintyprinciple.Theuncertaintyprincipleisn'tastatementabouttheaccuracyofourmeasuringequipment,butaboutthenatureofthesystemitselfourassumptionthatthecarhadadefinitepositionandspeedwasincorrect.Onascaleofcarsandpeople,theseuncertaintiesaretoosmalltonotice,butwhendealingwithatomsandelectronstheybecomecritical.[33]

Heisenberggave,asanillustration,themeasurementofthepositionandmomentumofanelectronusingaphotonoflight.Inmeasuringtheelectron'sposition,thehigherthefrequencyofthephoton,themoreaccurateisthemeasurementofthepositionoftheimpact,butthegreateristhedisturbanceoftheelectron,whichabsorbsarandomamountofenergy,renderingthemeasurementobtainedofitsmomentumincreasinglyuncertain(momentumisvelocitymultipliedbymass),foroneisnecessarilymeasuringitspostimpactdisturbedmomentumfromthecollisionproductsandnotitsoriginalmomentum.Withaphotonoflowerfrequency,thedisturbance(andhenceuncertainty)inthemomentumisless,butsoistheaccuracyofthemeasurementofthepositionoftheimpact.[34]

Theuncertaintyprincipleshowsmathematicallythattheproductoftheuncertaintyinthepositionandmomentumofaparticle(momentumisvelocitymultipliedbymass)couldneverbelessthanacertainvalue,andthatthisvalueisrelatedtoPlanck'sconstant.

Wavefunctioncollapse

Wavefunctioncollapseisaforcedexpressionforwhateverjusthappenedwhenitbecomesappropriatetoreplacethedescriptionofanuncertainstateofasystembyadescriptionofthesysteminadefinitestate.Explanationsforthenatureoftheprocessofbecomingcertainarecontroversial.Atanytimebeforeaphoton"showsup"onadetectionscreenitcanonlybedescribedbyasetofprobabilitiesforwhereitmightshowup.Whenitdoesshowup,forinstanceintheCCDofanelectroniccamera,thetimeandthespacewhereitinteractedwiththedeviceareknownwithinverytightlimits.However,thephotonhasdisappeared,andthewavefunctionhasdisappearedwithit.Initsplacesomephysicalchangeinthedetectionscreenhasappeared,e.g.,anexposedspotinasheetofphotographicfilm,orachangeinelectricpotentialinsomecellofaCCD.

Eigenstatesandeigenvalues

Foramoredetailedintroductiontothissubject,see:Introductiontoeigenstates

Becauseoftheuncertaintyprinciple,statementsaboutboththepositionandmomentumofparticlescanonlyassignaprobabilitythatthepositionormomentumwillhavesomenumericalvalue.Therefore,itisnecessarytoformulateclearlythedifferencebetweenthestateofsomethingthatisindeterminate,suchasanelectroninaprobabilitycloud,

7/27/2015 IntroductiontoquantummechanicsWikipedia,thefreeencyclopedia

https://en.wikipedia.org/wiki/Introduction_to_quantum_mechanics 12/21

WolfgangPauli

andthestateofsomethinghavingadefinitevalue.Whenanobjectcandefinitelybe"pinneddown"insomerespect,itissaidtopossessaneigenstate.

IntheSternGerlachexperimentdiscussedabove,thespinoftheatomabouttheverticalaxishastwoeigenstates:upanddown.Beforemeasuringit,wecanonlysaythatanyindividualatomhasequalprobabilityofbeingfoundtohavespinuporspindown.Themeasurementprocesscausesthewavefunctiontocollapseintooneofthetwostates.

Theeigenstatesofspinabouttheverticalaxisarenotsimultaneouslyeigenstatesofspinaboutthehorizontalaxis,sothisatomhasequalprobabilityofbeingfoundtohaveeithervalueofspinaboutthehorizontalaxis.Asdescribedinthesectionabove,measuringthespinaboutthehorizontalaxiscanallowanatomwhichwasspinuptobecomespindown:measuringitsspinaboutthehorizontalaxiscollapsesitswavefunctionintooneoftheeigenstatesofthismeasurement,whichmeansitisnolongerinaneigenstateofspinabouttheverticalaxis,socantakeeithervalue.

ThePauliexclusionprinciple

In1924,WolfgangPauliproposedanewquantumdegreeoffreedom(orquantumnumber),withtwopossiblevalues,toresolveinconsistenciesbetweenobservedmolecularspectraandthepredictionsofquantummechanics.Inparticular,thespectrumofatomichydrogenhadadoublet,orpairoflinesdifferingbyasmallamount,whereonlyonelinewasexpected.Pauliformulatedhisexclusionprinciple,statingthat"Therecannotexistanatominsuchaquantumstatethattwoelectronswithin[it]havethesamesetofquantumnumbers."[35]

Ayearlater,UhlenbeckandGoudsmitidentifiedPauli'snewdegreeoffreedomwiththepropertycalledspinwhoseeffectswereobservedintheSternGerlachexperiment.

Applicationtothehydrogenatom

Bohr'smodeloftheatomwasessentiallyaplanetaryone,withtheelectronsorbitingaroundthenuclear"sun."However,theuncertaintyprinciplestatesthatanelectroncannotsimultaneouslyhaveanexactlocationandvelocityinthewaythataplanetdoes.Insteadofclassicalorbits,electronsaresaidtoinhabitatomicorbitals.Anorbitalisthe"cloud"ofpossiblelocationsinwhichanelectronmightbefound,adistributionofprobabilitiesratherthanapreciselocation.[35]Eachorbitalisthreedimensional,ratherthanthetwodimensionalorbit,andisoftendepictedasathreedimensionalregionwithinwhichthereisa95percentprobabilityoffindingtheelectron.[36]

Schrdingerwasabletocalculatetheenergylevelsofhydrogenbytreatingahydrogenatom'selectronasawave,representedbythe"wavefunction",inanelectricpotentialwell,V,createdbytheproton.ThesolutionstoSchrdinger'sequationaredistributionsofprobabilitiesforelectronpositionsandlocations.Orbitalshavearangeofdifferentshapesinthreedimensions.Theenergiesofthedifferentorbitalscanbecalculated,andtheyaccuratelymatchtheenergylevelsoftheBohrmodel.

