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## Hydrogen Atom

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**Hydrogen Atom**PHY 361 2008-03-19**review of Lz**operator, eigenfunction, eigenvalues rotational kinetic energy traveling and standing waves spherical coordinates definition Laplacian operator Schrödinger’s equation in spherical coordinates separation of angular variables: L2 and Lz differential equations spherical harmonics and eigenvalues vector model of quantum angular momentum radial wavefunctions effective radial potential – centrifugal `force’ radial wave functions hydrogenic orbitals Outline**Cylindrical vs. Spherical Coordinates**Laplacian: Schrödinger Equation: Lz2 / 2I L2 / 2I**Spherical Harmonics**L2 Ylm=l(l+1)Ylm s 1 Lz Ylm= m Ylm x, y p z x, y x2+y2, xy xz, yz d 3z2-1 xz, yz x2+y2, xy f …**Vector model of quantized angular momentum**l = 0, 1, 2, … m = -1, -l+1, … l-1, l**Hydrogenic orbitals**http://www.orbitals.com/orb/