2016-03-29 18:00:53 +00:00
|
|
|
|
Developed Theory for The Hydrogen Atom
|
2016-03-31 05:44:29 +00:00
|
|
|
|
|
|
|
|
|
|
Theory of the Hydrogen Atom
|
|
|
|
|
|
━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━
|
|
|
|
|
|
For any hydrogenic ion with nuclear charge Z
|
|
|
|
|
|
V(𝐫) = -Ze²/r
|
|
|
|
|
|
|
|
|
|
|
|
V(𝐫) = -e²/r
|
|
|
|
|
|
|
|
|
|
|
|
Eigenfunctions in spherical coordinates:
|
|
|
|
|
|
|
|
|
|
|
|
Ψₑₗₘ(r,θ,φ) = Rₑₗ(r) Yₗₘ(θ,φ) = Uₑₗ/r Yₗₘ(θ,φ)
|
|
|
|
|
|
|
|
|
|
|
|
Derived from first principles the wave equation of an electron
|
|
|
|
|
|
in the hydrogen atom.
|
|
|
|
|
|
|
|
|
|
|
|
- Start with Dirac Notation
|
|
|
|
|
|
|
|
|
|
|
|
- transform to eigenvalue equation in position space
|
|
|
|
|
|
Replace with known special quantities
|
|
|
|
|
|
|
|
|
|
|
|
- replace pieces with function F and find derivatives
|
|
|
|
|
|
F = ∑ Cₖρᵏ
|
|
|
|
|
|
|
|
|
|
|
|
pic 3 first take had factors that needed to be fixed
|
|
|
|
|
|
|
|
|
|
|
|
- Led to the Laguerre Polynomials
|
|
|
|
|
|
These are infinite, though, so
|
|
|
|
|
|
|
|
|
|
|
|
- Force Laguerre Polynomial solutions to truncate
|
|
|
|
|
|
|
|
|
|
|
|
|