WithinSchrdinger'spicture,eachelectronhasfourproperties:

1. An"orbital"designation,indicatingwhethertheparticlewaveisonethatisclosertothenucleuswithlessenergyoronethatisfartherfromthenucleuswithmoreenergy

2. The"shape"oftheorbital,sphericalorotherwise3. The"inclination"oftheorbital,determiningthemagneticmomentoftheorbitalaroundthezaxis.4. The"spin"oftheelectron.

Thecollectivenameforthesepropertiesisthequantumstateoftheelectron.Thequantumstatecanbedescribedbygivinganumbertoeachofthesepropertiestheseareknownastheelectron'squantumnumbers.Thequantumstateoftheelectronisdescribedbyitswavefunction.ThePauliexclusionprincipledemandsthatnotwoelectronswithinanatommayhavethesamevaluesofallfournumbers.

7/27/2015 IntroductiontoquantummechanicsWikipedia,thefreeencyclopedia

https://en.wikipedia.org/wiki/Introduction_to_quantum_mechanics 13/21

Theshapesofthefirstfiveatomicorbitals:1s,2s,2px,2py,and2pz.Thecoloursshowthephaseofthewavefunction.

Thefirstpropertydescribingtheorbitalistheprincipalquantumnumber,n,whichisthesameasinBohr'smodel.ndenotestheenergylevelofeachorbital.Thepossiblevaluesfornareintegers:

Thenextquantumnumber,theazimuthalquantumnumber,denotedl,describestheshapeoftheorbital.Theshapeisaconsequenceoftheangularmomentumoftheorbital.Theangularmomentumrepresentstheresistanceofaspinningobjecttospeedinguporslowingdownundertheinfluenceofexternalforce.Theazimuthalquantumnumberrepresentstheorbitalangularmomentumofanelectronarounditsnucleus.Thepossiblevaluesforlareintegersfrom0ton1:

Theshapeofeachorbitalhasitsownletteraswell.Thefirstshapeisdenotedbytheletters(amnemonicbeing"sphere").Thenextshapeisdenotedbytheletterpandhastheformofadumbbell.Theotherorbitalshavemorecomplicatedshapes(seeatomicorbital),andaredenotedbythelettersd,f,andg.

Thethirdquantumnumber,themagneticquantumnumber,describesthemagneticmomentoftheelectron,andisdenotedbyml(orsimplym).Thepossiblevaluesformlareintegersfromltol:

Themagneticquantumnumbermeasuresthecomponentoftheangularmomentuminaparticulardirection.Thechoiceofdirectionisarbitrary,conventionallythezdirectionischosen.

Thefourthquantumnumber,thespinquantumnumber(pertainingtothe"orientation"oftheelectron'sspin)isdenotedms,withvalues+12or12.

ThechemistLinusPaulingwrote,bywayofexample:

Inthecaseofaheliumatomwithtwoelectronsinthe1sorbital,thePauliExclusionPrinciplerequiresthatthetwoelectronsdifferinthevalueofonequantumnumber.Theirvaluesofn,l,andmlarethesame.Accordinglytheymustdifferinthevalueofms,whichcanhavethevalueof+12foroneelectronand12fortheother."[35]

Itistheunderlyingstructureandsymmetryofatomicorbitals,andthewaythatelectronsfillthem,thatleadstotheorganisationoftheperiodictable.Thewaytheatomicorbitalsondifferentatomscombinetoformmolecularorbitalsdeterminesthestructureandstrengthofchemicalbondsbetweenatoms.

Diracwaveequation

In1928,PaulDiracextendedthePauliequation,whichdescribedspinningelectrons,toaccountforspecialrelativity.Theresultwasatheorythatdealtproperlywithevents,suchasthespeedatwhichanelectronorbitsthenucleus,occurringatasubstantialfractionofthespeedoflight.Byusingthesimplestelectromagneticinteraction,Diracwasabletopredictthevalueofthemagneticmomentassociatedwiththeelectron'sspin,andfoundtheexperimentally

7/27/2015 IntroductiontoquantummechanicsWikipedia,thefreeencyclopedia

https://en.wikipedia.org/wiki/Introduction_to_quantum_mechanics 14/21

PaulDirac(19021984)

Superpositionoftwoquantumcharacteristics,andtworesolutionpossibilities.

observedvalue,whichwastoolargetobethatofaspinningchargedspheregovernedbyclassicalphysics.Hewasabletosolveforthespectrallinesofthehydrogenatom,andtoreproducefromphysicalfirstprinciplesSommerfeld'ssuccessfulformulaforthefinestructureofthehydrogenspectrum.

Dirac'sequationssometimesyieldedanegativevalueforenergy,forwhichheproposedanovelsolution:hepositedtheexistenceofanantielectronandofadynamicalvacuum.Thisledtothemanyparticlequantumfieldtheory.

