Introduction to quantum mechanics - Wikipedia

Quantum mechanics is the study of matter and its interactions with energy on the scale of atomic and subatomic particles. By contrast, classical physics ... Introductiontoquantummechanics FromWikipedia,thefreeencyclopedia Jumptonavigation Jumptosearch Non-technicalintroductiontoquantumphysics Thisarticleisanon-technicalintroductiontothesubject.Forthemainencyclopediaarticle,seeQuantummechanics. ForthebookbyDavidJ.Griffiths,seeIntroductiontoQuantumMechanics(book). PartofaseriesofarticlesaboutQuantummechanics i ℏ ∂ ∂ t | ψ ( t ) ⟩ = H ^ | ψ ( t ) ⟩ {\displaystylei\hbar{\frac{\partial}{\partialt}}|\psi(t)\rangle={\hat{H}}|\psi(t)\rangle} Schrödingerequation Introduction Glossary History Background Classicalmechanics Oldquantumtheory Bra–ketnotation Hamiltonian Interference Fundamentals Complementarity Decoherence Entanglement Energylevel Measurement Nonlocality Quantumnumber State Superposition Symmetry Tunnelling Uncertainty Wavefunction Collapse Experiments Bell'sinequality Davisson–Germer Double-slit Elitzur–Vaidman Franck–Hertz Leggett–Garginequality Mach–Zehnder Popper Quantumeraser Delayed-choice Schrödinger'scat Stern–Gerlach Wheeler'sdelayed-choice Formulations Overview Heisenberg Interaction Matrix Phase-space Schrödinger Sum-over-histories(pathintegral) Equations Dirac Klein–Gordon Pauli Rydberg Schrödinger Interpretations Overview Bayesian Consistenthistories Copenhagen deBroglie–Bohm Ensemble Hidden-variable Local Many-worlds Objectivecollapse Quantumlogic Relational Transactional Advancedtopics Relativisticquantummechanics Quantumfieldtheory Quantuminformationscience Quantumcomputing Quantumchaos Densitymatrix Scatteringtheory Quantumstatisticalmechanics Quantummachinelearning Scientists Aharonov Bell Bethe Blackett Bloch Bohm Bohr Born Bose deBroglie Compton Dirac Davisson Debye Ehrenfest Einstein Everett Fock Fermi Feynman Glauber Gutzwiller Heisenberg Hilbert Jordan Kramers Pauli Lamb Landau Laue Moseley Millikan Onnes Planck Rabi Raman Rydberg Schrödinger Simmons Sommerfeld vonNeumann Weyl Wien Wigner Zeeman Zeilinger vte Quantummechanicsisthestudyofmatteranditsinteractionswithenergyonthescaleofatomicandsubatomicparticles.Bycontrast,classicalphysicsexplainsmatterandenergyonlyonascalefamiliartohumanexperience,includingthebehaviorofastronomicalbodiessuchasthemoon.Classicalphysicsisstillusedinmuchofmodernscienceandtechnology.However,towardstheendofthe19thcentury,scientistsdiscoveredphenomenainboththelarge(macro)andthesmall(micro)worldsthatclassicalphysicscouldnotexplain.[1]Thedesiretoresolveinconsistenciesbetweenobservedphenomenaandclassicaltheoryledtotwomajorrevolutionsinphysicsthatcreatedashiftintheoriginalscientificparadigm:thetheoryofrelativityandthedevelopmentofquantummechanics.[2]Thisarticledescribeshowphysicistsdiscoveredthelimitationsofclassicalphysicsanddevelopedthemainconceptsofthequantumtheorythatreplaceditintheearlydecadesofthe20thcentury.Itdescribestheseconceptsinroughlytheorderinwhichtheywerefirstdiscovered.Foramorecompletehistoryofthesubject,seeHistoryofquantummechanics. Lightbehavesinsomeaspectslikeparticlesandinotheraspectslikewaves.Matter—the"stuff"oftheuniverseconsistingofparticlessuchaselectronsandatoms—exhibitswavelikebehaviortoo.Somelightsources,suchasneonlights,giveoffonlycertainspecificfrequenciesoflight,asmallsetofdistinctpurecolorsdeterminedbyneon'satomicstructure.Quantummechanicsshowsthatlight,alongwithallotherformsofelectromagneticradiation,comesindiscreteunits,calledphotons,andpredictsitsspectralenergies(correspondingtopurecolors),andtheintensitiesofitslightbeams.Asinglephotonisaquantum,orsmallestobservableparticle,oftheelectromagneticfield.Apartialphotonisneverexperimentallyobserved.Morebroadly,quantummechanicsshowsthatmanypropertiesofobjects,suchasposition,speed,andangularmomentum,thatappearedcontinuousinthezoomed-outviewofclassicalmechanics,turnouttobe(intheverytiny,zoomed-inscaleofquantummechanics)quantized.Suchpropertiesofelementaryparticlesarerequiredtotakeononeofasetofsmall,discreteallowablevalues,andsincethegapbetweenthesevaluesisalsosmall,thediscontinuitiesareonlyapparentatverytiny(atomic)scales. Manyaspectsofquantummechanicsarecounterintuitive[3]andcanseemparadoxicalbecausetheydescribebehaviorquitedifferentfromthatseenatlargerscales.InthewordsofquantumphysicistRichardFeynman,quantummechanicsdealswith"natureasSheis—absurd".[4]Oneprinciple"paradox"istheapparentinconsistencybetweenNewton'slawsandquantummechanicswhichcanbeexplainedusingEhrenfest'stheorem,whichshowsthattheaveragevaluesobtainedfromquantummechanics(e.g.positionandmomentum)obeyclassicallaws.[5]However,Ehrenfest'stheoremisfarfromcapableofexplainingallthecounterintuitivephenomena(quantumweirdness)thathasbeenobserved,butratherisamathematicalexpressionofthecorrespondenceprinciple. Forexample,theuncertaintyprincipleofquantummechanicsmeansthatthemorecloselyonepinsdownonemeasurement(suchasthepositionofaparticle),thelessaccurateanothercomplementarymeasurementpertainingtothesameparticle(suchasitsspeed)mustbecome. Anotherexampleisentanglement,inwhichameasurementofanytwo-valuedstateofaparticle(suchaslightpolarizedupordown)madeoneitheroftwo"entangled"particlesthatareveryfarapartcausesasubsequentmeasurementontheotherparticletoalwaysbetheotherofthetwovalues(suchaspolarizedintheoppositedirection). Afinalexampleissuperfluidity,inwhichacontainerofliquidhelium,cooleddowntonearabsolutezerointemperaturespontaneouslyflows(slowly)upandovertheopeningofitscontainer,againsttheforceofgravity. Contents 1Thefirstquantumtheory:MaxPlanckandblack-bodyradiation 2Photons:thequantizationoflight 2.1Thephotoelectriceffect 2.2Consequencesoflightbeingquantized 3Thequantizationofmatter:theBohrmodeloftheatom 4Wave–particleduality 4.1Thedouble-slitexperiment 4.2ApplicationtotheBohrmodel 5Spin 6Developmentofmodernquantummechanics 7Copenhageninterpretation 7.1Uncertaintyprinciple 7.2Wavefunctioncollapse 7.3Eigenstatesandeigenvalues 7.4ThePauliexclusionprinciple 7.5Applicationtothehydrogenatom 8Diracwaveequation 9Quantumentanglement 10Quantumfieldtheory 11Quantumelectrodynamics 12StandardModel 13Interpretations 14Applications 15Seealso 16Notes 17References 18Bibliography 19Furtherreading 20Externallinks Thefirstquantumtheory:MaxPlanckandblack-bodyradiation[edit] Mainarticle:Ultravioletcatastrophe Hotmetalwork.Theyellow-orangeglowisthevisiblepartofthethermalradiationemittedduetothehightemperature.Everythingelseinthepictureisglowingwiththermalradiationaswell,butlessbrightlyandatlongerwavelengthsthanthehumaneyecandetect.Afar-infraredcameracanobservethisradiation. Thermalradiationiselectromagneticradiationemittedfromthesurfaceofanobjectduetotheobject'sinternalenergy.Ifanobjectisheatedsufficiently,itstartstoemitlightattheredendofthespectrum,asitbecomesredhot. Heatingitfurthercausesthecolortochangefromredtoyellow,white,andblue,asitemitslightatincreasinglyshorterwavelengths(higherfrequencies).Aperfectemitterisalsoaperfectabsorber:whenitiscold,suchanobjectlooksperfectlyblack,becauseitabsorbsallthelightthatfallsonitandemitsnone.Consequently,anidealthermalemitterisknownasablackbody,andtheradiationitemitsiscalledblack-bodyradiation. Predictionsoftheamountofthermalradiationofdifferentfrequenciesemittedbyabody.CorrectvaluespredictedbyPlanck'slaw(green)contrastedagainsttheclassicalvaluesofRayleigh-Jeanslaw(red)andWienapproximation(blue). Bythelate19thcentury,thermalradiationhadbeenfairlywellcharacterizedexperimentally.[note1]However,classicalphysicsledtotheRayleigh–Jeanslaw,which,asshowninthefigure,agreeswithexperimentalresultswellatlowfrequencies,butstronglydisagreesathighfrequencies.Physicistssearchedforasingletheorythatexplainedalltheexperimentalresults. ThefirstmodelthatwasabletoexplainthefullspectrumofthermalradiationwasputforwardbyMaxPlanckin1900.[6]Heproposedamathematicalmodelinwhichthethermalradiationwasinequilibriumwithasetofharmonicoscillators.Toreproducetheexperimentalresults,hehadtoassumethateachoscillatoremittedanintegernumberofunitsofenergyatitssinglecharacteristicfrequency,ratherthanbeingabletoemitanyarbitraryamountofenergy.Inotherwords,theenergyemittedbyanoscillatorwasquantized.[note2]Thequantumofenergyforeachoscillator,accordingtoPlanck,wasproportionaltothefrequencyoftheoscillator;theconstantofproportionalityisnowknownasthePlanckconstant.ThePlanckconstant,usuallywrittenash,hasthevalueof6.63×10−34 Js.So,theenergyEofanoscillatoroffrequencyfisgivenby E = n h f , where n = 1 , 2 , 3 , … {\displaystyleE=nhf,\quad{\text{where}}\quadn=1,2,3,\ldots} [7] Tochangethecolorofsucharadiatingbody,itisnecessarytochangeitstemperature.Planck'slawexplainswhy:increasingthetemperatureofabodyallowsittoemitmoreenergyoverall,andmeansthatalargerproportionoftheenergyistowardsthevioletendofthespectrum. Planck'slawwasthefirstquantumtheoryinphysics,andPlanckwontheNobelPrizein1918"inrecognitionoftheservicesherenderedtotheadvancementofPhysicsbyhisdiscoveryofenergyquanta".[8]Atthetime,however,Planck'sviewwasthatquantizationwaspurelyaheuristicmathematicalconstruct,ratherthan(asisnowbelieved)afundamentalchangeinourunderstandingoftheworld.[9] Photons:thequantizationoflight[edit] AlbertEinsteinc.1905 In1905,AlbertEinsteintookanextrastep.Hesuggestedthatquantizationwasnotjustamathematicalconstruct,butthattheenergyinabeamoflightactuallyoccursinindividualpackets,whicharenowcalledphotons.