1Hydrogen atomSWE Æ 2 equations Eq 1 Solution = 3D rotor-hbar22 meB1sin q ∑∑q Jsin q ∑∑qN +1sin2 q ikjj∑2∑f2y{zzFQF = bQFb=hbar22 me l Hl + 1L(,) () ()lllmmllmYθφθφ=Θ ΦEngel 7.33Hydrogen atomSWE Æ 2 equations Eq 2βfrom 3D rotorV(r) from Coulombhbar22 meB1r2 ddr Jr2 ddrN + HE - VHrLLFR =bR:-hbar22 meB1r2 ddr Jr2 ddrNF+ Bhbar22 me r2 lHl + 1L -e24 pe0 rF> R = ER2R(r) eqn gives atom energy Boundary condition R(r) Æ 0 as rgets large Enquantizedn≥l + 1n= 1, 2, 3, … Doesn’t include Etranslationof CMEn=-me e48 e02 h2 n2=-2.18 μ 10-18Jn2=-13.60 eVn2Properties of En En= -13.60 (1)-3.40 (2)-1.51 (3)-0.85 (4) E∞= 0En=-me e48 e02 h2 n2=-2.18 μ 10-18Jn2=-13.60 eVn2H atom has “bound” statesE < 0“free” (unmoving) electronE = 03H atom spectroscopy Rydberg formula for line spectra IP(H) = 13.6 eV = ΔE1,∞ Isotope effect (tiny) on line positionsnm,n= R J1n2-1m2NOrbitals, ψ(r,θ,φ) R(r)Y(θ, φ) R Æn& l Y Æl& mln12 3l00, 1 0, 1, 2ml00, {0, ±1} 0, {0,±1},{0,±1,±2}gn14 94Orbitals, ψ(r,θ,φ)d±1(Y2±2)R323dx2-y2± i3dxy±223d±1(Y2±1)R323dxz± i3dyz±123dz2(Y20)R323dz2023p±1(Y1±1)R313px± i3py±113pz(Y10)R313pz013s (Y00)R303s003p±1(Y1±1)pz(Y10)s (Y00)s (Y00)YlmR212px± i2py±112R212pz012R202s002R101s001RnlψnlmmllnVeffective Centripetal + Coulomb:-hbar22 meB1r2 ddr Jr2 ddrNF+ Bhbar22 me r2 lHl + 1L -e24 pe0 rF> R = ER1×10-102×10-103×10-104×10-105×10-106×10-10-2×10-18-1×10-181×10-182×10-18l= 0, n= 1, 2, 3, …l= 2, n= 3, 4, 5, …l= 1, n= 2, 3, 4, …R(0) = 0R(0) ≠ 05R10(radial factor for 1s) a0= 0.529 Å (= 1 bohr) = ε0h2/ π mee2R10 = 2 ikjj1a0y{zz3ê2 −rêa0rR101×10-102×10-103×10-104×10-101×10152×10153×10154×10155×10156×1015R20(radial factor for 2s)rR201×10-102×10-103×10-104×10-101×10152×10153×10154×10155×10156×1015R20 =1è!!!!8 ikjj1a0y{zz3ê2 J2 −ra0N −rêH2 a0Lradial node @ r = 2 a062×10-104×10-106×10-108×10-101×10-91.2 ×10-95×10141×10151.5 ×10152×1015R30(radial factor for 3s)rR302 radial nodesR30 =281 è!!!!3 ikjj1a0y{zz3ê2 ikjjjj27 − 18ra0+ 2r2a02y{zzzz −rêH3 a0LGraph scales have beenexpanded ~3xÆdiffusefunctionRn0vs. n2×10-104×10-106×10-108×10-101×10-95×10141×10151.5 ×10152×1015R10R20R30These functions must be orthogonal.To accomplish this, they:1) have different # nodes2) avoid each other7ψ2= R2Y2 (Y00)2= 1/4π Which is R102? vertical scales vary2×10-104×10-106×10-108×10-101×10-95×10301×10311.5 ×10312×10312.5 ×10313×10312×10-104×10-106×10-108×10-101×10-91×10302×10303×10304×10302×10-104×10-106×10-108×10-101×10-92×10294×10296×10298×10291×1030Radial distribution function r2R2 Probability at r for any θ& φ Sphere radius r has surface area 4πr22×10-104×10-106×10-108×10-101×10-91×1092×1093×1094×1098Probability of finding electron within rof nucleus2×10-104×10-106×10-108×10-101×10-90.20.40.60.81P10 =‡0xr2 R102
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