Quantumentanglement

ThePauliexclusionprinciplesaysthattwoelectronsinonesystemcannotbeinthesamestate.Natureleavesopenthepossibility,however,thattwoelectronscanhavebothstates"superimposed"overeachofthem.Recallthatthewavefunctionsthatemergesimultaneouslyfromthedoubleslitsarriveatthedetectionscreeninastateofsuperposition.Nothingiscertainuntilthesuperimposedwaveforms"collapse",Atthatinstantanelectronshowsupsomewhereinaccordancewiththeprobabilitythatisthesquareoftheabsolutevalueofthesumofthecomplexvaluedamplitudesofthetwosuperimposedwaveforms.Thesituationthereisalreadyveryabstract.Aconcretewayofthinkingaboutentangledphotons,photonsinwhichtwocontrarystatesaresuperimposedoneachoftheminthesameevent,isasfollows:

Imaginethatthesuperpositionofastatethatcanbementallylabeledasblueandanotherstatethatcanbementallylabeledasredwillthenappear(inimagination,ofcourse)asapurplestate.Twophotonsareproducedastheresultofthesameatomicevent.Perhapstheyareproducedbytheexcitationofacrystalthatcharacteristicallyabsorbsaphotonofacertainfrequencyandemitstwophotonsofhalftheoriginalfrequency.Sothetwophotonscomeout"purple."Iftheexperimenternowperformssomeexperimentthatwilldeterminewhetheroneofthephotonsiseitherblueorred,thenthatexperimentchangesthephotoninvolvedfromonehavingasuperpositionof"blue"and"red"characteristicstoaphotonthathasonlyoneofthosecharacteristics.TheproblemthatEinsteinhadwithsuchanimaginedsituationwasthatifoneofthesephotonshadbeenkeptbouncingbetweenmirrorsinalaboratoryonearth,andtheotheronehadtraveledhalfwaytotheneareststar,whenitstwinwasmadetorevealitselfaseitherblueorred,thatmeantthatthedistantphotonnowhadtoloseits"purple"statustoo.Sowheneveritmightbeinvestigatedafteritstwinhadbeenmeasured,itwouldnecessarilyshowupintheoppositestatetowhateveritstwinhadrevealed.

Intryingtoshowthatquantummechanicswasnotacompletetheory,Einsteinstartedwiththetheory'spredictionthattwoormoreparticlesthathaveinteractedinthepastcanappearstronglycorrelatedwhentheirvariouspropertiesarelatermeasured.Hesoughttoexplainthisseeminginteractioninaclassicalway,throughtheircommonpast,andpreferablynotbysome"spookyactionatadistance."Theargumentisworkedoutinafamouspaper,Einstein,Podolsky,andRosen(1935abbreviatedEPR),settingoutwhatisnowcalledtheEPRparadox.Assumingwhatisnowusuallycalledlocalrealism,EPRattemptedtoshowfromquantumtheorythataparticlehasbothpositionandmomentumsimultaneously,whileaccordingtotheCopenhageninterpretation,onlyoneofthosetwopropertiesactuallyexistsandonlyatthemomentthatitisbeingmeasured.EPRconcludedthatquantumtheoryisincompleteinthatitrefusestoconsiderphysicalpropertieswhichobjectivelyexistinnature.(Einstein,Podolsky,&Rosen1935iscurrentlyEinstein'smostcitedpublicationinphysicsjournals.)Inthesameyear,ErwinSchrdingerusedtheword

7/27/2015 IntroductiontoquantummechanicsWikipedia,thefreeencyclopedia

https://en.wikipedia.org/wiki/Introduction_to_quantum_mechanics 15/21

"entanglement"anddeclared:"Iwouldnotcallthatonebutratherthecharacteristictraitofquantummechanics."[37]

Thequestionofwhetherentanglementisarealconditionisstillindispute.[38]TheBellinequalitiesarethemostpowerfulchallengetoEinstein'sclaims.

Quantumfieldtheory

Theideaofquantumfieldtheorybeganinthelate1920swithBritishphysicistPaulDirac,whenheattemptedtoquantisetheelectromagneticfieldaprocedureforconstructingaquantumtheorystartingfromaclassicaltheory.

Afieldinphysicsis"aregionorspaceinwhichagiveneffect(suchasmagnetism)exists."[39]Othereffectsthatmanifestthemselvesasfieldsaregravitationandstaticelectricity.[40]In2008,physicistRichardHammondwrotethat

Sometimeswedistinguishbetweenquantummechanics(QM)andquantumfieldtheory(QFT).QMreferstoasysteminwhichthenumberofparticlesisfixed,andthefields(suchastheelectromechanicalfield)arecontinuousclassicalentities.QFT...goesastepfurtherandallowsforthecreationandannihilationofparticles....

Headded,however,thatquantummechanicsisoftenusedtoreferto"theentirenotionofquantumview."[41]:108

In1931,Diracproposedtheexistenceofparticlesthatlaterbecameknownasantimatter.[42]DiracsharedtheNobelPrizeinPhysicsfor1933withSchrdinger,"forthediscoveryofnewproductiveformsofatomictheory."[43]

Onitsface,quantumfieldtheoryallowsinfinitenumbersofparticles,andleavesituptothetheoryitselftopredicthowmanyandwithwhichprobabilitiesornumberstheyshouldexist.Whendevelopedfurther,thetheoryoftencontradictsobservation,sothatitscreationandannihilationoperatorscanbeempiricallytieddown.Furthermore,empiricalconservationlawslikethatofmassenergysuggestcertainconstraintsonthemathematicalformofthetheory,whicharemathematicallyspeakingfinicky.Thelatterfactbothservestomakequantumfieldtheoriesdifficulttohandle,buthasalsoleadtofurtherrestrictionsonadmissibleformsofthetheorythecomplicationsarementionedbelowundertherubrikofrenormalization.

Quantumelectrodynamics

Quantumelectrodynamics(QED)isthenameofthequantumtheoryoftheelectromagneticforce.UnderstandingQEDbeginswithunderstandingelectromagnetism.Electromagnetismcanbecalled"electrodynamics"becauseitisadynamicinteractionbetweenelectricalandmagneticforces.Electromagnetismbeginswiththeelectriccharge.