[10]Theenergyofasinglephotonoflightoffrequency f {\displaystylef} isgivenbythefrequencymultipliedbyPlanck'sconstant h {\displaystyleh} (anextremelytinypositivenumber): E = h f {\displaystyleE=hf} Forcenturies,scientistshaddebatedbetweentwopossibletheoriesoflight:wasitawaveordiditinsteadcompriseastreamoftinyparticles?Bythe19thcentury,thedebatewasgenerallyconsideredtohavebeensettledinfavorofthewavetheory,asitwasabletoexplainobservedeffectssuchasrefraction,diffraction,interference,andpolarization.[11]JamesClerkMaxwellhadshownthatelectricity,magnetism,andlightareallmanifestationsofthesamephenomenon:theelectromagneticfield.Maxwell'sequations,whicharethecompletesetoflawsofclassicalelectromagnetism,describelightaswaves:acombinationofoscillatingelectricandmagneticfields.Becauseofthepreponderanceofevidenceinfavorofthewavetheory,Einstein'sideasweremetinitiallywithgreatskepticism.Eventually,however,thephotonmodelbecamefavored.Oneofthemostsignificantpiecesofevidenceinitsfavorwasitsabilitytoexplainseveralpuzzlingpropertiesofthephotoelectriceffect,describedinthefollowingsection.Nonetheless,thewaveanalogyremainedindispensableforhelpingtounderstandothercharacteristicsoflight:diffraction,refraction,andinterference. Thephotoelectriceffect[edit] Lightisshoneuponthesurfacefromtheleft.Ifthelightfrequencyishighenough,i.e.ifitdeliverssufficientenergy,negativelychargedelectronsareejectedfromthemetal. Mainarticle:Photoelectriceffect In1887,HeinrichHertzobservedthatwhenlightwithsufficientfrequencyhitsametallicsurface,thesurfaceemitselectrons.[12]In1902,PhilippLenarddiscoveredthatthemaximumpossibleenergyofanejectedelectronisrelatedtothefrequencyofthelight,nottoitsintensity:ifthefrequencyistoolow,noelectronsareejectedregardlessoftheintensity.Strongbeamsoflighttowardtheredendofthespectrummightproducenoelectricalpotentialatall,whileweakbeamsoflighttowardthevioletendofthespectrumwouldproducehigherandhighervoltages.Thelowestfrequencyoflightthatcancauseelectronstobeemitted,calledthethresholdfrequency,isdifferentfordifferentmetals.Thisobservationisatoddswithclassicalelectromagnetism,whichpredictsthattheelectron'senergyshouldbeproportionaltotheintensityoftheincidentradiation.[13]: 24 Sowhenphysicistsfirstdiscovereddevicesexhibitingthephotoelectriceffect,theyinitiallyexpectedthatahigherintensityoflightwouldproduceahighervoltagefromthephotoelectricdevice. Einsteinexplainedtheeffectbypostulatingthatabeamoflightisastreamofparticles("photons")andthat,ifthebeamisoffrequencyf,theneachphotonhasanenergyequaltohf.[12]Anelectronislikelytobestruckonlybyasinglephoton,whichimpartsatmostanenergyhftotheelectron.[12]Therefore,theintensityofthebeamhasnoeffect[note3]andonlyitsfrequencydeterminesthemaximumenergythatcanbeimpartedtotheelectron.[12] Toexplainthethresholdeffect,Einsteinarguedthatittakesacertainamountofenergy,calledtheworkfunctionanddenotedbyφ,toremoveanelectronfromthemetal.[12]Thisamountofenergyisdifferentforeachmetal.Iftheenergyofthephotonislessthantheworkfunction,thenitdoesnotcarrysufficientenergytoremovetheelectronfromthemetal.Thethresholdfrequency,f0,isthefrequencyofaphotonwhoseenergyisequaltotheworkfunction: φ = h f 0 . {\displaystyle\varphi=hf_{0}.} Iffisgreaterthanf0,theenergyhfisenoughtoremoveanelectron.Theejectedelectronhasakineticenergy,EK,whichis,atmost,equaltothephoton'senergyminustheenergyneededtodislodgetheelectronfromthemetal: E K = h f − φ = h ( f − f 0 ) . {\displaystyleE_{K}=hf-\varphi=h(f-f_{0}).} Einstein'sdescriptionoflightasbeingcomposedofparticlesextendedPlanck'snotionofquantizedenergy,whichisthatasinglephotonofagivenfrequency,f,deliversaninvariantamountofenergy,hf.Inotherwords,individualphotonscandelivermoreorlessenergy,butonlydependingontheirfrequencies.Innature,singlephotonsarerarelyencountered.TheSunandemissionsourcesavailableinthe19thcenturyemitvastnumbersofphotonseverysecond,andsotheimportanceoftheenergycarriedbyeachphotonwasnotobvious.Einstein'sideathattheenergycontainedinindividualunitsoflightdependsontheirfrequencymadeitpossibletoexplainexperimentalresultsthathadseemedcounterintuitive.However,althoughthephotonisaparticle,itwasstillbeingdescribedashavingthewave-likepropertyoffrequency.Effectively,theaccountoflightasaparticleisinsufficient,anditswave-likenatureisstillrequired.[14][note4] Consequencesoflightbeingquantized[edit] Therelationshipbetweenthefrequencyofelectromagneticradiationandtheenergyofeachphotoniswhyultravioletlightcancausesunburn,butvisibleorinfraredlightcannot.Aphotonofultravioletlightdeliversahighamountofenergy—enoughtocontributetocellulardamagesuchasoccursinasunburn.Aphotonofinfraredlightdeliverslessenergy—onlyenoughtowarmone'sskin.So,aninfraredlampcanwarmalargesurface,perhapslargeenoughtokeeppeoplecomfortableinacoldroom,butitcannotgiveanyoneasunburn.[16] Allphotonsofthesamefrequencyhaveidenticalenergy,andallphotonsofdifferentfrequencieshaveproportionally(order1,Ephoton=hf)differentenergies.[17]However,althoughtheenergyimpartedbyphotonsisinvariantatanygivenfrequency,theinitialenergystateoftheelectronsinaphotoelectricdevicebeforeabsorptionoflightisnotnecessarilyuniform.Anomalousresultsmayoccurinthecaseofindividualelectrons.Forinstance,anelectronthatwasalreadyexcitedabovetheequilibriumlevelofthephotoelectricdevicemightbeejectedwhenitabsorbeduncharacteristicallylow-frequencyillumination.Statistically,however,thecharacteristicbehaviorofaphotoelectricdevicereflectsthebehaviorofthevastmajorityofitselectrons,whichareattheirequilibriumlevel.Thispointhelpsclarifythedistinctionbetweenthestudyofsmallindividualparticlesinquantumdynamicsandthestudyofmassiveindividualparticlesinclassicalphysics.[citationneeded] Thequantizationofmatter:theBohrmodeloftheatom[edit] Bythedawnofthe20thcentury,theevidencerequiredamodeloftheatomwithadiffusecloudofnegativelychargedelectronssurroundingasmall,dense,positivelychargednucleus.Thesepropertiessuggestedamodelinwhichelectronscirclethenucleuslikeplanetsorbitingasun.[note5]However,itwasalsoknownthattheatominthismodelwouldbeunstable:accordingtoclassicaltheory,orbitingelectronsareundergoingcentripetalacceleration,andshouldthereforegiveoffelectromagneticradiation,thelossofenergyalsocausingthemtospiraltowardthenucleus,collidingwithitinafractionofasecond. Asecond,relatedpuzzlewastheemissionspectrumofatoms.Whenagasisheated,itgivesofflightonlyatdiscretefrequencies.Forexample,thevisiblelightgivenoffbyhydrogenconsistsoffourdifferentcolors,asshowninthepicturebelow.Theintensityofthelightatdifferentfrequenciesisalsodifferent.Bycontrast,whitelightconsistsofacontinuousemissionacrossthewholerangeofvisiblefrequencies.Bytheendofthenineteenthcentury,asimpleruleknownasBalmer'sformulashowedhowthefrequenciesofthedifferentlinesrelatedtoeachother,thoughwithoutexplainingwhythiswas,ormakinganypredictionabouttheintensities.Theformulaalsopredictedsomeadditionalspectrallinesinultravioletandinfraredlightthathadnotbeenobservedatthetime.Theselineswerelaterobservedexperimentally,raisingconfidenceinthevalueoftheformula. Emissionspectrumofhydrogen.Whenexcited,hydrogengasgivesofflightinfourdistinctcolors(spectrallines)inthevisiblespectrum,aswellasanumberoflinesintheinfraredandultraviolet. Themathematicalformuladescribinghydrogen'semissionspectrum In1885theSwissmathematicianJohannBalmerdiscoveredthateachwavelengthλ(lambda)inthevisiblespectrumofhydrogenisrelatedtosomeintegernbytheequation λ = B ( n 2 n 2 − 4 ) n = 3 , 4 , 5 , 6 {\displaystyle\lambda=B\left({\frac{n^{2}}{n^{2}-4}}\right)\qquad\qquadn=3,4,5,6} whereBisaconstantBalmerdeterminedisequalto364.56 nm. In1888JohannesRydberggeneralizedandgreatlyincreasedtheexplanatoryutilityofBalmer'sformula.HepredictedthatλisrelatedtotwointegersnandmaccordingtowhatisnowknownastheRydbergformula:[18] 1 λ = R ( 1 m 2 − 1 n 2 ) , {\displaystyle{\frac{1}{\lambda}}=R\left({\frac{1}{m^{2}}}-{\frac{1}{n^{2}}}\right),} whereRistheRydbergconstant,equalto0.0110 nm−1,andnmustbegreaterthanm. Rydberg'sformulaaccountsforthefourvisiblewavelengthsofhydrogenbysettingm=2andn=3,4,5,6.Italsopredictsadditionalwavelengthsintheemissionspectrum:form=1andforn>1,theemissionspectrumshouldcontaincertainultravioletwavelengths,andform=3andn>3,itshouldalsocontaincertaininfraredwavelengths.Experimentalobservationofthesewavelengthscametwodecadeslater:in1908LouisPaschenfoundsomeofthepredictedinfraredwavelengths,andin1914TheodoreLymanfoundsomeofthepredictedultravioletwavelengths.[18] BothBalmerandRydberg'sformulasinvolveintegers:inmodernterms,theyimplythatsomepropertyoftheatomisquantized.Understandingexactlywhatthispropertywas,andwhyitwasquantized,wasamajorpartofthedevelopmentofquantummechanics,asshownintherestofthisarticle. TheBohrmodeloftheatom,showinganelectrontransitioningfromoneorbittoanotherbyemittingaphoton In1913NielsBohrproposedanewmodeloftheatomthatincludedquantizedelectronorbits:electronsstillorbitthenucleusmuchasplanetsorbitaroundthesun,buttheyarepermittedtoinhabitonlycertainorbits,nottoorbitatanyarbitrarydistance.[19]Whenanatomemitted(orabsorbed)energy,theelectrondidnotmoveinacontinuoustrajectoryfromoneorbitaroundthenucleustoanother,asmightbeexpectedclassically.Instead,theelectronwouldjumpinstantaneouslyfromoneorbittoanother,givingofftheemittedlightintheformofaphoton.[20]Thepossibleenergiesofphotonsgivenoffbyeachelementweredeterminedbythedifferencesinenergybetweentheorbits,andsotheemissionspectrumforeachelementwouldcontainanumberoflines.