diff --git a/lecture_notes/3-28/3-28-overview.txt b/lecture_notes/3-28/3-28-overview.txt deleted file mode 100644 index fdbe063..0000000 --- a/lecture_notes/3-28/3-28-overview.txt +++ /dev/null @@ -1 +0,0 @@ -Developed Theory for The Hydrogen Atom diff --git a/lecture_notes/3-28/hydrogen atom b/lecture_notes/3-28/hydrogen atom index 8ca5ece..18ea2f0 100644 --- a/lecture_notes/3-28/hydrogen atom +++ b/lecture_notes/3-28/hydrogen atom @@ -1,43 +1,17 @@ -Theory of the Hydrogen Atom -━━━━━━━━━━━━━━━━━━━━━━━━━━━━ - V(𝐫) = -e²/r - - For any hydrogenic ion with nuclear charge Z - V(𝐫) = -Ze²/r - - Eigenfunctions in spherical coordinates: - - Ψₑ﹐ₗ﹐ₘ(r,θ,φ) = Rₑ﹐ₗ(r) Yₗ﹐ₘ(θ,φ) = Uₑ﹐ₗ/r Yₗ﹐ₘ(θ,φ) - - Derived from first principles the wave equation of an electron in the - hydrogen atom. - - - Start with Dirac Notation - - Replace with known general functions - - transform to eigenvalue equation in position space - - 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 - -Derivation -━━━━━━━━━━ +Derivation of Electron States in Hydrogen Atom +━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ The Hydrogen atom follows the central potential development from the previous lecture. - ❬r,θ,φ❙Ψ❭ = R(r)ₑ,ₗ yₗ﹐ₘ(θ,φ) + ❬r,θ,φ❙Ψ❭ = R(r)ₑ,ₗ yₗₘ(θ,φ) - Rₑ﹐ₗ(r) = Uₑ﹐ₗ(r) /r - - ρ ≡ √⎛8͟m͟ │E│⎞r - ⎝ ħ² ⎠ - - λ = Z͟e͟² √⎛_͟m͟_͟ ⎞ - ħ ⎝2│E│⎠ + Rₑₗ(r) = Uₑₗ(r) /r + + ρ ≡ √⎛8͟m͟ │E│⎞r + ⎝ ħ² ⎠ + + λ = Z͟e͟² √⎛_͟m͟_͟ ⎞ + ħ ⎝2│E│⎠ d͟²͟ U - 1͟ l(l+1) + ⎛λ͟ - 1͟⎞U = 0 dρ² ρ² ⎝ρ 4⎠ @@ -56,31 +30,35 @@ Derivation U(ρ) = A exp(-ρ/2) + B exp(ρ/2) = A exp(-ρ/2) (ρ→∞, B=0) - U(ρ) = ρˡ⁺¹ exp(-ρ/2) Fₑ﹐ₗ(ρ) + U(ρ) = ρˡ⁺¹ exp(-ρ/2) Fₑₗ(ρ) - d͟ U = (l+1)ρˡ exp(-ρ/2) Fₑ﹐ₗ(ρ) + d͟ U = (l+1)ρˡ exp(-ρ/2) Fₑₗ(ρ) dρ - -½ ρˡ⁺¹ exp(-ρ/2) Fₑ﹐ₗ(ρ) - + ρˡ⁺¹ exp(-ρ/2) d͟ Fₑ﹐ₗ(ρ) - dρ + -½ ρˡ⁺¹ exp(-ρ/2) Fₑₗ(ρ) + + + + ρˡ⁺¹ exp(-ρ/2) d͟ Fₑₗ(ρ) + dρ d͟ U = ⎛l͟+͟1͟ - 1͟ ⎞ U + ρˡ⁺¹ exp(-ρ/2) d͟ F(ρ) dρ ⎝ ρ 2 ⎠ dρ - d͟²͟ U = -(l͟+͟1͟) U + ⎛l͟+͟1͟ - 1͟⎞² U - dρ² ρ² ⎝ ρ 2⎠ - + 2⎛l͟+͟1͟ - 1͟⎞ρˡ⁺¹ exp(-ρ/2) d͟F͟ - ⎝ ρ 2⎠ dρ - + ρˡ⁺¹ exp(-ρ/2) d͟²͟ F(ρ) - dρ² + d͟²͟ U = -(l͟+͟1͟) U + ⎛l͟+͟1͟ - 1͟⎞² U + dρ² ρ² ⎝ ρ 2⎠ + + + 2⎛l͟+͟1͟ - 1͟⎞ρˡ⁺¹ exp(-ρ/2) d͟F͟ + ⎝ ρ 2⎠ dρ + + + ρˡ⁺¹ exp(-ρ/2) d͟²͟ F(ρ) + dρ² - ⎡−͟ħ͟² d² + l͟ (l+1)ħ² - Z͟e͟²⎤Uₑ﹐ₗ(r) = E Uₑ﹐ₗ(r) + ⎡−͟ħ͟² d² + l͟ (l+1)ħ² - Z͟e͟²⎤Uₑₗ(r) = E Uₑₗ(r) ⎣2m dr² 2mr² r ⎦ ρ = √⎛8͟m͟ │E│⎞r diff --git a/lecture_notes/3-28/overview b/lecture_notes/3-28/overview index fdbe063..fdb53b2 100644 --- a/lecture_notes/3-28/overview +++ b/lecture_notes/3-28/overview @@ -1 +1,32 @@ Developed Theory for The Hydrogen Atom + + 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. + + - 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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 \ No newline at end of file