Electricchargesarethesourcesof,andcreate,electricfields.Anelectricfieldisafieldwhichexertsaforceonanyparticlesthatcarryelectriccharges,atanypointinspace.Thisincludestheelectron,proton,andevenquarks,amongothers.Asaforceisexerted,electricchargesmove,acurrentflowsandamagneticfieldisproduced.Thechangingmagneticfield,inturncauseselectriccurrent(oftenmovingelectrons).Thephysicaldescriptionofinteractingchargedparticles,electricalcurrents,electricalfields,andmagneticfieldsiscalledelectromagnetism.

In1928PaulDiracproducedarelativisticquantumtheoryofelectromagnetism.Thiswastheprogenitortomodernquantumelectrodynamics,inthatithadessentialingredientsofthemoderntheory.However,theproblemofunsolvableinfinitiesdevelopedinthisrelativisticquantumtheory.Yearslater,renormalizationlargelysolvedthisproblem.Initiallyviewedasasuspect,provisionalprocedurebysomeofitsoriginators,renormalizationeventuallywasembracedasanimportantandselfconsistenttoolinQEDandotherfieldsofphysics.Also,inthelate1940sFeynman'sdiagramsdepictedallpossibleinteractionspertainingtoagivenevent.Thediagramsshowedthattheelectromagneticforceistheinteractionsofphotonsbetweeninteractingparticles.

7/27/2015 IntroductiontoquantummechanicsWikipedia,thefreeencyclopedia

https://en.wikipedia.org/wiki/Introduction_to_quantum_mechanics 16/21

AnexampleofapredictionofquantumelectrodynamicswhichhasbeenverifiedexperimentallyistheLambshift.Thisreferstoaneffectwherebythequantumnatureoftheelectromagneticfieldcausestheenergylevelsinanatomoriontodeviateslightlyfromwhattheywouldotherwisebe.Asaresult,spectrallinesmayshiftorsplit.

Similarly,withinafreelypropagatingelectromagneticwave,thecurrentcanalsobejustanabstractdisplacementcurrent,insteadofinvolvingchargecarriers.InQED,itsfulldescriptionmakesessentialuseofshortlivedvirtualparticles.There,QEDagainvalidatesanearlier,rathermysteriousconcept.

StandardModel

Inthe1960sphysicistsrealizedthatQEDbrokedownatextremelyhighenergies.FromthisinconsistencytheStandardModelofparticlephysicswasdiscovered,whichremediedthehigherenergybreakdownintheory.Itisanother,extendedquantumfieldtheorywhichunifiestheelectromagneticandweakinteractionsintoonetheory.Thisiscalledtheelectroweaktheory.

AdditionallytheStandardModelcontainsahighenergyunificationoftheelectroweaktheorywiththestrongforce,describedbyquantumchromodynamics.Italsopostulatesaconnectionwithgravityasyetanothergaugetheory,buttheconnectionisasof2015stillpoorlyunderstood.Thetheory'spredictionoftheHiggsparticletoexplaininertialmasshasstoodrecentempiricaltestsattheLargehadroncollider,andthustheStandardmodelisnowconsideredthebasicandmoreorlesscompletedescriptionofparticlephysicsasweknowit.

Interpretations

Thephysicalmeasurements,equations,andpredictionspertinenttoquantummechanicsareallconsistentandholdaveryhighlevelofconfirmation.However,thequestionofwhattheseabstractmodelssayabouttheunderlyingnatureoftherealworldhasreceivedcompetinganswers.

Applications

Applicationsofquantummechanicsincludethelaser,thetransistor,theelectronmicroscope,andmagneticresonanceimaging.Aspecialclassofquantummechanicalapplicationsisrelatedtomacroscopicquantumphenomenasuchassuperfluidheliumandsuperconductors.Thestudyofsemiconductorsledtotheinventionofthediodeandthetransistor,whichareindispensableformodernelectronics.

Ineventhesimplelightswitch,quantumtunnellingisabsolutelyvital,asotherwisetheelectronsintheelectriccurrentcouldnotpenetratethepotentialbarriermadeupofalayerofoxide.FlashmemorychipsfoundinUSBdrivesalsousequantumtunnelling,toerasetheirmemorycells.[44]

Seealso

MacroscopicquantumphenomenaPhilosophyofphysics

QuantumcomputerVirtualparticle

Notes1. Anumberofformulaehadbeencreatedwhichwereabletodescribesomeoftheexperimentalmeasurementsofthermal

radiation:howthewavelengthatwhichtheradiationisstrongestchangeswithtemperatureisgivenbyWien'sdisplacementlaw,theoverallpoweremittedperunitareaisgivenbytheStefanBoltzmannlaw.ThebesttheoreticalexplanationoftheexperimentalresultswastheRayleighJeanslaw,whichagreeswithexperimentalresultswellatlargewavelengths(or,

7/27/2015 IntroductiontoquantummechanicsWikipedia,thefreeencyclopedia

https://en.wikipedia.org/wiki/Introduction_to_quantum_mechanics 17/21

equivalently,lowfrequencies),butstronglydisagreesatshortwavelengths(orhighfrequencies).Infact,atshortwavelengths,classicalphysicspredictedthatenergywillbeemittedbyahotbodyataninfiniterate.Thisresult,whichisclearlywrong,isknownastheultravioletcatastrophe.

2. ThewordquantumcomesfromtheLatinwordfor"howmuch"(asdoesquantity).Somethingwhichisquantized,liketheenergyofPlanck'sharmonicoscillators,canonlytakespecificvalues.Forexample,inmostcountriesmoneyiseffectivelyquantized,withthequantumofmoneybeingthelowestvaluecoinincirculation.Mechanicsisthebranchofsciencethatdealswiththeactionofforcesonobjects.So,quantummechanicsisthepartofmechanicsthatdealswithobjectsforwhichparticularpropertiesarequantized.

3. Actually,therecanbeintensitydependenteffects,butatintensitiesachievablewithnonlasersources,theseeffectsareunobservable.