[21] NielsBohrasayoungman Startingfromonlyonesimpleassumptionabouttherulethattheorbitsmustobey,theBohrmodelwasabletorelatetheobservedspectrallinesintheemissionspectrumofhydrogentopreviouslyknownconstants.InBohr'smodel,theelectronwasnotallowedtoemitenergycontinuouslyandcrashintothenucleus:onceitwasintheclosestpermittedorbit,itwasstableforever.Bohr'smodeldidnotexplainwhytheorbitsshouldbequantizedinthatway,norwasitabletomakeaccuratepredictionsforatomswithmorethanoneelectron,ortoexplainwhysomespectrallinesarebrighterthanothers. SomefundamentalassumptionsoftheBohrmodelweresoonprovenwrong—butthekeyresultthatthediscretelinesinemissionspectraareduetosomepropertyoftheelectronsinatomsbeingquantizediscorrect.ThewaythattheelectronsactuallybehaveisstrikinglydifferentfromBohr'satom,andfromwhatweseeintheworldofoureverydayexperience;thismodernquantummechanicalmodeloftheatomisdiscussedbelow. AmoredetailedexplanationoftheBohrmodel Bohrtheorizedthattheangularmomentum,L,ofanelectronisquantized: L = n h 2 π = n ℏ {\displaystyleL=n{\frac{h}{2\pi}}=n\hbar} wherenisanintegerandhandħarethePlanckconstantandPlanckreducedconstantrespectively.Startingfromthisassumption,Coulomb'slawandtheequationsofcircularmotionshowthatanelectronwithnunitsofangularmomentumorbitsaprotonatadistancergivenby r = n 2 h 2 4 π 2 k e m e 2 {\displaystyler={\frac{n^{2}h^{2}}{4\pi^{2}k_{e}me^{2}}}} , wherekeistheCoulombconstant,misthemassofanelectron,andeisthechargeonanelectron. Forsimplicitythisiswrittenas r = n 2 a 0 , {\displaystyler=n^{2}a_{0},\!} wherea0,calledtheBohrradius,isequalto0.0529 nm. TheBohrradiusistheradiusofthesmallestallowedorbit. Theenergyoftheelectron[note6]canalsobecalculated,andisgivenby E = − k e e 2 2 a 0 1 n 2 {\displaystyleE=-{\frac{k_{\mathrm{e}}e^{2}}{2a_{0}}}{\frac{1}{n^{2}}}} . ThusBohr'sassumptionthatangularmomentumisquantizedmeansthatanelectroncaninhabitonlycertainorbitsaroundthenucleusandthatitcanhaveonlycertainenergies.Aconsequenceoftheseconstraintsisthattheelectrondoesnotcrashintothenucleus:itcannotcontinuouslyemitenergy,anditcannotcomeclosertothenucleusthana0(theBohrradius). Anelectronlosesenergybyjumpinginstantaneouslyfromitsoriginalorbittoalowerorbit;theextraenergyisemittedintheformofaphoton.Conversely,anelectronthatabsorbsaphotongainsenergy,henceitjumpstoanorbitthatisfartherfromthenucleus. Eachphotonfromglowingatomichydrogenisduetoanelectronmovingfromahigherorbit,withradiusrn,toalowerorbit,rm.TheenergyEγofthisphotonisthedifferenceintheenergiesEnandEmoftheelectron: E γ = E n − E m = k e e 2 2 a 0 ( 1 m 2 − 1 n 2 ) {\displaystyleE_{\gamma}=E_{n}-E_{m}={\frac{k_{\mathrm{e}}e^{2}}{2a_{0}}}\left({\frac{1}{m^{2}}}-{\frac{1}{n^{2}}}\right)} SincePlanck'sequationshowsthatthephoton'senergyisrelatedtoitswavelengthbyEγ=hc/λ,thewavelengthsoflightthatcanbeemittedaregivenby 1 λ = k e e 2 2 a 0 h c ( 1 m 2 − 1 n 2 ) . {\displaystyle{\frac{1}{\lambda}}={\frac{k_{\mathrm{e}}e^{2}}{2a_{0}hc}}\left({\frac{1}{m^{2}}}-{\frac{1}{n^{2}}}\right).} ThisequationhasthesameformastheRydbergformula,andpredictsthattheconstantRshouldbegivenby R = k e e 2 2 a 0 h c . {\displaystyleR={\frac{k_{\mathrm{e}}e^{2}}{2a_{0}hc}}.} Therefore,theBohrmodeloftheatomcanpredicttheemissionspectrumofhydrogenintermsoffundamentalconstants.[note7]However,itwasnotabletomakeaccuratepredictionsformulti-electronatoms,ortoexplainwhysomespectrallinesarebrighterthanothers. Wave–particleduality[edit] Mainarticle:Wave–particleduality LouisdeBrogliein1929.DeBrogliewontheNobelPrizeinPhysicsforhispredictionthatmatteractsasawave,madeinhis1924PhDthesis. Justaslighthasbothwave-likeandparticle-likeproperties,matteralsohaswave-likeproperties.[22] Matterbehavingasawavewasfirstdemonstratedexperimentallyforelectrons:abeamofelectronscanexhibitdiffraction,justlikeabeamoflightorawaterwave.[note8]Similarwave-likephenomenawerelatershownforatomsandevenmolecules. Thewavelength,λ,associatedwithanyobjectisrelatedtoitsmomentum,p,throughthePlanckconstant,h:[23][24] p = h λ . {\displaystylep={\frac{h}{\lambda}}.} Therelationship,calledthedeBrogliehypothesis,holdsforalltypesofmatter:allmatterexhibitspropertiesofbothparticlesandwaves. Theconceptofwave–particledualitysaysthatneithertheclassicalconceptof"particle"norof"wave"canfullydescribethebehaviorofquantum-scaleobjects,eitherphotonsormatter.Wave–particledualityisanexampleoftheprincipleofcomplementarityinquantumphysics.[25][26][27][28][29]Anelegantexampleofwave-particleduality,thedouble-slitexperiment,isdiscussedinthesectionbelow. Thedouble-slitexperiment[edit] Mainarticle:Double-slitexperiment Thediffractionpatternproducedwhenlightisshonethroughoneslit(top)andtheinterferencepatternproducedbytwoslits(bottom).Themuchmorecomplexpatternfromtwoslits,withitssmall-scaleinterferencefringes,demonstratesthewave-likepropagationoflight. Thedouble-slitexperimentforaclassicalparticle,awave,andaquantumparticledemonstratingwave-particleduality Inthedouble-slitexperiment,asoriginallyperformedbyThomasYoungin1803,[30]andthenAugustinFresneladecadelater,[30]abeamoflightisdirectedthroughtwonarrow,closelyspacedslits,producinganinterferencepatternoflightanddarkbandsonascreen.Ifoneoftheslitsiscoveredup,onemightnaïvelyexpectthattheintensityofthefringesduetointerferencewouldbehalvedeverywhere.Infact,amuchsimplerpatternisseen,adiffractionpatterndiametricallyoppositetheopenslit.Thesamebehaviorcanbedemonstratedinwaterwaves,andsothedouble-slitexperimentwasseenasademonstrationofthewavenatureoflight. Variationsofthedouble-slitexperimenthavebeenperformedusingelectrons,atoms,andevenlargemolecules,[31][32]andthesametypeofinterferencepatternisseen.Thusithasbeendemonstratedthatallmatterpossessesbothparticleandwavecharacteristics. Evenifthesourceintensityisturneddown,sothatonlyoneparticle(e.g.photonorelectron)ispassingthroughtheapparatusatatime,thesameinterferencepatterndevelopsovertime.Thequantumparticleactsasawavewhenpassingthroughthedoubleslits,butasaparticlewhenitisdetected.Thisisatypicalfeatureofquantumcomplementarity:aquantumparticleactsasawaveinanexperimenttomeasureitswave-likeproperties,andlikeaparticleinanexperimenttomeasureitsparticle-likeproperties.Thepointonthedetectorscreenwhereanyindividualparticleshowsupistheresultofarandomprocess.However,thedistributionpatternofmanyindividualparticlesmimicsthediffractionpatternproducedbywaves. ApplicationtotheBohrmodel[edit] DeBroglieexpandedtheBohrmodeloftheatombyshowingthatanelectroninorbitaroundanucleuscouldbethoughtofashavingwave-likeproperties.Inparticular,anelectronisobservedonlyinsituationsthatpermitastandingwavearoundanucleus.Anexampleofastandingwaveisaviolinstring,whichisfixedatbothendsandcanbemadetovibrate.Thewavescreatedbyastringedinstrumentappeartooscillateinplace,movingfromcresttotroughinanup-and-downmotion.Thewavelengthofastandingwaveisrelatedtothelengthofthevibratingobjectandtheboundaryconditions.Forexample,becausetheviolinstringisfixedatbothends,itcancarrystandingwavesofwavelengths 2 l n {\displaystyle{\frac{2l}{n}}} ,wherelisthelengthandnisapositiveinteger.DeBrogliesuggestedthattheallowedelectronorbitswerethoseforwhichthecircumferenceoftheorbitwouldbeanintegernumberofwavelengths.Theelectron'swavelength,therefore,determinesthatonlyBohrorbitsofcertaindistancesfromthenucleusarepossible.Inturn,atanydistancefromthenucleussmallerthanacertainvalue,itwouldbeimpossibletoestablishanorbit.TheminimumpossibledistancefromthenucleusiscalledtheBohrradius.[33] DeBroglie'streatmentofquantumeventsservedasastartingpointforSchrödingerwhenhesetouttoconstructawaveequationtodescribequantum-theoreticalevents. Spin[edit] Mainarticle:Spin(physics) Seealso:Stern–Gerlachexperiment QuantumspinversusclassicalmagnetintheStern–Gerlachexperiment In1922,OttoSternandWaltherGerlachshotsilveratomsthroughaninhomogeneousmagneticfield.Relativetoitsnorthernpole,pointingup,down,orsomewhereinbetween,inclassicalmechanics,amagnetthrownthroughamagneticfieldmaybedeflectedasmallorlargedistanceupwardsordownwards.TheatomsthatSternandGerlachshotthroughthemagneticfieldactedsimilarly.However,whilethemagnetscouldbedeflectedvariabledistances,theatomswouldalwaysbedeflectedaconstantdistanceeitherupordown.Thisimpliedthatthepropertyoftheatomthatcorrespondstothemagnet'sorientationmustbequantized,takingoneoftwovalues(eitherupordown),asopposedtobeingchosenfreelyfromanyangle. RalphKronigoriginatedthetheorythatparticlessuchasatomsorelectronsbehaveasiftheyrotate,or"spin",aboutanaxis.Spinwouldaccountforthemissingmagneticmoment,[clarificationneeded]andallowtwoelectronsinthesameorbitaltooccupydistinctquantumstatesifthey"spun"inoppositedirections,thussatisfyingtheexclusionprinciple.Thequantumnumberrepresentedthesense(positiveornegative)ofspin. ThechoiceoftheorientationofthemagneticfieldusedintheStern–Gerlachexperimentisarbitrary.Intheanimationshownhere,thefieldisverticalandsotheatomsaredeflectedeitherupordown.Ifthemagnetisrotatedaquarterturn,theatomsaredeflectedeitherleftorright.Usingaverticalfieldshowsthatthespinalongtheverticalaxisisquantized,andusingahorizontalfieldshowsthatthespinalongthehorizontalaxisisquantized. Ifinsteadofhittingadetectorscreen,oneofthebeamsofatomscomingoutoftheStern–Gerlachapparatusispassedintoanother(inhomogeneous)magneticfieldorientedinthesamedirection,alloftheatomsaredeflectedthesamewayinthissecondfield.However,ifthesecondfieldisorientedat90°tothefirst,thenhalfoftheatomsaredeflectedonewayandhalftheothersothattheatom'sspinaboutthehorizontalandverticalaxesareindependentofeachother.However,ifoneofthesebeams(e.g.theatomsthatweredeflectedupthenleft)ispassedintoathirdmagneticfield,orientedthesamewayasthefirst,halfoftheatomsgoonewayandhalftheother,eventhoughtheyallwentinthesamedirectionoriginally.Theactionofmeasuringtheatoms'spinconcerningahorizontalfieldhaschangedtheirspinconcerningaverticalfield. TheStern–Gerlachexperimentdemonstratesseveralimportantfeaturesofquantummechanics: Afeatureofthenaturalworldhasbeendemonstratedtobequantized,andabletotakeonlycertaindiscretevalues. Particlespossessanintrinsicangularmomentumthatiscloselyanalogoustotheangularmomentumofaclassicallyspinningobject. Measurementchangesthesystembeingmeasuredinquantummechanics.Onlythespinofanobjectinonedirectioncanbeknown,andobservingthespininanotherdirectiondestroystheoriginalinformationaboutthespin. Quantummechanicsisprobabilistic:whetherthespinofanyindividualatomsentintotheapparatusispositiveornegativeisrandom. Developmentofmodernquantummechanics[edit] In1925,WernerHeisenbergattemptedtosolveoneoftheproblemsthattheBohrmodelleftunanswered,explainingtheintensitiesofthedifferentlinesinthehydrogenemissionspectrum.Throughaseriesofmathematicalanalogies,hewroteoutthequantum-mechanicalanalogfortheclassicalcomputationofintensities.