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Hydrogen Atom in Spherical Coordinates (III) Differential equation for F(") Solution: Quantization: Differential equation for T(!) Solution: Quantization: Differential equation for R(r) Solution: Quantization: Eigenenergies of one electron atoms Eigenvalues (energies) for the solutions: Hydrogen (note: energy equation is the same as Bohr’s) Three quantum numbers: [principal] energy [azimuthal] angular momentum: s, p, d, f, .. [magnetic] orientation in space (note: one more quantum number to come … Spin !) Because the energy of oneelectron atoms depends on n only, we have degeneracy; i.e. several solutions (with different l, ml) having the same energy. Eigenfunctions of the One-Electron Atom General form: Components: Spherical Harmonics: (combines the angular parts) Radial Functions Rnl(r) Radial Probability Functions Angular Functions : Spherical Harmonics