4. Einstein'sphotoelectriceffectequationcanbederivedandexplainedwithoutrequiringtheconceptof"photons".Thatis,theelectromagneticradiationcanbetreatedasaclassicalelectromagneticwave,aslongastheelectronsinthematerialaretreatedbythelawsofquantummechanics.Theresultsarequantitativelycorrectforthermallightsources(thesun,incandescentlamps,etc)bothfortherateofelectronemissionaswellastheirangulardistribution.Formoreonthispoint,see[12]

5. Theclassicalmodeloftheatomiscalledtheplanetarymodel,orsometimestheRutherfordmodelafterErnestRutherfordwhoproposeditin1911,basedontheGeigerMarsdengoldfoilexperimentwhichfirstdemonstratedtheexistenceofthenucleus.

6. Inthiscase,theenergyoftheelectronisthesumofitskineticandpotentialenergies.Theelectronhaskineticenergybyvirtueofitsactualmotionaroundthenucleus,andpotentialenergybecauseofitselectromagneticinteractionwiththenucleus.

7. Themodelcanbeeasilymodifiedtoaccountfortheemissionspectrumofanysystemconsistingofanucleusandasingleelectron(thatis,ionssuchasHe+orO7+whichcontainonlyoneelectron)butcannotbeextendedtoanatomwithtwoelectronslikeneutralhelium.

8. ElectrondiffractionwasfirstdemonstratedthreeyearsafterdeBrogliepublishedhishypothesis.AttheUniversityofAberdeen,GeorgeThomsonpassedabeamofelectronsthroughathinmetalfilmandobserveddiffractionpatterns,aswouldbepredictedbythedeBrogliehypothesis.AtBellLabs,DavissonandGermerguidedanelectronbeamthroughacrystallinegrid.DeBrogliewasawardedtheNobelPrizeinPhysicsin1929forhishypothesisThomsonandDavissonsharedtheNobelPrizeforPhysicsin1937fortheirexperimentalwork.

9. ForasomewhatmoresophisticatedlookathowHeisenbergtransitionedfromtheoldquantumtheoryandclassicalphysicstothenewquantummechanics,seeHeisenberg'sentrywaytomatrixmechanics.

References1. QuantumMechanics(http://www.pbs.org/transistor/science/info/quantum.html)fromNationalPublicRadio2. Kuhn,ThomasS.TheStructureofScientificRevolutions.Fourthed.ChicagoLondon:TheUniversityofChicagoPress,

2012.Print.3. Feynman,RichardP.(1988).QED:thestrangetheoryoflightandmatter(1stPrincetonpbk.,seventhprintingwith

corrections.ed.).Princeton,N.J.:PrincetonUniversityPress.p.10.ISBN9780691024172.4. Thisresultwaspublished(inGerman)asPlanck,Max(1901)."UeberdasGesetzderEnergieverteilungimNormalspectrum"

(http://www.physik.uniaugsburg.de/annalen/history/historicpapers/1901_309_553563.pdf)(PDF).Ann.Phys.309(3):55363.Bibcode:1901AnP...309..553P(http://adsabs.harvard.edu/abs/1901AnP...309..553P).doi:10.1002/andp.19013090310(https://dx.doi.org/10.1002%2Fandp.19013090310)..Englishtranslation:"OntheLawofDistributionofEnergyintheNormalSpectrum(http://dbhs.wvusd.k12.ca.us/webdocs/ChemHistory/Planck1901/Planck1901.html)".

5. FrancisWestonSears(1958).Mechanics,WaveMotion,andHeat(http://books.google.com/books?q=%22Mechanics%2C+Wave+Motion%2C+and+Heat%22+%22where+n+%3D+1%2C%22&btnG=Search+Books).AddisonWesley.p.537.

6. "TheNobelPrizeinPhysics1918"(http://nobelprize.org/nobel_prizes/physics/laureates/1918/).NobelFoundation.Retrieved20090801.

7. Kragh,Helge(1December2000)."MaxPlanck:thereluctantrevolutionary"(http://physicsworld.com/cws/article/print/373).PhysicsWorld.com.

8. Einstein,Albert(1905)."bereinendieErzeugungundVerwandlungdesLichtesbetreffendenheuristischenGesichtspunkt"(http://www.zbp.univie.ac.at/dokumente/einstein1.pdf)(PDF).AnnalenderPhysik17(6):132148.Bibcode:1905AnP...322..132E(http://adsabs.harvard.edu/abs/1905AnP...322..132E).doi:10.1002/andp.19053220607(https://dx.doi.org/10.1002%2Fandp.19053220607).,translatedintoEnglishasOnaHeuristicViewpointConcerningtheProductionandTransformationofLight(http://lorentz.phl.jhu.edu/AnnusMirabilis/AeReserveArticles/eins_lq.pdf).Theterm"photon"wasintroducedin1926.

9. Taylor,J.R.Zafiratos,C.D.Dubson,M.A.(2004).ModernPhysicsforScientistsandEngineers.PrenticeHall.pp.1279.ISBN0135897890.

10. StephenHawking,TheUniverseinaNutshell,Bantam,2001.11. DickeandWittke,IntroductiontoQuantumMechanics,p.12

7/27/2015 IntroductiontoquantummechanicsWikipedia,thefreeencyclopedia

https://en.wikipedia.org/wiki/Introduction_to_quantum_mechanics 18/21

12. NTRS.NASA.gov(http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19680009569_1968009569.pdf)13. Taylor,J.R.Zafiratos,C.D.Dubson,M.A.(2004).ModernPhysicsforScientistsandEngineers.PrenticeHall.pp.147

8.ISBN0135897890.14. McEvoy,J.P.Zarate,O.(2004).IntroducingQuantumTheory.Totem\Books.pp.7089,especiallyp.89.ISBN184046

5778.15. WorldBookEncyclopedia,page6,2007.16. DickeandWittke,IntroductiontoQuantumMechanics,p.10f.17. J.P.McEvoyandOscarZarate(2004).IntroducingQuantumTheory.TotemBooks.p.110f.ISBN1840465778.18. Aczel,AmirD.,Entanglement,p.51f.(Penguin,2003)ISBN978155192647619. J.P.McEvoyandOscarZarate(2004).IntroducingQuantumTheory.TotemBooks.p.114.ISBN1840465778.20. Zettili,Nouredine(2009).QuantumMechanics:ConceptsandApplications(https://books.google.com/books?

id=6jXlpJCSz98C&pg=PA26&dq=%22complementarity+principle%22+%22waveparticle+duality%22).JohnWileyandSons.pp.2627.ISBN0470026782.