[34]Shortlyafterward,Heisenberg'scolleagueMaxBornrealizedthatHeisenberg'smethodofcalculatingtheprobabilitiesfortransitionsbetweenthedifferentenergylevelscouldbestbeexpressedbyusingthemathematicalconceptofmatrices.[note9] Inthesameyear,buildingondeBroglie'shypothesis,ErwinSchrödingerdevelopedtheequationthatdescribesthebehaviorofaquantum-mechanicalwave.[35]Themathematicalmodel,calledtheSchrödingerequationafteritscreator,iscentraltoquantummechanics,definesthepermittedstationarystatesofaquantumsystem,anddescribeshowthequantumstateofaphysicalsystemchangesintime.[36]Thewaveitselfisdescribedbyamathematicalfunctionknownasa"wavefunction".Schrödingersaidthatthewavefunctionprovidesthe"meansforpredictingtheprobabilityofmeasurementresults".[37] Schrödingerwasabletocalculatetheenergylevelsofhydrogenbytreatingahydrogenatom'selectronasaclassicalwave,movinginawelloftheelectricalpotentialcreatedbytheproton.ThiscalculationaccuratelyreproducedtheenergylevelsoftheBohrmodel. InMay1926,SchrödingerprovedthatHeisenberg'smatrixmechanicsandhisownwavemechanicsmadethesamepredictionsaboutthepropertiesandbehavioroftheelectron;mathematically,thetwotheorieshadanunderlyingcommonform.Yetthetwomendisagreedontheinterpretationoftheirmutualtheory.Forinstance,Heisenbergacceptedthetheoreticalpredictionofjumpsofelectronsbetweenorbitalsinanatom,[38]butSchrödingerhopedthatatheorybasedoncontinuouswave-likepropertiescouldavoidwhathecalled(asparaphrasedbyWilhelmWien)"thisnonsenseaboutquantumjumps".[39]Intheend,Heisenberg'sapproachwonout,andquantumjumpswereconfirmed.[40] Copenhageninterpretation[edit] Mainarticle:Copenhageninterpretation TheNielsBohrInstituteinCopenhagen,whichwasafocalpointforresearchersinquantummechanicsandrelatedsubjectsinthe1920sand1930s.Mostoftheworld'sbestknowntheoreticalphysicistsspenttimethere. Bohr,Heisenberg,andotherstriedtoexplainwhattheseexperimentalresultsandmathematicalmodelsreallymean.Theirdescription,knownastheCopenhageninterpretationofquantummechanics,aimedtodescribethenatureofrealitythatwasbeingprobedbythemeasurementsanddescribedbythemathematicalformulationsofquantummechanics. ThemainprinciplesoftheCopenhageninterpretationare: Asystemiscompletelydescribedbyawavefunction,usuallyrepresentedbytheGreekletter ψ {\displaystyle\psi} ("psi").(Heisenberg) How ψ {\displaystyle\psi} changesovertimeisgivenbytheSchrödingerequation.[clarificationneeded] Thedescriptionofnatureisessentiallyprobabilistic.Theprobabilityofanevent—forexample,whereonthescreenaparticleshowsupinthedouble-slitexperiment—isrelatedtothesquareoftheabsolutevalueoftheamplitudeofitswavefunction.(Bornrule,duetoMaxBorn,whichgivesaphysicalmeaningtothewavefunctionintheCopenhageninterpretation:theprobabilityamplitude) Itisnotpossibletoknowthevaluesofallofthepropertiesofthesystematthesametime;thosepropertiesthatarenotknownwithprecisionmustbedescribedbyprobabilities.(Heisenberg'suncertaintyprinciple) Matter,likeenergy,exhibitsawave-particleduality.Anexperimentcandemonstratetheparticle-likepropertiesofmatter,oritswave-likeproperties;butnotbothatthesametime.(ComplementarityprincipleduetoBohr) Measuringdevicesareessentiallyclassicaldevicesandmeasureclassicalpropertiessuchaspositionandmomentum. Thequantummechanicaldescriptionoflargesystemsshouldcloselyapproximatetheclassicaldescription.(CorrespondenceprincipleofBohrandHeisenberg) Variousconsequencesoftheseprinciplesarediscussedinmoredetailinthefollowingsubsections. Uncertaintyprinciple[edit] Mainarticle:Uncertaintyprinciple WernerHeisenbergattheageof26.HeisenbergwontheNobelPrizeinPhysicsin1932fortheworkhedidataroundthistime.[41] Supposeitisdesiredtomeasurethepositionandspeedofanobject—forexample,acargoingthrougharadarspeedtrap.Itcanbeassumedthatthecarhasadefinitepositionandspeedataparticularmomentintime.Howaccuratelythesevaluescanbemeasureddependsonthequalityofthemeasuringequipment.Iftheprecisionofthemeasuringequipmentisimproved,itprovidesaresultclosertothetruevalue.Itmightbeassumedthatthespeedofthecaranditspositioncouldbeoperationallydefinedandmeasuredsimultaneously,aspreciselyasmightbedesired. In1927,Heisenbergprovedthatthislastassumptionisnotcorrect.[42]Quantummechanicsshowsthatcertainpairsofphysicalproperties,forexample,positionandspeed,cannotbesimultaneouslymeasured,nordefinedinoperationalterms,toarbitraryprecision:themorepreciselyonepropertyismeasured,ordefinedinoperationalterms,thelesspreciselycantheother.Thisstatementisknownastheuncertaintyprinciple.Theuncertaintyprincipleisnotonlyastatementabouttheaccuracyofourmeasuringequipmentbut,moredeeply,isabouttheconceptualnatureofthemeasuredquantities—theassumptionthatthecarhadsimultaneouslydefinedpositionandspeeddoesnotworkinquantummechanics.Onascaleofcarsandpeople,theseuncertaintiesarenegligible,butwhendealingwithatomsandelectronstheybecomecritical.[43] Heisenberggave,asanillustration,themeasurementofthepositionandmomentumofanelectronusingaphotonoflight.Inmeasuringtheelectron'sposition,thehigherthefrequencyofthephoton,themoreaccurateisthemeasurementofthepositionoftheimpactofthephotonwiththeelectron,butthegreateristhedisturbanceoftheelectron.Thisisbecausefromtheimpactwiththephoton,theelectronabsorbsarandomamountofenergy,renderingthemeasurementobtainedofitsmomentumincreasinglyuncertain(momentumisvelocitymultipliedbymass),foroneisnecessarilymeasuringitspost-impactdisturbedmomentumfromthecollisionproductsandnotitsoriginalmomentum(momentumwhichshouldbesimultaneouslymeasuredwithposition).Withaphotonoflowerfrequency,thedisturbance(andhenceuncertainty)inthemomentumisless,butsoistheaccuracyofthemeasurementofthepositionoftheimpact.[44] Attheheartoftheuncertaintyprincipleisafactthatforanymathematicalanalysisinthepositionandvelocitydomains,achievingasharper(moreprecise)curveinthepositiondomaincanonlybedoneattheexpenseofamoregradual(lessprecise)curveinthespeeddomain,andviceversa.Moresharpnessinthepositiondomainrequirescontributionsfrommorefrequenciesinthespeeddomaintocreatethenarrowercurve,andviceversa.Itisafundamentaltradeoffinherentinanysuchrelatedorcomplementarymeasurements,butisonlyreallynoticeableatthesmallest(Planck)scale,nearthesizeofelementaryparticles. Theuncertaintyprincipleshowsmathematicallythattheproductoftheuncertaintyinthepositionandmomentumofaparticle(momentumisvelocitymultipliedbymass)couldneverbelessthanacertainvalue,andthatthisvalueisrelatedtoPlanck'sconstant. Wavefunctioncollapse[edit] Mainarticle:Wavefunctioncollapse Wavefunctioncollapsemeansthatameasurementhasforcedorconvertedaquantum(probabilisticorpotential)stateintoadefinitemeasuredvalue.Thisphenomenonisonlyseeninquantummechanicsratherthanclassicalmechanics. Forexample,beforeaphotonactually"showsup"onadetectionscreenitcanbedescribedonlywithasetofprobabilitiesforwhereitmightshowup.Whenitdoesappear,forinstanceintheCCDofanelectroniccamera,thetimeandspacewhereitinteractedwiththedeviceareknownwithinverytightlimits.However,thephotonhasdisappearedintheprocessofbeingcaptured(measured),anditsquantumwavefunctionhasdisappearedwithit.Initsplace,somemacroscopicphysicalchangeinthedetectionscreenhasappeared,e.g.,anexposedspotinasheetofphotographicfilm,orachangeinelectricpotentialinsomecellofaCCD. Eigenstatesandeigenvalues[edit] Furtherinformation:Introductiontoeigenstates Becauseoftheuncertaintyprinciple,statementsaboutboththepositionandmomentumofparticlescanassignonlyaprobabilitythatthepositionormomentumhassomenumericalvalue.Therefore,itisnecessarytoformulateclearlythedifferencebetweenthestateofsomethingindeterminate,suchasanelectroninaprobabilitycloud,andthestateofsomethinghavingadefinitevalue.Whenanobjectcandefinitelybe"pinned-down"insomerespect,itissaidtopossessaneigenstate. IntheStern–Gerlachexperimentdiscussedabove,thespinoftheatomabouttheverticalaxishastwoeigenstates:upanddown.Beforemeasuringit,wecanonlysaythatanyindividualatomhasanequalprobabilityofbeingfoundtohavespinuporspindown.Themeasurementprocesscausesthewavefunctiontocollapseintooneofthetwostates. Theeigenstatesofspinabouttheverticalaxisarenotsimultaneouslyeigenstatesofspinaboutthehorizontalaxis,sothisatomhasanequalprobabilityofbeingfoundtohaveeithervalueofspinaboutthehorizontalaxis.Asdescribedinthesectionabove,measuringthespinaboutthehorizontalaxiscanallowanatomthatwasspunuptospindown:measuringitsspinaboutthehorizontalaxiscollapsesitswavefunctionintooneoftheeigenstatesofthismeasurement,whichmeansitisnolongerinaneigenstateofspinabouttheverticalaxis,socantakeeithervalue. ThePauliexclusionprinciple[edit] WolfgangPauli Mainarticle:Pauliexclusionprinciple In1924,WolfgangPauliproposedanewquantumdegreeoffreedom(orquantumnumber),withtwopossiblevalues,toresolveinconsistenciesbetweenobservedmolecularspectraandthepredictionsofquantummechanics.Inparticular,thespectrumofatomichydrogenhadadoublet,orpairoflinesdifferingbyasmallamount,whereonlyonelinewasexpected.Pauliformulatedhisexclusionprinciple,stating,"Therecannotexistanatominsuchaquantumstatethattwoelectronswithin[it]havethesamesetofquantumnumbers."