1How does angular momentum affect radial distribution?1×10-102×10-103×10-104×10-105×10-106×10-10-2×10-18-1×10-181×10-182×10-18Veffectivel= 2, n= 3, 4, 5, …l= 1, n= 2, 3, 4, …R(0) = 0l= 0, n= 1, 2, 3, …R(0) ≠ 0:-hbar22 meB1r2 ddr Jr2 ddrNF+ Bhbar22 me r2 lHl + 1L -e24 pe0 rF> R = ERR10 = 2 ikjj1a0y{zz3ê2 −zR20 =1è!!!!8 ikjj1a0y{zz3ê2 H2 − zL −zê2R21 =1è!!!!!!!24 ikjj1a0y{zz3ê2 z −zê2R30 =281 è!!!!3 ikjj1a0y{zz3ê2 H27 − 18 z + 2 z2L −zê3R31 =481 è!!!!6 ikjj1a0y{zz3ê2 H6 z − z2L −zê3R32 =481 è!!!!!!!30 ikjj1a0y{zz3ê2 z2 −zê3z = r/a0a0= 0.529 Å = 52.9 pmDecay rate?# nodes?R(0) = 0?25 10 15 20 25-2×1014-1×10141×10142×10143×10145 10 15 20 251×10142×10143×10144×10145 10 15 20 252×10134×10136×10138×10131×1014Rnl(z)5 10 15 20 251×1092×1093×1094×1095 10 15 20 255×1081×1091.5 ×1092×1092.5 ×1093×1093.5 ×1095 10 15 20 255×1081×1091.5 ×1092×109(a0z)2Rnl235 10 1520250.20.40.60.815 10 1520250.20.40.60.8151015 20 250.20.40.60.81∫(a0z)2Rnl2dza00xWhich RnlAre Orthogonal?5 10 15 20 25-2×1014-1×10141×10142×10143×10145 10 15 20 251×10142×10143×10144×10145 10 15 20 252×10134×10136×10138×10131×1014Rnl(z)4Eigenfunctions of …:-hbar22 meB1r2 ddr Jr2 ddrNF+ Bhbar22 me r2 lHl + 1L -e24 pe0 rF> R = ERR10, R20, R30, …, are orthogonalR21, R31, …, are orthogonalR32, R42, …, are orthogonal∫(a0z)2RnlRn’l’ dza00∞Shouldn’t eigenfunctions of the Hamiltonian be orthogonal?•Are ψ(1s) and ψ(2s) orthogonal?•Are ψ(2s) and ψ(2p)
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