21. Selleri,Franco(2012).WaveParticleDuality(https://books.google.com/books?id=r8bkBwAAQBAJ&pg=PA41&dq=%22complementarity+principle%22+%22waveparticle+duality%22).SpringerScienceandBusinessMedia.p.41.ISBN1461533325.

22. Podgorsak,ErvinB.(2013).CompendiumtoRadiationPhysicsforMedicalPhysicists(https://books.google.com/books?id=7zfBBAAAQBAJ&pg=PA88&dq=%22complementarity+principle%22+%22waveparticle+duality%22).SpringerScienceandBusinessMedia.p.88.ISBN3642201865.

23. Halliday,DavidResnick,Robert(2013).FundamentalsofPhysics,10thEd.(https://books.google.com/books?id=nQZyAgAAQBAJ&pg=SL9PA21&dq=%22complementarity+principle%22+%22waveparticle+duality%22)JohnWileyandSons.p.1272.ISBN1118230612.

24. Myers,RustyL.(2006).TheBasicsofPhysics(https://books.google.com/books?id=KnynjL44pI4C&pg=PA172&dq=%22complementarity+principle%22+%22waveparticle+duality%22).GreenwoodPublishingGroup.p.172.ISBN0313328579.

25. IntroducingQuantumTheory,p.8726. VanderWaerden,B.L.(1967).SourcesofQuantumMechanics(inGermantranslatedtoEnglish).Mineola,NewYork:

DoverPublications.pp.261276."ReceivedJuly29,1925"SeeWernerHeisenberg'spaper,"QuantumTheoreticalReinterpretationofKinematicandMechanicalRelations"pp.261276

27. NobelPrizeOrganization."ErwinSchrdingerBiographical"(http://www.nobelprize.org/nobel_prizes/physics/laureates/1933/schrodingerbio.html).Retrieved28March2014."Hisgreatdiscovery,Schrdinger'swaveequation,wasmadeattheendofthisepochduringthefirsthalfof1926."

28. "SchrodingerEquation(Physics),"EncyclopdiaBritannica(http://www.britannica.com/EBchecked/topic/528298/Schrodingerequation)

29. ErwinSchrdinger,"ThePresentSituationinQuantumMechanics",p.9."ThistranslationwasoriginallypublishedinProceedingsoftheAmericanPhilosophicalSociety,124,32338,andthenappearedasSectionI.11ofPartIofQuantumTheoryandMeasurement(J.A.WheelerandW.H.Zurek,eds.,PrincetonuniversityPress,NewJersey1983).Thispapercanbedownloadedfromhttp://www.tuharburg.de/rzt/rzt/it/QM/cat.html."

30. W.Moore,Schrdinger:LifeandThought,CambridgeUniversityPress(1989),p.222.Seep.227forSchrdinger'sownwords.

31. Heisenberg'sNobelPrizecitation(http://nobelprize.org/nobel_prizes/physics/laureates/1932/)32. HeisenbergfirstpublishedhisworkontheuncertaintyprincipleintheleadingGermanphysicsjournalZeitschriftfrPhysik:

Heisenberg,W.(1927)."berdenanschaulichenInhaltderquantentheoretischenKinematikundMechanik".Z.Phys.43(34):172198.Bibcode:1927ZPhy...43..172H(http://adsabs.harvard.edu/abs/1927ZPhy...43..172H).doi:10.1007/BF01397280(https://dx.doi.org/10.1007%2FBF01397280).

33. NobelPrizeinPhysicspresentationspeech,1932(http://nobelprize.org/nobel_prizes/physics/laureates/1932/press.html)34. "Uncertaintyprinciple,"EncyclopdiaBritannica(http://www.britannica.com/EBchecked/topic/614029/uncertaintyprinciple)35. LinusPauling,TheNatureoftheChemicalBond,p.4736. "Orbital(chemistryandphysics),"EncyclopdiaBritannica(http://www.britannica.com/EBchecked/topic/431159/orbital)37. E.Schrdinger,ProceedingsoftheCambridgePhilosophicalSociety,31(1935),p.555,says:"Whentwosystems,ofwhich

weknowthestatesbytheirrespectiverepresentation,enterintoatemporaryphysicalinteractionduetoknownforcesbetweenthemandwhenafteratimeofmutualinfluencethesystemsseparateagain,thentheycannolongerbedescribedasbefore,viz.,byendowingeachofthemwitharepresentativeofitsown.Iwouldnotcallthatonebutratherthecharacteristictraitofquantummechanics."

