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Download Hydrogen Atom in Spherical Coordinates (III) Eigenenergies of one
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Transcript
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
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