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27 lines
1017 B
Plaintext
27 lines
1017 B
Plaintext
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Developed Theory for The Hydrogen Atom
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Theory of the Hydrogen Atom
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━━━━━━━━━━━━━━━━━━━━━━━━━━━━
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For any hydrogenic ion with nuclear charge Z
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V(𝐫) = -Ze²/r
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V(𝐫) = -e²/r
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Eigenfunctions in spherical coordinates:
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Ψₑₗₘ(r,θ,φ) = Rₑₗ(r) Yₗₘ(θ,φ) = Uₑₗ/r Yₗₘ(θ,φ)
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Derived from first principles the wave equation of an electron in the
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hydrogen atom.
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- Start with Dirac Notation
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- Replace with known general functions
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- transform to eigenvalue equation in position space
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- replace pieces with function F and find derivatives
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F = ∑ Cₖρᵏ
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pic 3 first take had factors that needed to be fixed
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- Led to the Laguerre Polynomials
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These are infinite, though, so
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- Force Laguerre Polynomial solutions to truncate
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