[45] Ayearlater,UhlenbeckandGoudsmitidentifiedPauli'snewdegreeoffreedomwiththepropertycalledspinwhoseeffectswereobservedintheStern–Gerlachexperiment. Applicationtothehydrogenatom[edit] Mainarticle:Atomicorbitalmodel Bohr'smodeloftheatomwasessentiallyaplanetaryone,withtheelectronsorbitingaroundthenuclear"sun".However,theuncertaintyprinciplestatesthatanelectroncannotsimultaneouslyhaveanexactlocationandvelocityinthewaythataplanetdoes.Insteadofclassicalorbits,electronsaresaidtoinhabitatomicorbitals.Anorbitalisthe"cloud"ofpossiblelocationsinwhichanelectronmightbefound,adistributionofprobabilitiesratherthanapreciselocation.[45]Eachorbitalisthreedimensional,ratherthanthetwo-dimensionalorbit,andisoftendepictedasathree-dimensionalregionwithinwhichthereisa95percentprobabilityoffindingtheelectron.[46] Schrödingerwasabletocalculatetheenergylevelsofhydrogenbytreatingahydrogenatom'selectronasawave,representedbythe"wavefunction"Ψ,inanelectricpotentialwell,V,createdbytheproton.ThesolutionstoSchrödinger'sequation[clarificationneeded]aredistributionsofprobabilitiesforelectronpositionsandlocations.Orbitalshavearangeofdifferentshapesinthreedimensions.Theenergiesofthedifferentorbitalscanbecalculated,andtheyaccuratelymatchtheenergylevelsoftheBohrmodel. WithinSchrödinger'spicture,eachelectronhasfourproperties: An"orbital"designation,indicatingwhethertheparticle-waveisonethatisclosertothenucleuswithlessenergyoronethatisfartherfromthenucleuswithmoreenergy; The"shape"oftheorbital,sphericalorotherwise; The"inclination"oftheorbital,determiningthemagneticmomentoftheorbitalaroundthez-axis. The"spin"oftheelectron. Thecollectivenameforthesepropertiesisthequantumstateoftheelectron.Thequantumstatecanbedescribedbygivinganumbertoeachoftheseproperties;theseareknownastheelectron'squantumnumbers.Thequantumstateoftheelectronisdescribedbyitswavefunction.ThePauliexclusionprincipledemandsthatnotwoelectronswithinanatommayhavethesamevaluesofallfournumbers.Theshapesofatomicorbitals.Rows:1s,2p,3dand4f.Fromlefttoright m = − l , … , l {\displaystylem=-l,\ldots,l} .Thecolorsshowthephaseofthewavefunction. Thefirstpropertydescribingtheorbitalistheprincipalquantumnumber,n,whichisthesameasinBohr'smodel.ndenotestheenergylevelofeachorbital.Thepossiblevaluesfornareintegers: n = 1 , 2 , 3 … {\displaystylen=1,2,3\ldots} Thenextquantumnumber,theazimuthalquantumnumber,denotedl,describestheshapeoftheorbital.Theshapeisaconsequenceoftheangularmomentumoftheorbital.Theangularmomentumrepresentstheresistanceofaspinningobjecttospeedinguporslowingdownundertheinfluenceofexternalforce.Theazimuthalquantumnumberrepresentstheorbitalangularmomentumofanelectronarounditsnucleus.Thepossiblevaluesforlareintegersfrom0ton−1(wherenistheprincipalquantumnumberoftheelectron): l = 0 , 1 , … , n − 1. {\displaystylel=0,1,\ldots,n-1.} Theshapeofeachorbitalisusuallyreferredtobyaletter,ratherthanbyitsazimuthalquantumnumber.Thefirstshape(l=0)isdenotedbytheletters(amnemonicbeing"sphere").Thenextshapeisdenotedbytheletterpandhastheformofadumbbell.Theotherorbitalshavemorecomplicatedshapes(seeatomicorbital),andaredenotedbythelettersd,f,g,etc. Thethirdquantumnumber,themagneticquantumnumber,describesthemagneticmomentoftheelectron,andisdenotedbyml(orsimplym).Thepossiblevaluesformlareintegersfrom−ltol(wherelistheazimuthalquantumnumberoftheelectron): m l = − l , − ( l − 1 ) , … , 0 , … , ( l − 1 ) , l . {\displaystylem_{l}=-l,-(l-1),\ldots,0,\ldots,(l-1),l.} Themagneticquantumnumbermeasuresthecomponentoftheangularmomentuminaparticulardirection.Thechoiceofdirectionisarbitrary;conventionallythez-directionischosen. Thefourthquantumnumber,thespinquantumnumber(pertainingtothe"orientation"oftheelectron'sspin)isdenotedms,withvalues+1⁄2or−1⁄2. ThechemistLinusPaulingwrote,bywayofexample: Inthecaseofaheliumatomwithtwoelectronsinthe1sorbital,thePauliExclusionPrinciplerequiresthatthetwoelectronsdifferinthevalueofonequantumnumber.Theirvaluesofn,l,andmlarethesame.Accordinglytheymustdifferinthevalueofms,whichcanhavethevalueof+1⁄2foroneelectronand−1⁄2fortheother."[45] Itistheunderlyingstructureandsymmetryofatomicorbitals,andthewaythatelectronsfillthem,thatleadstotheorganizationoftheperiodictable.Thewaytheatomicorbitalsondifferentatomscombinetoformmolecularorbitalsdeterminesthestructureandstrengthofchemicalbondsbetweenatoms. Diracwaveequation[edit] Mainarticle:Diracequation PaulDirac(1902–1984) In1928,PaulDiracextendedthePauliequation,whichdescribedspinningelectrons,toaccountforspecialrelativity.Theresultwasatheorythatdealtproperlywithevents,suchasthespeedatwhichanelectronorbitsthenucleus,occurringatasubstantialfractionofthespeedoflight.Byusingthesimplestelectromagneticinteraction,Diracwasabletopredictthevalueofthemagneticmomentassociatedwiththeelectron'sspinandfoundtheexperimentallyobservedvalue,whichwastoolargetobethatofaspinningchargedspheregovernedbyclassicalphysics.HewasabletosolveforthespectrallinesofthehydrogenatomandtoreproducefromphysicalfirstprinciplesSommerfeld'ssuccessfulformulaforthefinestructureofthehydrogenspectrum. Dirac'sequationssometimesyieldedanegativevalueforenergy,forwhichheproposedanovelsolution:hepositedtheexistenceofanantielectronandadynamicalvacuum.Thisledtothemany-particlequantumfieldtheory. Quantumentanglement[edit] Mainarticle:Quantumentanglement Superpositionoftwoquantumcharacteristics,andtworesolutionpossibilities ThePauliexclusionprinciplesaysthattwoelectronsinonesystemcannotbeinthesamestate.Natureleavesopenthepossibility,however,thattwoelectronscanhavebothstates"superimposed"overeachofthem.Recallthatthewavefunctionsthatemergesimultaneouslyfromthedoubleslitsarriveatthedetectionscreeninastateofsuperposition.Nothingiscertainuntilthesuperimposedwaveforms"collapse".Atthatinstant,anelectronshowsupsomewhereinaccordancewiththeprobabilitythatisthesquareoftheabsolutevalueofthesumofthecomplex-valuedamplitudesofthetwosuperimposedwaveforms.Thesituationthereisalreadyveryabstract.Aconcretewayofthinkingaboutentangledphotons,photonsinwhichtwocontrarystatesaresuperimposedoneachoftheminthesameevent,isasfollows: Imaginethatwehavetwocolor-codedstatesofphotons:onestatelabeledblueandanotherstatelabeledred.Letthesuperpositionoftheredandthebluestateappear(inimagination)asapurplestate.Weconsideracaseinwhichtwophotonsareproducedastheresultofonesingleatomicevent.Perhapstheyareproducedbytheexcitationofacrystalthatcharacteristicallyabsorbsaphotonofacertainfrequencyandemitstwophotonsofhalftheoriginalfrequency.Inthiscase,thephotonsareinterconnectedviatheirsharedorigininasingleatomicevent.Thissetupresultsinsuperimposedstatesofthephotons.Sothetwophotonscomeoutpurple.Iftheexperimenternowperformssomeexperimentthatdetermineswhetheroneofthephotonsiseitherblueorred,thenthatexperimentchangesthephotoninvolvedfromonehavingasuperpositionofblueandredcharacteristicstoaphotonthathasonlyoneofthosecharacteristics.TheproblemthatEinsteinhadwithsuchanimaginedsituationwasthatifoneofthesephotonshadbeenkeptbouncingbetweenmirrorsinalaboratoryonearth,andtheotheronehadtraveledhalfwaytotheneareststarwhenitstwinwasmadetorevealitselfaseitherblueorred,thatmeantthatthedistantphotonnowhadtoloseitspurplestatustoo.Sowheneveritmightbeinvestigatedafteritstwinhadbeenmeasured,itwouldnecessarilyshowupintheoppositestatetowhateveritstwinhadrevealed. Intryingtoshowthatquantummechanicswasnotacompletetheory,Einsteinstartedwiththetheory'spredictionthattwoormoreparticlesthathaveinteractedinthepastcanappearstronglycorrelatedwhentheirvariouspropertiesarelatermeasured.Hesoughttoexplainthisseeminginteractionclassically,throughtheircommonpast,andpreferablynotbysome"spookyactionatadistance".Theargumentisworkedoutinafamouspaper,Einstein,Podolsky,andRosen(1935;abbreviatedEPR)settingoutwhatisnowcalledtheEPRparadox.Assumingwhatisnowusuallycalledlocalrealism,EPRattemptedtoshowfromquantumtheorythataparticlehasbothpositionandmomentumsimultaneously,whileaccordingtotheCopenhageninterpretation,onlyoneofthosetwopropertiesactuallyexistsandonlyatthemomentthatitisbeingmeasured.EPRconcludedthatquantumtheoryisincompleteinthatitrefusestoconsiderphysicalpropertiesthatobjectivelyexistinnature.(Einstein,Podolsky,&Rosen1935iscurrentlyEinstein'smostcitedpublicationinphysicsjournals.)Inthesameyear,ErwinSchrödingerusedtheword"entanglement"anddeclared:"Iwouldnotcallthatonebutratherthecharacteristictraitofquantummechanics."