38. "QuantumNonlocalityandthePossibilityofSuperluminalEffects",JohnG.Cramer,npl.washington.edu(http://www.npl.washington.edu/npl/int_rep/qm_nl.html)

39. "Mechanics,"MerriamWebsterOnlineDictionary(http://www.merriamwebster.com/dictionary/field)40. "Field"(http://www.britannica.com/EBchecked/topic/206162/field),EncyclopdiaBritannica41. RichardHammond,TheUnknownUniverse,NewPageBooks,2008.ISBN9781601630032

7/27/2015 IntroductiontoquantummechanicsWikipedia,thefreeencyclopedia

https://en.wikipedia.org/wiki/Introduction_to_quantum_mechanics 19/21

Bibliography

Bernstein,Jeremy(2005)."MaxBornandthequantumtheory".AmericanJournalofPhysics73(11):999.Bibcode:2005AmJPh..73..999B(http://adsabs.harvard.edu/abs/2005AmJPh..73..999B).doi:10.1119/1.2060717(https://dx.doi.org/10.1119%2F1.2060717).Beller,Mara(2001).QuantumDialogue:TheMakingofaRevolution.UniversityofChicagoPress.Bohr,Niels(1958).AtomicPhysicsandHumanKnowledge.JohnWiley&Sons].ASINB00005VGVF(https://www.amazon.com/dp/B00005VGVF).ISBN0486479285.OCLC530611(https://www.worldcat.org/oclc/530611).deBroglie,Louis(1953).TheRevolutioninPhysics.NoondayPress.LCCN53010401(http://lccn.loc.gov/53010401).Bronner,PatrickStrunz,AndreasSilberhorn,ChristineMeyn,JanPeter(2009)."Demonstratingquantumrandomwithsinglephotons".EuropeanJournalofPhysics30(5):11891200.Bibcode:2009EJPh...30.1189B(http://adsabs.harvard.edu/abs/2009EJPh...30.1189B).doi:10.1088/01430807/30/5/026(https://dx.doi.org/10.1088%2F01430807%2F30%2F5%2F026).Einstein,Albert(1934).EssaysinScience.PhilosophicalLibrary.ISBN0486470113.LCCN55003947(http://lccn.loc.gov/55003947).Feigl,HerbertBrodbeck,May(1953).ReadingsinthePhilosophyofScience.AppletonCenturyCrofts.ISBN0390304883.LCCN53006438(http://lccn.loc.gov/53006438).Feynman,RichardP.(1949)."SpaceTimeApproachtoQuantumElectrodynamics"(http://www.physics.princeton.edu/~mcdonald/examples/QED/feynman_pr_76_769_49.pdf)(PDF).PhysicalReview76(6):769789.Bibcode:1949PhRv...76..769F(http://adsabs.harvard.edu/abs/1949PhRv...76..769F).doi:10.1103/PhysRev.76.769(https://dx.doi.org/10.1103%2FPhysRev.76.769).Feynman,RichardP.(1990).QED,TheStrangeTheoryofLightandMatter.PenguinBooks.ISBN9780140125054.Fowler,Michael(1999).TheBohrAtom.UniversityofVirginia.Heisenberg,Werner(1958).PhysicsandPhilosophy.HarperandBrothers.ISBN0061305499.LCCN99010404(http://lccn.loc.gov/99010404).Lakshmibala,S.(2004)."Heisenberg,MatrixMechanicsandtheUncertaintyPrinciple".Resonance,JournalofScienceEducation9(8).Liboff,RichardL.(1992).IntroductoryQuantumMechanics(2nded.).Lindsay,RobertBruceMargenau,Henry(1957).FoundationsofPhysics.Dover.ISBN0918024188.LCCN57014416(http://lccn.loc.gov/57014416).McEvoy,J.P.Zarate,Oscar.IntroducingQuantumTheory.ISBN1874166374.Nave,CarlRod(2005)."QuantumPhysics"(http://hyperphysics.phyastr.gsu.edu/hbase/quacon.html#quacon).HyperPhysics.GeorgiaStateUniversity.Peat,F.David(2002).FromCertaintytoUncertainty:TheStoryofScienceandIdeasintheTwentyFirstCentury.JosephHenryPress.Reichenbach,Hans(1944).PhilosophicFoundationsofQuantumMechanics.UniversityofCaliforniaPress.ISBN0486404595.LCCNa44004471(http://lccn.loc.gov/a44004471).Schlipp,PaulArthur(1949).AlbertEinstein:PhilosopherScientist.TudorPublishingCompany.LCCN50005340(http://lccn.loc.gov/50005340).ScientificAmericanReader,1953.Sears,FrancisWeston(1949).Optics(3rded.).AddisonWesley.ISBN0195046013.LCCN51001018(http://lccn.loc.gov/51001018).Shimony,A.(1983)."(titlenotgivenincitation)".FoundationsofQuantumMechanicsintheLightofNewTechnology(S.Kamefuchietal.,eds.).Tokyo:JapanPhysicalSociety.p.225.citedin:Popescu,SanduDanielRohrlich(1996)."ActionandPassionataDistance:AnEssayinHonorofProfessorAbnerShimony".arXiv:quantph/9605004(https://arxiv.org/abs/quantph/9605004)[quantph(https://arxiv.org/archive/quantph)].

42. ThePhysicalWorldwebsite(http://www.physicalworld.org/restless_universe/html/ru_dira.html)43. "TheNobelPrizeinPhysics1933"(http://nobelprize.org/nobel_prizes/physics/laureates/1933/).NobelFoundation.Retrieved

20071124.44. Durrani,Z.A.K.Ahmed,H.(2008).VijayKumar,ed.Nanosilicon.Elsevier.p.345.ISBN9780080445281.

7/27/2015 IntroductiontoquantummechanicsWikipedia,thefreeencyclopedia

https://en.wikipedia.org/wiki/Introduction_to_quantum_mechanics 20/21

TheWikibookQuantumMechanicshasapageonthetopicof:IntroductiontoQuantumMechanics

Tavel,MortonTavel,Judith(illustrations)(2002).Contemporaryphysicsandthelimitsofknowledge(http://books.google.com/?id=SELS0HbIhjYC&pg=PA200&dq=Wave+function+collapse).RutgersUniversityPress.ISBN9780813530772.VanVleck,J.H.,1928,"TheCorrespondencePrincipleintheStatisticalInterpretationofQuantumMechanics",Proc.Nat.Acad.Sci.14:179.WestmorelandBenjaminSchumacher(1998)."QuantumEntanglementandtheNonexistenceofSuperluminalSignals".arXiv:quantph/9801014(https://arxiv.org/abs/quantph/9801014)[quantph(https://arxiv.org/archive/quantph)].Wheeler,JohnArchibaldFeynman,RichardP.(1949)."ClassicalElectrodynamicsinTermsofDirectInterparticleAction".ReviewsofModernPhysics21(3):425433.Bibcode:1949RvMP...21..425W(http://adsabs.harvard.edu/abs/1949RvMP...21..425W).doi:10.1103/RevModPhys.21.425(https://dx.doi.org/10.1103%2FRevModPhys.21.425).

