医学物理学9.docx
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医学物理学9.docx
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医学物理学9
Chapter9
IntroductiontoQuantumMechanics
Inthepreviouschapters,wehavelearnedfundamentallawsofmechanics,fluiddynamics,vibrationsandwaves,specialrelativity,surfacephenomenaofliquidandOpticsthatarepartsofclassicalphysicstheories.Intheabovecourses,thephysicalquantitiesusedarecontinuous,suchasmomentum,angularmomentumandenergies.Thesetheoriesarevalidinlargescales,butnotproperiftheyareemployedinsmallscalelikewithinatoms.Inthischapter,wewilldiscussanewandimportanttheory,calledQuantumMechanics,whichisvalidinaverysmallregionbutitslargescalederivationcouldalsogiveclassicalresults.
Itisknownthatquantumtheoryisaveryimportanttheorynotonlyinphysics,butalsoinmanybranchesofscience,suchaslifescienceandmedicalscience.
Theconceptofquantumtheoryhasgreatdifferencewithclassicalphysics.Evenwecansay,someconceptsareoppositeandcontradicted(continuousanddiscreteforexample)!
Asastartingpointtoquantumtheory,theparticle’snatureoflightisusuallyexplainedindetail.Thisisbecausethequantumtheorystemmed(originated)fromthestudyofthispart.
§9.1BlackbodyradiationandPlanckhypothesis
9.1.1Historicalrecall:
studiesonlight
•Withoutquestion,geometricopticsistheearliesttheorytodescribethephenomenonoflightanditisstillusedtoday.Asyouknow,thereareseveralbasiclawsandequationsingeometricopticsandtheyarestillusefultodayinexplainingtheopticalinstrument.Youmightnoticethatlightpathsareallstraightlinesingeometricoptics.
•Intheearlyof18thcentury,basedontheexistingknowledgeoflightatthattime,Newtonproposedhiscorpusculartheoryoflighttoexplainthelighttravelsalongastraightlineinthespaceoruniformmedium.Insuchatheory,lightisregardedasastreamofparticlesthatsetupdisturbancesinthe‘aether’(whichisrecognizedasthemediumoflightpropagation)ofspace.Newton’scorpuscular(微粒的)theoryisthefirstonetotrytoexplainthenatureoflightbutthereexistmanyotherinterestingandbeautifuleffectsthatcannotbeexplainedbyNewton’scorpusculartheorysuchaslightbendingaroundcorners.Thereforeitisnotacorrecttheory.
•ItisknownthattheGeometricopticsisjustaphenomenon,macroscopicandapplicationstudyonlight.Thatlighttravelsinastraightlineinspaceormediumisjustaphysicalmodelusedtostudylightingeometricopticsanditisthesuperficialphenomenonoflightandithasnothingtodowiththepropertyoflight.
•Thewavenatureoflightisbasedontheexperimentsofinterferenceanddiffractionoflight.SuchaphenomenoncanbeexplainedbyHuygen’sprincipleandthesephenomenaputNewton’scorpusculartheoryoflightintohistory.
•ThecompletewavetheoryoflightwasfinallygivenbyMaxwell(JamesClerk)whoshowedthatlightformedthepartofelectromagneticwaves.ThefamousexperimentstoshowlighthaswavepropertyareThomasYang’sdouble-slitinterferenceexperimentandFraunhoferSingleslitdiffractionexperiment.In1801,ThomasYoungdidadouble-slitinterferenceexperimentoflightandshowedthatawavetheorywasessentialtointerpretthistypeofphenomenon.Andfromthen,Newton’scorpusculartheoryoflighthasbeenconsideredasawrongtheoryfortheexplanationofthenatureoflight.AnotherfamousexperimentwasdonebyFraunhoferSingle-slitdiffraction.Itshouldbealsoexplainedbywavetheoryoflight.
•Inthemiddleof19thcentury,lightsarerecognizedaspartofelectromagneticspectrumanditsspaceandtimedependencefollowsthelawsgiveninMaxwellequationswhichwereconsideredasacompletetheoryforelectromagneticwaves.Therefore,lightbelongstoelectromagneticwavesanditdoeshavethewaveproperties.
•MostphysiciststhoughtPhysicsskyscraperhadbeencompletelybuiltandsomeminorproblemslikeblackbodyradiationandlightspeedwouldbesoonfoundoutusingexistingphysicstheory.However,Geniusphysiciststhoughtthattheblackbodyradiationandlightspeedwerethetwopatchesofcloudsinphysicssky.Infact,the“minor”problemshadshakenthe“skyscraper”ofclassicalphysics.
•Inordertoexplainthequantumnatureoflight,wehavetounderstandhowthefollowingthreeproblemstobesolved:
blackbodyradiation,photoelectriceffectandComptonEffect.Amongthethreeproblems,blackbodyradiationisthemostimportantone.
9.1.2Basicconceptsofblackbodyradiation
Typesofheatenergytransmissionareconduction,convectionandradiation.
Conductionistransferofheatenergybymolecularvibrationsnotbyactualmotionofmaterial.Forexample,ifyouholdoneendofanironrod(铁杆)andtheotherendoftherodisputonaflame,youwillfeelhotsometimelater.Wecansaythattheheatenergyreachesyourhandbyheatconduction.