[47]EversinceIrishphysicistJohnStewartBelltheoreticallyandexperimentallydisprovedthe"hiddenvariables"theoryofEinstein,Podolsky,andRosen,mostphysicistshaveacceptedentanglementasarealphenomenon.[48]However,thereissomeminoritydispute.[49]TheBellinequalitiesarethemostpowerfulchallengetoEinstein'sclaims. Quantumfieldtheory[edit] Mainarticle:Quantumfieldtheory Theideaofquantumfieldtheorybeganinthelate1920swithBritishphysicistPaulDirac,whenheattemptedtoquantizetheenergyoftheelectromagneticfield;justlikeinquantummechanicstheenergyofanelectroninthehydrogenatomwasquantized.Quantizationisaprocedureforconstructingaquantumtheorystartingfromaclassicaltheory. Merriam-Websterdefinesafieldinphysicsas"aregionorspaceinwhichagiveneffect(suchasmagnetism)exists".[50]Othereffectsthatmanifestthemselvesasfieldsaregravitationandstaticelectricity.[51]In2008,physicistRichardHammondwrote: Sometimeswedistinguishbetweenquantummechanics(QM)andquantumfieldtheory(QFT).QMreferstoasysteminwhichthenumberofparticlesisfixed,andthefields(suchastheelectromechanicalfield)arecontinuousclassicalentities.QFT ...goesastepfurtherandallowsforthecreationandannihilationofparticles ... Headded,however,thatquantummechanicsisoftenusedtoreferto"theentirenotionofquantumview".[52]: 108  In1931,Diracproposedtheexistenceofparticlesthatlaterbecameknownasantimatter.[53]DiracsharedtheNobelPrizeinPhysicsfor1933withSchrödinger"forthediscoveryofnewproductiveformsofatomictheory".[54] Onitsface,quantumfieldtheoryallowsinfinitenumbersofparticlesandleavesituptothetheoryitselftopredicthowmanyandwithwhichprobabilitiesornumberstheyshouldexist.Whendevelopedfurther,thetheoryoftencontradictsobservation,sothatitscreationandannihilationoperatorscanbeempiricallytieddown.[clarificationneeded]Furthermore,empiricalconservationlawssuchasthatofmass–energysuggestcertainconstraintsonthemathematicalformofthetheory,whicharemathematicallyspeakingfinicky.Thelatterfactmakesquantumfieldtheoriesdifficulttohandle,buthasalsoledtofurtherrestrictionsonadmissibleformsofthetheory;thecomplicationsarementionedbelowundertherubricofrenormalization. Quantumelectrodynamics[edit] Mainarticle:Quantumelectrodynamics Quantumelectrodynamics(QED)isthenameofthequantumtheoryoftheelectromagneticforce.UnderstandingQEDbeginswithunderstandingelectromagnetism.Electromagnetismcanbecalled"electrodynamics"becauseitisadynamicinteractionbetweenelectricalandmagneticforces.Electromagnetismbeginswiththeelectriccharge. Electricchargesarethesourcesofandcreate,electricfields.Anelectricfieldisafieldthatexertsaforceonanyparticlesthatcarryelectriccharges,atanypointinspace.Thisincludestheelectron,proton,andevenquarks,amongothers.Asaforceisexerted,electricchargesmove,acurrentflows,andamagneticfieldisproduced.Thechangingmagneticfield,inturn,causeselectriccurrent(oftenmovingelectrons).Thephysicaldescriptionofinteractingchargedparticles,electricalcurrents,electricalfields,andmagneticfieldsiscalledelectromagnetism. In1928PaulDiracproducedarelativisticquantumtheoryofelectromagnetism.Thiswastheprogenitortomodernquantumelectrodynamics,inthatithadessentialingredientsofthemoderntheory.However,theproblemofunsolvableinfinitiesdevelopedinthisrelativisticquantumtheory.Yearslater,renormalizationlargelysolvedthisproblem.Initiallyviewedasaprovisional,suspectprocedurebysomeofitsoriginators,renormalizationeventuallywasembracedasanimportantandself-consistenttoolinQEDandotherfieldsofphysics.Also,inthelate1940sFeynman'sdiagramsdepictedallpossibleinteractionsonagivenevent.Thediagramsshowedinparticularthattheelectromagneticforceistheexchangeofphotonsbetweeninteractingparticles.[55] TheLambshiftisanexampleofaquantumelectrodynamicspredictionthathasbeenexperimentallyverified.Itisaneffectwherebythequantumnatureoftheelectromagneticfieldmakestheenergylevelsinanatomoriondeviateslightlyfromwhattheywouldotherwisebe.Asaresult,spectrallinesmayshiftorsplit. Similarly,withinafreelypropagatingelectromagneticwave,thecurrentcanalsobejustanabstractdisplacementcurrent,insteadofinvolvingchargecarriers.InQED,itsfulldescriptionmakesessentialuseofshort-livedvirtualparticles.There,QEDagainvalidatesanearlier,rathermysteriousconcept. StandardModel[edit] Mainarticle:StandardModel Inthe1960sphysicistsrealizedthatQEDbrokedownatextremelyhighenergies.[citationneeded]FromthisinconsistencytheStandardModelofparticlephysicswasdiscovered,whichremediedthehigherenergybreakdownintheory.Itisanotherextendedquantumfieldtheorythatunifiestheelectromagneticandweakinteractionsintoonetheory.Thisiscalledtheelectroweaktheory. Additionally,theStandardModelcontains[citationneeded]ahighenergyunificationoftheelectroweaktheorywiththestrongforce,describedbyquantumchromodynamics.Italsopostulatesaconnectionwithgravityasyetanothergaugetheory,buttheconnectionisasof2015stillpoorlyunderstood.Thetheory'ssuccessfulpredictionoftheHiggsparticletoexplaininertialmasswasconfirmedbytheLargeHadronCollider,[56]andthustheStandardmodelisnowconsideredthebasicandmoreorlesscompletedescriptionofparticlephysicsasweknowit. Interpretations[edit] Mainarticle:Interpretationsofquantummechanics Thephysicalmeasurements,equations,andpredictionspertinenttoquantummechanicsareallconsistentandholdaveryhighlevelofconfirmation.However,thequestionofwhattheseabstractmodelssayabouttheunderlyingnatureoftherealworldhasreceivedcompetinganswers.Theseinterpretationsarewidelyvaryingandsometimessomewhatabstract.Forinstance,theCopenhageninterpretationstatesthatbeforeameasurement,statementsaboutaparticle'spropertiesarecompletelymeaningless,whileintheMany-worldsinterpretationdescribestheexistenceofamultiversemadeupofeverypossibleuniverse.[57] Applications[edit] Mainarticle:Quantummechanics:applications Applicationsofquantummechanicsincludethelaser,thetransistor,theelectronmicroscope,andmagneticresonanceimaging.Aspecialclassofquantummechanicalapplicationsisrelatedtomacroscopicquantumphenomenasuchassuperfluidheliumandsuperconductors.Thestudyofsemiconductorsledtotheinventionofthediodeandthetransistor,whichareindispensableformodernelectronics. Ineventhesimplelightswitch,quantumtunnelingisabsolutelyvital,asotherwisetheelectronsintheelectriccurrentcouldnotpenetratethepotentialbarriermadeupofalayerofoxide.FlashmemorychipsfoundinUSBdrivesalsousequantumtunneling,toerasetheirmemorycells.[58] Seealso[edit] Physicsportal Einstein'sthoughtexperiments Macroscopicquantumphenomena Philosophyofphysics Quantumcomputing Virtualparticle Listoftextbooksonclassicalandquantummechanics Notes[edit] ^Severalformulashadbeencreatedthatcoulddescribesomeoftheexperimentalmeasurementsofthermalradiation:howthewavelengthatwhichtheradiationisstrongestchangeswithtemperatureisgivenbyWien'sdisplacementlaw,theoverallpoweremittedperunitareaisgivenbytheStefan–Boltzmannlaw.ThebesttheoreticalexplanationoftheexperimentalresultswastheRayleigh–Jeanslaw,whichagreeswithexperimentalresultswellatlargewavelengths(or,equivalently,lowfrequencies),butstronglydisagreesatshortwavelengths(orhighfrequencies).Infact,atshortwavelengths,classicalphysicspredictedthatenergywillbeemittedbyahotbodyataninfiniterate.Thisresult,whichisclearlywrong,isknownastheultravioletcatastrophe. ^ThewordquantumcomesfromtheLatinwordfor"howmuch"(asdoesquantity).Somethingthatisquantized,astheenergyofPlanck'sharmonicoscillators,canonlytakespecificvalues.Forexample,inmostcountries,moneyiseffectivelyquantized,withthequantumofmoneybeingthelowest-valuecoinincirculation.Mechanicsisthebranchofsciencethatdealswiththeactionofforcesonobjects.So,quantummechanicsisthepartofmechanicsthatdealswithobjectsforwhichparticularpropertiesarequantized. ^Actually,therecanbeintensity-dependenteffects,butatintensitiesachievablewithnon-lasersources,theseeffectsareunobservable. ^Einstein'sphotoelectriceffectequationcanbederivedandexplainedwithoutrequiringtheconceptof"photons".Thatis,theelectromagneticradiationcanbetreatedasaclassicalelectromagneticwave,aslongastheelectronsinthematerialaretreatedbythelawsofquantummechanics.Theresultsarequantitativelycorrectforthermallightsources(thesun,incandescentlamps,etc)bothfortherateofelectronemissionaswellastheirangulardistribution.Formoreonthispoint,see[15] ^Theclassicalmodeloftheatomiscalledtheplanetarymodel,orsometimestheRutherfordmodel—afterErnestRutherfordwhoproposeditin1911,basedontheGeiger–Marsdengoldfoilexperiment,whichfirstdemonstratedtheexistenceofthenucleus. ^Inthiscase,theenergyoftheelectronisthesumofitskineticandpotentialenergies.Theelectronhaskineticenergybyvirtueofitsactualmotionaroundthenucleus,andpotentialenergybecauseofitselectromagneticinteractionwiththenucleus. ^Themodelcanbeeasilymodifiedtoaccountfortheemissionspectrumofanysystemconsistingofanucleusandasingleelectron(thatis,ionssuchasHe+orO7+,whichcontainonlyoneelectron)butcannotbeextendedtoanatomwithtwoelectronssuchasneutralhelium. ^ElectrondiffractionwasfirstdemonstratedthreeyearsafterdeBrogliepublishedhishypothesis.AttheUniversityofAberdeen,GeorgeThomsonpassedabeamofelectronsthroughathinmetalfilmandobserveddiffractionpatterns,aswouldbepredictedbythedeBrogliehypothesis.AtBellLabs,DavissonandGermerguidedanelectronbeamthroughacrystallinegrid.DeBrogliewasawardedtheNobelPrizeinPhysicsin1929forhishypothesis;ThomsonandDavissonsharedtheNobelPrizeforPhysicsin1937fortheirexperimentalwork. ^ForasomewhatmoresophisticatedlookathowHeisenbergtransitionedfromtheoldquantumtheoryandclassicalphysicstothenewquantummechanics,seeHeisenberg'sentrywaytomatrixmechanics. References[edit] ^"QuantumMechanics".NationalPublicRadio.Retrieved22June2016. ^Kuhn,ThomasS.TheStructureofScientificRevolutions.Fourthed.Chicago;London:TheUniversityofChicagoPress,2012.Print. ^"IntroductiontoQuantumMechanics".Socratease.Archivedfromtheoriginalon15September2017. ^Feynman,RichardP.