Bound States in 3 DimensionsFrom 1-d to 3-d3-d HamiltonianBass-AckwardsFor a particle of mass, m,ContinuingAppropriate Choice of Separation ConstantRadial SolutionAngular SolutionAnother fortuitous choice!And Voila!The Azimuthal DependenceBack to P(q)Some cleanin’ upAnd nowFinally,Properties of Spherical HarmonicsConnection Between Spherical Harmonics and Angular MomentumAnd thusRecallBack SubstitutingUsing these, we can find new relationships for the various components of angular momentumThe Importance of these ResultsThe Rigid RotorThe rigid rotor HamiltonianSolving for the Energy eigenvaluesDipole Selection RulesFor most atomic and molecular systems, Vi looks likeSpherical Coordinates AgainSkipping to the resultSlide 31Slide 32Slide 33Slide 34Radial SolutionsThe Hollow SphereThm: {(l+1)/x - d/dx}Ul=Ul+1Proof cont’dConclusionsRaisin’ itThe SolutionEnergy Levels of Hollow SphereSlide 43Normalizin’What if V=V1 when r<R and V=V2 when r>RSpherical Hankel FunctionsThe Hydrogen AtomChange of VariablesRe-writing the SEAs r goes to ∞As r goes to 0The Hydrogen Atom Radial Wave functionOur friend, the confluent hypergeometric equationIn our case,The point of the matter is to solve for E, remember?An asideSlide 57Finally!Some Hydrogen WavefunctionsIf n=1 and l=m=0, then gnd stateSlide 61Slide 62Slide 63Slide 64The Isotropic Harmonic OscillatorDefining k and lambdaAs r goes to infinitySlide 68Our main bud, the confluent hypergeometric equationSlide 70Since the confluent hypergeometric equation looks likeThe W functions must converge just like the L functionsBut we know s=E/h-bar/omegaFinal Wave Function1Bound States in 3 Dimensions2From 1-d to 3-d22222222222ˆˆˆ)(2RecallzyxpppthereforeizzyyxxipsozipandyipandxipandxVmpHwheretiHzyxx33-d Hamiltonian)(222rVmH4Bass-Ackwardszyxr22222222sin1sinsin11cossinsincossinrrrrrrrxryrx5For a particle of mass, m,  0),()()(2),(sin1cot)(),()(),(sin1cot)(),()(),()(0)(22222223222223222rYrUrVEmYrrUrYrUYrrUrYrUrYrULetrVEm22222sin1cotQLet6Continuing    ),(),()(2)()(),(),()()()(2)(0)()(2),(),()()(),(0),()()(2),()(),()(222222223YQYrrVEmrUrrUYQYrrUrUrVEmrUrUrVEmYQYrrUrUYrbymultiplyNowrYrUrVEmQYrrUrYrUOnly on rOnly on  and 7Appropriate Choice of Separation Constant         0)(12)(2)()(1)()(2)(1Constant1)(2)()(1Constant22222222rUllrmrVEmrUrrUl-lrUrVEmrUl-ll-lrrVEmrUrrUl-l1-dSch.Eq.Called centrifugal potential  22222222222222122112mrLILImrIILyClassicallrmLrmllllrm8Radial SolutionWhat happens next depends on V(r).We will leave this part of the Sch. Equation alone and concentrate on the angular part.9Angular Solution),()1(),()1(),(),(YllQYllYQY22222sin1cotQLet),()1(),(sin1),(cot),(22222YllYYY10Another fortuitous choice! Let Y(,)=P()())()()1()(sin)()(cot)()()()()()1()(sin)()(cot)()()(222222PllPPPPllPPPMultiply by)()(sin2P11And Voila!2222sin)1(cossinsinsin)1(cossinsinllPPPPllPPPPLet the separation constant = -m212The Azimuthal DependenceimimDeCesomm)(0223,2,1,012msoeim•  is azimuth angle and goes from 0 to 2 .• Demand that (+2)=()•Therefore•m is quantized•m carries the plus/minus sign so we write a general expressionimimeCCdCdandCe21)(211211)()()(22022013Back to P() 0sin)1(cot0sin)1(cossinsin0sin)1(cossinsinsin)1(cossinsinsin)1(cossinsin22222222222222PmllPPPmllPPmllPPPPllmPPPPmButllPPPP14Some cleanin’ up      21222222212212222121cot1111sincosxxdxdxdxdxdddxdxdxdxdddxdxdxddxdddxddxLet15And now      01)1(2101)1(1110sin)1(cot222222221221222222PxmlldxdPxdxPdxPxmlldxdPxxxdxdPxdxPdxPmllPPWhen m=0 then called Legendre EquationWhen m<>0 then called Associated Legendre EquationSolutions called Legendre Functions16Finally,Y()=P()()Called Spherical HarmonicsExplicit Form       2cos!2!121,2immlmmmlePmlmllYPhase factor does not affect normalizationWhich is chosen to agree with Condon & Shortley, Blatt & Weisskopf, Particle Data Group17Properties of Spherical Harmonics    mllmlmlmlmlmmlYYandddYYandYY1cos,)1(,2011**18Connection Between Spherical Harmonics and Angular