Convectionistransferofheatbyactualmotionofmaterial.Thehot-airfurnace,thehot-waterheatingsystem,andtheflowofbloodinthebodyareexamples.
RadiationTheheatreachingtheearthfromthesuncannotbetransferredeitherbyconductionorconvectionsincethespacebetweentheearthandthesunhasnomaterialmedium.Theenergyiscarriedbyelectromagneticwavesthatdonotrequireamaterialmediumforpropagation.Thekindofheattransferiscalledthermalradiation.Blackbodyradiationproblemwasfoundintheresearchofthermalradiation.
Blackbodyisdefinedasthebodythatcanabsorballenergiesthatfallonit.Thisislikeablackhole.Nolightsormaterialcangetawayfromitaslongasitistrapped.Alargecavitywithasmallholeonitswallcanbetakenasablackbody(cf.Fig.12-1themodelofblackbodyinyourtextbook).
Blackbodyradiation:
Anyradiationthatenterstheholeisabsorbedintheinteriorofthecavity,andtheradiationemittedfromtheholeiscalledblackbodyradiation.
9.1.3Twosuccessfullaws
1.StefanandBoltzmann’slaw:
itisfoundthattheradiationenergyisproportionaltothefourthpoweroftheassociatedtemperature.
1.
(9.1)
whereM(T)isradiationenergyandactuallyitistheareaundereachcurve,σiscalledStefan’sconstantdeterminedbyexperimentandTisabsolutetemperature.
2.
Wien’sdisplacementlaw:
thepeakofthecurveshiftstowardslongerwavelengthasthetemperaturefalls.(cf..Fig.12.2thespectrumcurvesofblackbodyradiationinyourtextbook)
2.
(9.2)
isthepeakvalueofthecurveM(T)andbiscalledWein’sconstant.Thislawisquiteusefulwhilemeasuringthetemperatureofablackbodywithaveryhightemperature.Youcanseetheexampleforhowtomeasurethetemperatureonthesurfaceofthesun.
•Theabovelawsdescribestheblackbodyradiationverywell.
9.1.4Problemsexistinblackbodyradiationandthreeformulas
•TheproblemexistingintherelationisbetweentheradiationpowerMλ(T)andthewavelengthλ.
•Blackbodyradiationhasnothingtodowithboththematerialusedintheblackbodyconcavewallandtheshapeoftheconcavewall.
•Twotypicalblackbodyradiationformulas:
oneisgivenbyRayleighandJeansandtheotherisgivenbyWein.
1.RayleighandJeans’formula:
In1890,RayleighandJeansobtainedaformulausingtheclassicalelectromagnetic(Maxwell)theoryandtheclassicalequipartitiontheoremofenergy(能均分定理)inthermotics(热学).Theformulaisgivenby
(9.3)
where
isradiationpower,C1isaconstantnumbertobedetermined,isthewavelengthofblackbodyradiation,Tistheabsolutetemperature.Rayleigh-Jeansformulawascorrectforverylongwavelengthinthefarinfraredbuthopelesslywronginthevisiblelightandultravioletregion.Maxwell’selectromagnetictheoryandthermodynamicsareknownascorrecttheory.Thefailureinexplainingblackbodyradiationpuzzledphysicists!
ItwasregardedasultravioletCatastrophe(disaster).
2.Wein’sformula:
Lateronin1896,Weinderiveanotherimportantformulausingthermodynamics.
(9.4)
whereC2andC3areconstantstobedetermined.Unfortunately,thisformulaisonlyvalidintheregionofshortwavelengths.
3.Plank’sempiricalformula:
In1900,afterstudyingtheabovetwoformulascarefully,Planckproposed(提出)anempiricalformula
Fig.9.1Blackbodyradiation
(9.5)
wherecislightvelocity,kisBoltzmen’sconstant,eisthebaseofnaturallogarithms.
Itissurprisingthattheexperienceformulacandescribethecurveofblackbodyradiationexactlyforallwavelengths.
9.1.5DerivationsfromPlanck’sformula
1.RayleighandJeans’formula;
Forverylargewavelength,theRayleigh-Jeans’formulacanbeobtainedfromPlanck’sformula;
2.Wein’sformula;
Forsmallerwavelengthofblackbodyradiation,theWein’sformulacanbeachievedalsofromPlanck’sexperienceformula;
3.StefanandBoltzmann’slaw;
IntegratingPlanck’sformulawithrespecttowavelength,theStefanandBoltzmann’slawcanbeobtainedaswell.
4.Wien’sdisplacementlaw.
Finally,accordingtothebasicmathematicaltheoryanddifferentiatingthePlanck’sformulawithrespecttowavelength,Wien’sdisplacementlawcanalsoderived!
9.1.6Planck’sHypotheses
1.Themoleculesandatomscomposingtheblackbodyconcavecanberegardedasthelinearharmonicoscillatorwithelectricalcharge;
2.Theoscillatorscanonlybeinaspecialenergystate.Alltheseenergiesmustbetheintegermultiplesofasmallestenergy(0=hν).ThereforetheenergiesoftheoscillatorsareE=nhνwithn=1,2,3,…
3.derivationofPlanck’sfumula(omitted)
4.hνwasnamedphotonbyEinsteinand
(9.6)
iscalledPlanck-Einsteinquantizationlaw.hiscalledPlanckconstant.
(9.7)
Therel
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