(1988).QED :thestrangetheoryoflightandmatter(1stPrincetonpbk.,seventhprintingwithcorrections. ed.).Princeton,NJ:PrincetonUniversityPress.pp. 10.ISBN 978-0691024172. ^"Remarksconcerningthestatus&someramificationsofEHRENFEST'STHEOREM"(PDF). ^Thisresultwaspublished(inGerman)asPlanck,Max(1901)."UeberdasGesetzderEnergieverteilungimNormalspectrum".Ann.Phys.309(3):553–63.Bibcode:1901AnP...309..553P.doi:10.1002/andp.19013090310..Englishtranslation:"OntheLawofDistributionofEnergyintheNormalSpectrum".Archivedfromtheoriginalon18April2008. ^FrancisWestonSears(1958).Mechanics,WaveMotion,andHeat.Addison-Wesley.p. 537. ^"TheNobelPrizeinPhysics1918".NobelFoundation.Retrieved1August2009. ^Kragh,Helge(1December2000)."MaxPlanck:thereluctantrevolutionary".PhysicsWorld.com. ^Einstein,Albert(1905)."ÜbereinendieErzeugungundVerwandlungdesLichtesbetreffendenheuristischenGesichtspunkt"(PDF).AnnalenderPhysik.17(6):132–48.Bibcode:1905AnP...322..132E.doi:10.1002/andp.19053220607.,translatedintoEnglishasOnaHeuristicViewpointConcerningtheProductionandTransformationofLightArchived11June2009attheWaybackMachine.Theterm"photon"wasintroducedin1926. ^"RevivaloftheWaveTheoryofLightintheEarlyNineteenth-Century".www.encyclopedia.com.Retrieved16October2018. ^abcdeTaylor,J.R.;Zafiratos,C.D.;Dubson,M.A.(2004).ModernPhysicsforScientistsandEngineers.PrenticeHall.pp. 127–29.ISBN 0135897890. ^Hawking,Stephen(6November2001)[November5,2001].TheUniverseinaNutshell.Vol. 55.Impey,C.D.BantamSpectra(publishedApril2002).p. 80~.doi:10.1063/1.1480788.ISBN 978-0553802023.Archivedfromtheoriginalon21September2020.Retrieved14December2020–viaRandomHouseAudiobooks.{{citebook}}:CS1maint:dateandyear(link)AltURL ^Dicke,RobertHenry;Wittke,JamesP.(1960).IntroductiontoQuantumMechanics.Addison-WesleyPublishingCompany.p. 12.ISBN 978-0201015102. ^Lamb,WillisE.,Jr.;Scully,MarlanO."ThePhotoelectricEffectWithoutPhotons"(PDF).NTRS.NASA.gov. ^JimLucas:'WhatIsUltravioletLight?',15September2017,atlivescience.comAccessed27December2017 ^'TwoEquationsGoverningLight'sBehavior:PartTwoE=hν'atchemteam.infoAccessed27December2017 ^abTaylor,J.R.;Zafiratos,C.D.;Dubson,M.A.(2004).ModernPhysicsforScientistsandEngineers.PrenticeHall.pp. 147–48.ISBN 0135897890. ^McEvoy,J.P.;Zarate,O.(2004).IntroducingQuantumTheory.TotemBooks.pp. 70–89,[89].ISBN 1840465778. ^WorldBook.Inc(2007)."22".WorldBookEncyclopedia(Electronicreproduction).TheWorldBookencyclopedia.Vol. 22(3 ed.).Chicago,Illinois:WorldBook.p. 6.ISBN 978-0716601074.OCLC 894799866.Archivedfromtheoriginalon30January2017.Retrieved14December2020.AltURL ^Wittke,J.P;Dicke,R.H(1June1961)[1960]."11".InHolladay,W.G.(ed.).IntroductiontoQuantumMechanics(eBook).Vol. 16.Nashville,Tennessee:ADDISONWESLEYLONGMANINC(published1January1978).p. 10.doi:10.1063/1.3057610.ISBN 978-0201015102.OCLC 53473.Retrieved14December2020–viaVanderbiltUniversity. ^McEvoy,J.P.;Zarate,O.(2004).IntroducingQuantumTheory.TotemBooks.pp. 110ff.ISBN 1840465778. ^Aczel,AmirD.,Entanglement,pp.51ff.(Penguin,2003)ISBN 978-1551926476 ^McEvoy,J.P.;Zarate,O.(2004).IntroducingQuantumTheory.TotemBooks.p. 114.ISBN 1840465778. ^Zettili,Nouredine(2009).QuantumMechanics:ConceptsandApplications.JohnWileyandSons.pp. 26–27.ISBN 978-0470026786. ^Selleri,Franco(2012).Wave-ParticleDuality.SpringerScienceandBusinessMedia.p. 41.ISBN 978-1461533320. ^Podgorsak,ErvinB.(2013).CompendiumtoRadiationPhysicsforMedicalPhysicists.SpringerScienceandBusinessMedia.p. 88.ISBN 978-3642201868. ^Halliday,David;Resnick,Robert(2013).FundamentalsofPhysics,10thEd.JohnWileyandSons.p. 1272.ISBN 978-1118230619. ^Myers,RustyL.(2006).TheBasicsofPhysics.GreenwoodPublishingGroup.pp. 172.ISBN 0313328579.complementarityprinciplewave-particleduality. ^abShamos,MorrisH(1January1987).GreatExperimentsinPhysics:FirsthandAccountsfromGalileotoEinstein.CourierCorporation.p. 108. ^Merali,Zeeya(21May2015)."Quantumphysics:Whatisreallyreal?".Nature.pp. 278–80.Bibcode:2015Natur.521..278M.doi:10.1038/521278a.Retrieved7January2017. ^Eibenberger,Sandra(2013)."Matter–waveinterferenceofparticlesselectedfromamolecularlibrarywithmassesexceeding10000amu".PhysicalChemistryChemicalPhysics.15(35):14696–700.arXiv:1310.8343.Bibcode:2013PCCP...1514696E.doi:10.1039/C3CP51500A.PMID 23900710.S2CID 3944699.[I]nathree-gratinginterferometer...Weobservehigh-contrastquantumfringepatternsofmolecules...having810atomsinasingleparticle. ^McEvoy,J.P.;Zarate,O.(2004).IntroducingQuantumTheory.TotemBooks.p. 87.ISBN 1840465778. ^VanderWaerden,B.L.(1967).SourcesofQuantumMechanics.Mineola,NY:DoverPublications.pp. 261–76.Received29July1925SeeWernerHeisenberg'spaper,"Quantum-TheoreticalRe-interpretationofKinematicandMechanicalRelations"pp.261–76 ^NobelPrizeOrganization."ErwinSchrödinger–Biographical".Retrieved28March2014.Hisgreatdiscovery,Schrödinger'swaveequation,wasmadeattheendofthisepoch-duringthefirsthalfof1926. ^"SchrodingerEquation(Physics)",EncyclopædiaBritannica ^ErwinSchrödinger,"ThePresentSituationinQuantumMechanics",p.9."ThistranslationwasoriginallypublishedinProceedingsoftheAmericanPhilosophicalSociety,124,323–38,andthenappearedasSectionI.11ofPartIofQuantumTheoryandMeasurement(J.A.WheelerandW.H.Zurek,eds.,PrincetonUniversityPress,NJ1983).Thispapercanbedownloadedhere:ErwinSchrödinger."ATranslationofSchrödinger's"CatParadoxPaper"".TranslatedbyJohnD.Trimmer.Archivedfromtheoriginalon13November2010. ^Heisenberg,W.(1955).Thedevelopmentoftheinterpretationofthequantumtheory,pp. 12–29inNielsBohrandtheDevelopmentofPhysics:EssaysdedicatedtoNielsBohrontheoccasionofhisseventiethbirthday,editedbyPauli,W.withtheassistanceofRosenfeld,L.andWeisskopf,V.,Pergamon,London,p. 13:"thesinglequantumjump...is"factual"innature". ^W.Moore,Schrödinger:LifeandThought,CambridgeUniversityPress(1989),p.222.Seep.227forSchrödinger'sownwords. ^"Physicistsfinallygettoseequantumjumpwithowneyes".TheNewYorkTimes.Retrieved30November2019. ^"TheNobelPrizeinPhysics1932".NobelPrize.org. ^HeisenbergfirstpublishedhisworkontheuncertaintyprincipleintheleadingGermanphysicsjournalZeitschriftfürPhysik:Heisenberg,W.(1927)."ÜberdenanschaulichenInhaltderquantentheoretischenKinematikundMechanik".Z.Phys.43(3–4):172–98.Bibcode:1927ZPhy...43..172H.doi:10.1007/BF01397280.S2CID 122763326. ^"TheNobelPrizeinPhysics1932".NobelPrize.org. ^"Uncertaintyprinciple",EncyclopædiaBritannica ^abcPauling,Linus(1960).TheNatureoftheChemicalBond(3rd ed.).Itahca,NY:CornellUniversityPress.p. 47.ISBN 0801403332.Retrieved1March2016. ^"Orbital(chemistryandphysics)",EncyclopædiaBritannica ^E.Schrödinger,ProceedingsoftheCambridgePhilosophicalSociety,31(1935),p.555,says:"Whentwosystems,ofwhichweknowthestatesbytheirrespectiverepresentation,enterintoatemporaryphysicalinteractionduetoknownforcesbetweenthemandwhenafteratimeofmutualinfluencethesystemsseparateagain,thentheycannolongerbedescribedasbefore,viz.,byendowingeachofthemwitharepresentativeofitsown.Iwouldnotcallthatonebutratherthecharacteristictraitofquantummechanics." ^DavidKaiser,IsQuantumEntanglementReal?,TheNewYorkTimes,Nov.2014. ^JohnG.Cramer."QuantumNonlocalityandthePossibilityofSuperluminalEffects".npl.washington.edu.Archivedfromtheoriginalon29December2010. ^"Mechanics",Merriam-WebsterOnlineDictionary ^"Field",EncyclopædiaBritannica ^RichardHammond,TheUnknownUniverse,NewPageBooks,2008.ISBN 978-1601630032 ^"FeaturedPhysicists–PaulDirac1902–1984".www.physicalworld.org. ^"TheNobelPrizeinPhysics1933".NobelFoundation.Retrieved24November2007. ^"ExchangeParticles".hyperphysics.phy-astr.gsu.edu.Retrieved16October2018. ^"TenyearsofLargeHadronColliderdiscoveriesbelowSwisscountrysidearejustthestartofdecodingtheuniverse".www.thelocal.ch.5October2018.Retrieved16October2018. ^"CopenhagenInterpretation".abyss.uoregon.edu.Retrieved16October2018. ^ Durrani,Z.A.K.;Ahmed,H.(2008).VijayKumar(ed.).Nanosilicon.Elsevier.p. 345.ISBN 978-0080445281. Bibliography[edit] Bernstein,Jeremy(2005)."MaxBornandthequantumtheory".AmericanJournalofPhysics.73(11):999–1008.Bibcode:2005AmJPh..73..999B.doi:10.1119/1.2060717. Beller,Mara(2001).QuantumDialogue:TheMakingofaRevolution.UniversityofChicagoPress. Bohr,Niels(1958).AtomicPhysicsandHumanKnowledge.JohnWiley&Sons].ISBN 0486479285.OCLC 530611. deBroglie,Louis(1953).TheRevolutioninPhysics.NoondayPress.LCCN 53010401. Bronner,Patrick;Strunz,Andreas;Silberhorn,Christine;Meyn,Jan-Peter(2009)."Demonstratingquantumrandomwithsinglephotons".EuropeanJournalofPhysics.30(5):1189–1200.Bibcode:2009EJPh...30.1189B.doi:10.1088/0143-0807/30/5/026. Einstein,Albert(1934).EssaysinScience.PhilosophicalLibrary.ISBN 0486470113.LCCN 55003947. Feigl,Herbert;Brodbeck,May(1953).ReadingsinthePhilosophyofScience.Appleton-Century-Crofts.ISBN 0390304883.LCCN 53006438. Feynman,RichardP.(1949)."Space-TimeApproachtoQuantumElectrodynamics".PhysicalReview.76(6):769–89.Bibcode:1949PhRv...76..769F.doi:10.1103/PhysRev.76.769. Feynman,RichardP.(1990).QED,TheStrangeTheoryofLightandMatter.PenguinBooks.ISBN 978-0140125054. Fowler,Michael(1999).TheBohrAtom.UniversityofVirginia.[ISBN missing] Heisenberg,Werner(1958).PhysicsandPhilosophy.HarperandBrothers.ISBN 0061305499.LCCN 99010404. Lakshmibala,S.