ContentsHydrogenatom1.Angular & radial component2.Variational principle3.Basisfunctions4.Secularequations5.Spin
Contents 7 Hydrogen atom 1. Angular & radial component 2. Variational principle 3. Basis functions 4. Secular equations 5. Spin
x=rsingcos1Hydrogenatomy=rsinesingz=rcosodsin drdd? Simplest chemical “thing"- Represented in spherical coordinates-Massive nucleus at origin00<x,Y.z<0<r<8-Mapmotionofelectron0<<0<2元-HatomSchrodingerequationv(r)+V(r)v(r)=Ev(r)2mee1Coulombpotentialdecaysslowlywithdistance.V(r):4元.re282r281ab1aonb1sineEu2μ2OrOrr2 sin0 00004元E0r2 sin? 0g2Separationofvariables-andpdon'tappearanywhereotherthantheLaplacianthreedegreeoffreedom=threequantumnumbery(r)=Ym (.0)R. (r)8
Hydrogen atom • Simplest chemical “thing” – Represented in spherical coordinates – Massive nucleus at origin* – Map motion of electron – H atom Schrödinger equation 8 Coulomb potential decays slowly with distance. Separation of variables – θ and φ don’t appear anywhere other than the Laplacian three degree of freedom = three quantum number
Hydrogenatom:angularcomponenty(r)=Ym (e,o)R, (r)Angular component- Yim are so-called “spherical harmonics", describe angularmotion/angularmomentum of electronAngularquantumnumbers- azimuthal / = O, 1, 2, ... "shape" of angular distribution magnetic m, =-l, -/+1, ., /"orientation"1=0.m=0."s"function1=1,m=0.±1,“Pz,Px,Py"functionssign change on opposite lobesof p
Hydrogen atom: angular component 9 • Angular component – Ylm are so-called “spherical harmonics”, describe angular motion/angular momentum of electron • Angular quantum numbers – azimuthal l = 0, 1, 2, . “shape” of angular distribution – magnetic ml = -l, -l+1, ., l “orientation” sign change on opposite lobes of p
Hydrogenatom:radialcomponente2h? d?11(1 + 1)rR(r)=ErR(r2m。dr24元r2m.r?Solve by ... looking up (analytical) answer!-13.6eVn =1, 2, 3, ..2a1nlonizationenergy of an H atom?1s → 2s energy?Thermalpopulationdistribution?10
Hydrogen atom: radial component 10 • Solve by . looking up (analytical) answer! • Ionization energy of an H atom? • 1s → 2s energy? • Thermal population distribution?