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1.3. CLASSICALMECHANICSINANUTSHELL 15 1.3 Classical Mechanicsin a Nutshell AllthemachineryoftheLeastActionPrinciple,theLagrangia ...
16 CHAPTER1. THECLASSICALSTATE TheHamiltonian H[{qi,pi}]≡ ∑ n pnq ̇n−L[{qi,q ̇i}] (1.40) UsuallytheHamiltonianhastheform H[p,q ...
1.3. CLASSICALMECHANICSINANUTSHELL 17 Example: ThePlanePendulum Ourpendulumisamassmattheendofaweightlessrigidrodoflengthl,which ...
18 CHAPTER1. THECLASSICALSTATE 6.Hamilton’sequations θ ̇ = ∂H ∂p = p ml^2 p ̇ = − ∂H ∂θ =−mglsin(θ) (1.51) whichareeasilyseentob ...
1.4. THECLASSICALSTATE 19 possible,bysolvingtheequationsofmotionexactly).Notethatthecoordinates{qa} alonearenotenoughtospecifyth ...
20 CHAPTER1. THECLASSICALSTATE ...
Chapter 2 Origins of Quantum Mechanics Wheredocorrectideascomefrom?Dotheydropfromthesky?No!Aretheyinnate inthemind?No!Theycomefr ...
22 CHAPTER2. ORIGINSOFQUANTUMMECHANICS Quantummechanicsemergedasanattempttoexplainthesephenomenaand,as inthebible,thestorybegins ...
2.1. BLACK-BODYRADIATION 23 Theenergydensityofradiationinthebox,asafunctionoffrequency,iseasily workedoutusingtheequipartitionpr ...
24 CHAPTER2. ORIGINSOFQUANTUMMECHANICS (otherwiseitsmass,accordingtospecialrelativity,wouldbeinfinite),sothisresult cannotpossib ...
2.2. THEPHOTOELECTRICEFFECT 25 whatissurprisingaboutitisthattheenergyoftheemittedelectronsisindependent oftheintensityoftheincid ...
26 CHAPTER2. ORIGINSOFQUANTUMMECHANICS Sincetherecanonlybeanintegernumberofphotonsnatanygivenfrequency,each ofenergy hf, theener ...
2.4. THEHEISENBERGMICROSCOPE 27 while,fromconservationofenergy,wehave E 1 +mc^2 =E 2 + √ p^2 ec^2 +m^2 c^4 (2.18) InthisformulaE ...
28 CHAPTER2. ORIGINSOFQUANTUMMECHANICS Thisimpliesthatthemoreaccuratelywe measuretheparticle’sposition(∆xvery small),themoreunce ...
2.5. THEBOHRATOM 29 ofaphysicalstate. Ofcourse,itcouldbethattherelation(2.28)issimplyapracti- callimitonmeasurement;aparticlemig ...
30 CHAPTER2. ORIGINSOFQUANTUMMECHANICS passedthroughaprismordiffractiongrating,itissplitintoarelativelysmallnumber ofbrightlines ...
2.5. THEBOHRATOM 31 Bohrto guessthat,ifelectron angularmomentumcan onlychangebyunitsof ̄h, thenthetotalangularmomentumoftheelect ...
32 CHAPTER2. ORIGINSOFQUANTUMMECHANICS Thisfinalexpressionnotonlymatchestheformoftheempiricalequationforspectral lines,butanexpl ...
2.6. DEBROGLIEWAVES 33 atom.Nowtheconditionforastandingwaveon,e.g.,acircularstring,issimplythat anintegralnumberofwavelengthsfit ...
34 CHAPTER2. ORIGINSOFQUANTUMMECHANICS ...
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