(2004)."Heisenberg,MatrixMechanicsandtheUncertaintyPrinciple".Resonance:JournalofScienceEducation.9(8):46–56.doi:10.1007/bf02837577.S2CID 29893512. Liboff,RichardL.(1992).IntroductoryQuantumMechanics(2nd ed.).Addison-WesleyPub.Co.ISBN 9780201547153.[ISBN missing] Lindsay,RobertBruce;Margenau,Henry(1957).FoundationsofPhysics.Dover.ISBN 0918024188.LCCN 57014416. McEvoy,J.P.;Zarate,Oscar(2004).IntroducingQuantumTheory.ISBN 1874166374. Nave,CarlRod(2005)."QuantumPhysics".HyperPhysics.GeorgiaStateUniversity. Peat,F.David(2002).FromCertaintytoUncertainty:TheStoryofScienceandIdeasintheTwenty-FirstCentury.JosephHenryPress. Reichenbach,Hans(1944).PhilosophicFoundationsofQuantumMechanics.UniversityofCaliforniaPress.ISBN 0486404595.LCCN a44004471. Schilpp,PaulArthur(1949).AlbertEinstein:Philosopher-Scientist.TudorPublishingCompany.LCCN 50005340. ScientificAmericanReader,1953. Sears,FrancisWeston(1949).Optics(3rd ed.).Addison-Wesley.ISBN 0195046013.LCCN 51001018. Shimony,A.(1983)."(titlenotgivenincitation)".FoundationsofQuantumMechanicsintheLightofNewTechnology(S.Kamefuchietal.,eds.).Tokyo:JapanPhysicalSociety.p. 225.;citedin:Popescu,Sandu;DanielRohrlich(1996)."ActionandPassionataDistance:AnEssayinHonorofProfessorAbnerShimony".arXiv:quant-ph/9605004. Tavel,Morton;Tavel,Judith(illustrations)(2002).Contemporaryphysicsandthelimitsofknowledge.RutgersUniversityPress.ISBN 978-0813530772. VanVleck,J.H.,1928,"TheCorrespondencePrincipleintheStatisticalInterpretationofQuantumMechanics",Proc.Natl.Acad.Sci.14:179. Westmoreland;BenjaminSchumacher(1998)."QuantumEntanglementandtheNonexistenceofSuperluminalSignals".arXiv:quant-ph/9801014. Wheeler,JohnArchibald;Feynman,RichardP.(1949)."ClassicalElectrodynamicsinTermsofDirectInterparticleAction"(PDF).ReviewsofModernPhysics.21(3):425–33.Bibcode:1949RvMP...21..425W.doi:10.1103/RevModPhys.21.425. Wieman,Carl;Perkins,Katherine(2005)."TransformingPhysicsEducation".PhysicsToday.58(11):36.Bibcode:2005PhT....58k..36W.doi:10.1063/1.2155756. Furtherreading[edit] Thissectionneedstobeupdated.Pleasehelpupdatethisarticletoreflectrecenteventsornewlyavailableinformation.(September2021) Thefollowingtitles,allbyworkingphysicists,attempttocommunicatequantumtheorytolaypeople,usingaminimumoftechnicalapparatus. JimAl-Khalili(2003).Quantum:AGuideforthePerplexed.Weidenfeld&Nicolson.ISBN 978-1780225340. Chester,Marvin(1987).PrimerofQuantumMechanics.JohnWiley.ISBN 0486428788. BrianCoxandJeffForshaw(2011)TheQuantumUniverse.AllenLane.ISBN 978-1846144325. RichardFeynman(1985).QED:TheStrangeTheoryofLightandMatter.PrincetonUniversityPress.ISBN 0691083886. Ford,Kenneth(2005).TheQuantumWorld.HarvardUniv.Press.Includeselementaryparticlephysics. Ghirardi,GianCarlo(2004).SneakingaLookatGod'sCards,GeraldMalsbary,trans.PrincetonUniv.Press.Themosttechnicaloftheworkscitedhere.Passagesusingalgebra,trigonometry,andbra–ketnotationcanbepassedoveronafirstreading. TonyHeyandWalters,Patrick(2003).TheNewQuantumUniverse.CambridgeUniv.Press.Includesmuchaboutthetechnologiesquantumtheoryhasmadepossible.ISBN 978-0521564571. VladimirG.Ivancevic,TijanaT.Ivancevic(2008).Quantumleap:fromDiracandFeynman,Acrosstheuniverse,tohumanbodyandmind.WorldScientificPublishingCompany.Providesanintuitiveintroductioninnon-mathematicaltermsandanintroductionincomparativelybasicmathematicalterms.ISBN 978-9812819277. J.P.McEvoyandOscarZarate(2004).IntroducingQuantumTheory.TotemBooks.ISBN 1840465778' N.DavidMermin(1990)."Spookyactionsatadistance:mysteriesoftheQT"inhisBoojumsallthewaythrough.CambridgeUniv.Press:110–76.Theauthorisararephysicistwhotriestocommunicatetophilosophersandhumanists.ISBN 978-0521388801. RolandOmnès(1999).UnderstandingQuantumMechanics.PrincetonUniv.Press.ISBN 978-0691004358. VictorStenger(2000).TimelessReality:Symmetry,Simplicity,andMultipleUniverses.BuffaloNY:PrometheusBooks.Chpts.5–8.ISBN 978-1573928595. MartinusVeltman(2003).FactsandMysteriesinElementaryParticlePhysics.WorldScientificPublishingCompany.ISBN 978-9812381491. Externallinks[edit] TheWikibookQuantumMechanicshasapageonthetopicof:IntroductiontoQuantumMechanics "MicroscopicWorld –IntroductiontoQuantumMechanics".byTakada,Kenjiro,EmeritusprofessoratKyushuUniversity TheQuantumExchange(tutorialsandopen-sourcelearningsoftware). AtomsandthePeriodicTable Singleanddoubleslitinterference Time-EvolutionofaWavepacketinaSquareWellAnanimateddemonstrationofawavepacketdispersionovertime. Carroll,SeanM."QuantumMechanics(anembarrassment)".SixtySymbols.BradyHaranfortheUniversityofNottingham. vteQuantummechanicsBackground Introduction History Timeline Classicalmechanics Oldquantumtheory Glossary Fundamentals Bornrule Bra–ketnotation Complementarity Densitymatrix Energylevel Groundstate Excitedstate Degeneratelevels Zero-pointenergy Entanglement Hamiltonian Interference Decoherence Measurement Nonlocality Quantumstate Superposition Tunnelling Scatteringtheory Symmetryinquantummechanics Uncertainty Wavefunction Collapse Wave–particleduality Formulations Formulations Heisenberg Interaction Matrixmechanics Schrödinger Pathintegralformulation Phasespace Equations Dirac Klein–Gordon Pauli Rydberg Schrödinger Interpretations Interpretations Bayesian Consistenthistories Copenhagen deBroglie–Bohm Ensemble Hidden-variable Local Many-worlds Objectivecollapse Quantumlogic Relational Transactional VonNeumann-Wigner Experiments Bell'sinequality Davisson–Germer Delayed-choicequantumeraser Double-slit Franck–Hertz Mach–Zehnderinterferometer Elitzur–Vaidman Popper Quantumeraser Stern–Gerlach Wheeler'sdelayedchoice Science Quantumbiology Quantumchemistry Quantumchaos Quantumcosmology Quantumdifferentialcalculus Quantumdynamics Quantumgeometry Quantummeasurementproblem Quantumstochasticcalculus Quantumspacetime Technology Quantumalgorithms Quantumamplifier Quantumbus Quantumcellularautomata Quantumfiniteautomata Quantumchannel Quantumcircuit Quantumcomplexitytheory Quantumcomputing Timeline Quantumcryptography Quantumelectronics Quantumerrorcorrection Quantumimaging Quantumimageprocessing Quantuminformation Quantumkeydistribution Quantumlogic Quantumlogicgates Quantummachine Quantummachinelearning Quantummetamaterial Quantummetrology Quantumnetwork Quantumneuralnetwork Quantumoptics Quantumprogramming Quantumsensing Quantumsimulator Quantumteleportation Extensions Casimireffect Quantumstatisticalmechanics Quantumfieldtheory History Quantumgravity Relativisticquantummechanics Related Schrödinger'scat inpopularculture Quantummysticism Category Physicsportal Commons vteIntroductorysciencearticles Introductiontoangularmomentum Introductiontoeigenstates Introductiontoelectromagnetism Introductiontoentropy Introductiontoevolution Introductiontogaugetheory Introductiontogeneralrelativity Introductiontogenetics IntroductiontoM-theory Introductiontothemathematicsofgeneralrelativity Introductiontoquantummechanics Introductiontosystolicgeometry Introductiontotheheaviestelements Introductiontoviruses vteChemicalbondingtheory Atomicorbital Quantummechanics Introductiontoquantummechanics TypesofbondsBysymmetry Sigma(σ) Pi(π) Delta(δ) Phi(φ) Bymultiplicity 1(single) 2(double) 3(triple) 4(quadruple) 5(quintuple) 6(sextuple) Byspin Triplet Singlet Exchange-coupled ValencebondtheoryConcepts Hybridorbital Resonance Lewisstructure Constituentunits Covalentbond Lonepair MolecularorbitaltheoryConcepts Molecularorbital LCAO MOdiagram Constituentunits BondingMO Non-bondingMO AntibondingMO vteElectronconfiguration Electronshell Atomicorbital Quantummechanics Introductiontoquantummechanics Quantumnumbers Principalquantumnumber(n) Azimuthalquantumnumber(ℓ) Magneticquantumnumber(m) Spinquantumnumber(s) Ground-stateconfigurations Periodictable(electronconfigurations) Electronconfigurationsoftheelements(datapage) Electronfilling Pauliexclusionprinciple Hund'srule Aufbauprinciple Electronpairing Electronpair Unpairedelectron Bondingparticipation Valenceelectron Coreelectron Electroncountingrules Octetrule 18-electronrule Retrievedfrom"https://en.wikipedia.org/w/index.php?title=Introduction_to_quantum_mechanics&oldid=1103301397" Categories:QuantummechanicsHiddencategories:WebarchivetemplatewaybacklinksCS1maint:dateandyearArticleswithshortdescriptionShortdescriptionisdifferentfromWikidataIntroductoryarticlesUsedmydatesfromJuly2020AllarticleswithunsourcedstatementsArticleswithunsourcedstatementsfromDecember2017WikipediaarticlesneedingclarificationfromJanuary2015WikipediaarticlesneedingclarificationfromNovember2019WikipediaarticlesneedingclarificationfromJune2018ArticleswithunsourcedstatementsfromJune2018PageswithmissingISBNsWikipediaarticlesinneedofupdatingfromSeptember2021AllWikipediaarticlesinneedofupdatingArticlescontainingvideoclips Navigationmenu Personaltools NotloggedinTalkContributionsCreateaccountLogin Namespaces ArticleTalk English Views ReadEditViewhistory More Search Navigation MainpageContentsCurrenteventsRandomarticleAboutWikipediaContactusDonate Contribute HelpLearntoeditCommunityportalRecentchangesUploadfile Tools WhatlinkshereRelatedchangesUploadfileSpecialpagesPermanentlinkPageinformationCitethispageWikidataitem Print/export DownloadasPDFPrintableversion Languages العربيةবাংলাFrançais한국어हिन्दीHrvatskiBahasaIndonesiaÍslenskaಕನ್ನಡमराठीBahasaMelayuਪੰਜਾਬੀپنجابیPortuguêsRomânăSlovenčinaSlovenščinaTürkçe中文 Editlinks