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Quantum Mechanics: The Hydrogen Atom

QuantumMechanics:TheHydrogenAtom12thApri l2008I. TheHydrogenAtomIn thisnextsection,we willtietogethertheelements of thelastseveralsectionsto arrive at a completedescriptionof thede nitionof ,taken as a completebasis,we willbe abletoconstructapproximationsto morecomplexwave ,theworkof thelastfewlectureshasfundamentallybeenam iedat establishinga foundationformorecomplexproblemsin termsofexactsolutionsforsmaller, TheRadialFunctionWe willstartby reiteratingtheSchrodingerequationin 3 Dsphericalcoordi-natesas (referto any standardtextto getthetransformationfromCartesianto sphericalcoordinatereferencesystems).Her e,we have notplacedthecon-straint of a constant distancesepartingthemassesof therigidrotor(referto lastlecture);furthermore,we willkeepin theformulationthepotentialV(r; ; ) ,in sphericalpolarcoordinates,^H(r; ; ) (r; ; ) =E (r; ; )becomes:" h22 1r2@@r r2@@r +1r2sin @@ sin @@ +1r2sin2 @2@ 2!

ANGULARMOMENTUM:DEPENDon"l" jLj = h p l(l+1) Lz component: DEPENDon"m" Lz = m h Total H atom wavefunctions are normalized and orthogonal: Z2ˇ 0 d˚ Zˇ 0 d sin Z1 0 drr2 nlm(r; ;˚) n 0 l0m0(r; ;˚) = nn0 l 0 mm0 LowesttotalHydrogen atom wavefunctions: n=1 andn=2 ( de ne ˙ Zr ao) Table 2. Hydrogen Atom Wavefunctions n l m nlm Orbital Name 1 0 ...

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