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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!

Quantum Mechanics: The Hydrogen Atom 12th April 2008 I. The Hydrogen Atom In this next section, we will tie together the elements of the last several sections to arrive at a complete description of the hydrogen atom. This will culminate in the de nition of the hydrogen-atom orbitals and associated energies.

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Transcription of Quantum Mechanics: The Hydrogen Atom

1 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!

2 +V(r; ; )# (r; ; ) =E (r; ; )Now,forthehydrogenatom,withoneelectronf oundin "orbits"(notethequotes!)aroundthenucleus of charge+1,we canincludeanelectrostaticpotentialwhich is essentiallytheCoulomb potentialbetweena positive andnegative charge:V(jrj) =V(r) = Ze24 orIt is important to notethattheCoulomb potentialas we havewrittenit hereis simplya functionof themagnitudeof theposi-tionvectorbetweenthe2 masses( ,betweentheelectronand1nucleus).Thereis noangulardependence!.Recall:^L2= h2 1r2sin @@ sin @@ +1r2sin @2@ 2!is thetotalangularmomentumsquaredoperator(f unctionof and only!).Thus,we canrewritetheSchrodingerequationas: h2 @@r r2@@r + 2 r2[V(r) E] (r; ; ) +^L2 (r; ; ) = 0 ThisdemonstratesthattheHamiltonianis separablesincethetermsinbracketsarefunct ionsofronly, andtheangularmomentumoperatorisonlya functionof and . Thus,thewavefunctioncanbe writtenin aformthatlendsto separationof theangularmomentumoperator: (r; ; ) =R(r)Yml( ; )Separation of V ariables^L2 Yml( ; ) = h2l(l+ 1)Yml( ; )l= 0;1;2;:::Accountingforseparationof variablesandtheangularmomentumresuls,the Schrodingerequationis transformedinto theRadialequationfortheHydrogenatom: h22 r2 ddr r2dR(r)dr +" h2l(l+ 1)2 r2V(r) E#R(r) = 0 Thesolutionsof theradialequationaretheHydrogenatomradia lwave-functions,R(r).

3 II. SolutionsandEnergiesThegeneralsolutionso f theradialequationareproductsof anexponentialanda (energies)are:E= Z2e28 oaon2= Z e48 2oh2n2n= 1;2;3;::Theconstantaois knownas theBohrRadius:2ao= 2oh2 e2 TheRadialeigenfunctionsare:Rnl(r) = "(n l 1)!2n[(n+l)!]3#12 2 Znao l+32rle ZrnaoL2l+1n+l 2 Zrnao TheL2l+1n+l 2 Zrnao aretheassociatedLaguerrefunctions. Thoseforn= 1andn= 2 areshown:n= 1;l= 0;L11= 1n= 2;l= 0;L12= 2! 2 Zrao n= 2;l= 1;L33= 3!How to normalize:SphericalHarmonics:Z2 0d Z 0d sin Ym l( ; )Yml( ; ) = 1 RadialWavefunctions:Z10dr r2R nl(r)Rnl(r) = 1 Thetotalhydrogenatomwavefunctionsare: (r; ; ) =RnlYml( ; )Table1. NomenclatureandRangesof H-AtomquantumnumbersNameSymbolAllowedVal uesprinciplequantumnumbern1,2,3,..angula rmomentumquantumnumberl0,1,2,..,n-1magne ticquantumnumberm0, 1; 2; 3;:::; l3 ENERGIES:DEPENDon"n"E= Z2e28 :DEPENDon "l"jLj= hpl(l+ 1)Lz component:DEPENDon"m"Lz=m hTotalH atomwavefunctionsarenormalizedandorthogo nal:Z2 0d Z 0d sin Z10drr2 nlm(r; ; ) n0l0m0(r; ; ) = nn0 ll0 mm0 LowesttotalHydrogenatomwavefunctions:n=1 andn=2( de ne Zrao)Table2.

4 HydrogenAtomWavefunctionsnlm nlmOrbitalName100 100=1p Zao 32e 1s200 200=1p32 Zao 32(2 )e 2 2s10 210=1p32 Zao 32 e 2cos 2pz1 1 21 1=1p64 Zao 32 e 2sin e i Transformingto realfunctionsvianormalizedlinearcombinat ions1 1 2px=1p32 Zao 32 e 2sin cos 2px=1p2( 21+1+ 21 1)1 1 2py=1p32 Zao 32 e 2sin cos 2py=1p2( 21+1 21 1)Thus,we have cometo thepoint wherewe canconnectwhatwealreadyknow frompreviousanalysis:4 Table3. Quantumnumbersn= 1;2;3;:::l= 0;1;2;:::;n 1m= 0; 1; 2;:::; lOrbital1001s2002s2102pz21+2px21-2pyWhat arethedegeneraciesof theHydrogenatomenergylevels?Recalltheyar edependent on theprinciplequantumnumber Spectroscopy of theHydrogenAtomTransitionsbetweentheener gystates(levels)of individualatomsgive riseto usedas analyticaltoolsto assesscompositionof instance,ourknowledgeof theatomiccompositionof thesunwas in partaidedby consideringthespectraof theHydrogenatom,earlyscientistsobservedt hattheemissionspectra(generatedby excitinghydrogenatomsfromthegroundto excitedstates),gave riseto speci clines; thequantummechanicalnatureof thehydrogenatomhelpsus linesin thehydrogenspectrum,namedafterthescien-t istswhoobservedandcharacterizedthem,canb e relatedto theenergiesassociatedwithtranssitionsfro mthevariousenergylev-elsof ,simpleenoughas it is,turnsoutto (andthusenergyviaE=h ) associatedwitha transitionfroma staten1to anothertatenis givenby:1 =RRydberg 1n21 1n2 n1= 2;Balmer Seriesn1= 1.

5 Lyman Seriesn1= 3;PaschenSeries5


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