Wieman,CarlPerkins,Katherine(2005)."TransformingPhysicsEducation".PhysicsToday58(11):36.Bibcode:2005PhT....58k..36W(http://adsabs.harvard.edu/abs/2005PhT....58k..36W).doi:10.1063/1.2155756(https://dx.doi.org/10.1063%2F1.2155756).

Furtherreading

Thefollowingtitles,allbyworkingphysicists,attempttocommunicatequantumtheorytolaypeople,usingaminimumoftechnicalapparatus.

JimAlKhalili(2003)Quantum:AGuideforthePerplexed.Weidenfield&Nicholson.ISBN9781780225340Chester,Marvin(1987)PrimerofQuantumMechanics.JohnWiley.ISBN0486428788BrianCoxandJeffForshaw(2011)TheQuantumUniverse.AllenLane.ISBN9781846144325RichardFeynman(1985)QED:TheStrangeTheoryofLightandMatter.PrincetonUniversityPress.ISBN0691083886Ford,Kenneth(2005)TheQuantumWorld.HarvardUniv.Press.Includeselementaryparticlephysics.Ghirardi,GianCarlo(2004)SneakingaLookatGod'sCards,GeraldMalsbary,trans.PrincetonUniv.Press.Themosttechnicaloftheworkscitedhere.Passagesusingalgebra,trigonometry,andbraketnotationcanbepassedoveronafirstreading.TonyHeyandWalters,Patrick(2003)TheNewQuantumUniverse.CambridgeUniv.Press.Includesmuchaboutthetechnologiesquantumtheoryhasmadepossible.ISBN9780521564571VladimirG.Ivancevic,TijanaT.Ivancevic(2008)Quantumleap:fromDiracandFeynman,acrosstheuniverse,tohumanbodyandmind.WorldScientificPublishingCompany.Providesanintuitiveintroductioninnonmathematicaltermsandanintroductionincomparativelybasicmathematicalterms.ISBN9789812819277N.DavidMermin(1990)"Spookyactionsatadistance:mysteriesoftheQT"inhisBoojumsallthewaythrough.CambridgeUniv.Press:110176.Theauthorisararephysicistwhotriestocommunicatetophilosophersandhumanists.ISBN9780521388801RolandOmns(1999)UnderstandingQuantumMechanics.PrincetonUniv.Press.ISBN9780691004358VictorStenger(2000)TimelessReality:Symmetry,Simplicity,andMultipleUniverses.BuffaloNY:PrometheusBooks.Chpts.58.ISBN9781573928595MartinusVeltman(2003)FactsandMysteriesinElementaryParticlePhysics.WorldScientificPublishingCompany.ISBN9789812381491J.P.McEvoyandOscarZarate(2004).IntroducingQuantumTheory.TotemBooks.ISBN1840465778

Externallinks

"MicroscopicWorldIntroductiontoQuantumMechanics.(http://www.kutl.kyushuu.ac.jp/seminar/MicroWorld1_E/MicroWorld_1_E.html)"byTakada,Kenjiro,EmeritusprofessoratKyushuUniversityQuantumTheory.(http://www.encyclopedia.com/doc/1E1quantumt.html)atencyclopedia.comThespookyquantum

7/27/2015 IntroductiontoquantummechanicsWikipedia,thefreeencyclopedia

https://en.wikipedia.org/wiki/Introduction_to_quantum_mechanics 21/21

(http://www.imamu.edu.sa/Scientific_selections/abstracts/Physics/THE%20SPOOKY%20QUANTUM.pdf)TheQuantumExchange(http://www.compadre.org/quantum)(tutorialsandopensourcelearningsoftware).AtomsandthePeriodicTable(http://www.chem1.com/acad/webtext/atoms/)Singleanddoubleslitinterference(http://intro.phys.psu.edu/class/251Labs/10_Interference_&_Diffraction/Single_and_DoubleSlit_Interference.pdf)TimeEvolutionofaWavepacketinaSquareWell(http://demonstrations.wolfram.com/TimeEvolutionOfAWavepacketInASquareWell/)Ananimateddemonstrationofawavepacketdispersionovertime.Experimentswithsinglephotons(http://www.didaktik.physik.unierlangen.de/quantumlab/english/)AnintroductionintoquantumphysicswithinteractiveexperimentsCarroll,SeanM.."QuantumMechanics(anembarrassment)"(http://www.sixtysymbols.com/videos/quantum_mechanics.htm).SixtySymbols.BradyHaranfortheUniversityofNottingham.Comprehensiveanimations(http://www.embd.be/quantummechanics/default.html)

Retrievedfrom"https://en.wikipedia.org/w/index.php?title=Introduction_to_quantum_mechanics&oldid=672417736"

Categories: Quantummechanics

Thispagewaslastmodifiedon21July2015,at12:25.TextisavailableundertheCreativeCommonsAttributionShareAlikeLicenseadditionaltermsmayapply.Byusingthissite,youagreetotheTermsofUseandPrivacyPolicy.WikipediaisaregisteredtrademarkoftheWikimediaFoundation,Inc.,anonprofitorganization.