What is this? Jerry Gilfoyle The Hydrogen Atom 1 / 18

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1 What is this? Jey Gilfoyle The Hydogen Atom 1 / 18

2 What is this? The Hydogen Atom Jey Gilfoyle The Hydogen Atom 1 / 18

3 What is this? The Hydogen Atom Jey Gilfoyle The Hydogen Atom 1 / 18

4 What is this? The Hydogen Atom 1 ( 1 λ = R H n f 2 1 ) n i 2 R H - Rydbeg constant Jey Gilfoyle The Hydogen Atom 1 / 18

5 Hydogen Eigenvalues 13.6 ev E n = n 2 Quantitative compaison fo Balme seies hydogen in units of σ. Line My Results (Å) NIST Results (Å) Nomalized Pecent Diffeence Diffeence α 6.64 ± β 4.85 ± γ 4.39 ± α : n = 3 n = 2 β : n = 4 n = 2 γ : n = 5 n = 2 Jey Gilfoyle The Hydogen Atom 2 / 18

6 n = 8, l = 3, m = 1 Jey Gilfoyle The Hydogen Atom 3 / 18

7 How do we build the quantum model? 1 What is the mechanical enegy? Jey Gilfoyle The Hydogen Atom 4 / 18

8 How do we build the quantum model? 1 What is the mechanical enegy? E = p2 2µ e2 = p2 2µ + L2 2µ 2 e2 µ = m pm e m p + m e m e Jey Gilfoyle The Hydogen Atom 4 / 18

9 How do we build the quantum model? 1 What is the mechanical enegy? E = p2 2µ e2 = p2 2µ + L2 2µ 2 e2 µ = m pm e m p + m e m e 2 What is the Schoedinge equation? Jey Gilfoyle The Hydogen Atom 4 / 18

10 How do we build the quantum model? 1 What is the mechanical enegy? E = p2 2µ e2 = p2 2µ + L2 2µ 2 e2 µ = m pm e m p + m e m e 2 What is the Schoedinge equation? ( 2 1 2µ sin θ 2 2µ 2 ϕ s ( ) e2 ϕ s( ) = Eϕ s ( ) θ sin θ θ sin 2 θ 2 2 φ ) ϕ s( ) e2 ϕs( ) = Eϕs( ) Jey Gilfoyle The Hydogen Atom 4 / 18

11 How do we build the quantum model? 1 What is the mechanical enegy? E = p2 2µ e2 = p2 2µ + L2 2µ 2 e2 µ = m pm e m p + m e m e 2 What is the Schoedinge equation? ( 2 1 2µ sin θ 2 2µ 2 ϕ s ( ) e2 ϕ s( ) = Eϕ s ( ) θ sin θ θ sin 2 θ 3 What do we know about the solution? 2 2 φ ) ϕ s( ) e2 ϕs( ) = Eϕs( ) Jey Gilfoyle The Hydogen Atom 4 / 18

12 How do we build the quantum model? 1 What is the mechanical enegy? E = p2 2µ e2 = p2 2µ + L2 2µ 2 e2 µ = m pm e m p + m e m e 2 What is the Schoedinge equation? ( 2 1 2µ sin θ 2 2µ 2 ϕ s ( ) e2 ϕ s( ) = Eϕ s ( ) θ sin θ θ sin 2 θ 3 What do we know about the solution? 2 2 φ ) ϕ s( ) e2 ϕs( ) = Eϕs( ) ϕ s ( ) = R()Θ(θ)Φ(φ) = R()Y m l (θ, φ) Jey Gilfoyle The Hydogen Atom 4 / 18

13 How do we build the quantum model? 1 What is the mechanical enegy? E = p2 2µ e2 = p2 2µ + L2 2µ 2 e2 µ = m pm e m p + m e m e 2 What is the Schoedinge equation? ( 2 1 2µ sin θ 2 2µ 2 ϕ s ( ) e2 ϕ s( ) = Eϕ s ( ) θ sin θ θ sin 2 θ 2 2 φ GO SOLVE IT! 3 What do we know about the solution? ) ϕ s( ) e2 ϕs( ) = Eϕs( ) ϕ s ( ) = R()Θ(θ)Φ(φ) = R()Y m l (θ, φ) Jey Gilfoyle The Hydogen Atom 4 / 18

14 Hydogen Bound State Eigenfunctions ϕ nlm (, θ, φ) = R nl ()Y m l (θ, φ) = (2κ) 3/2 A nl ρ l e ρ/2 F nl (ρ)y m l (θ, φ) Jey Gilfoyle The Hydogen Atom 5 / 18

15 Hydogen Bound State Eigenfunctions ϕ nlm (, θ, φ) = R nl ()Y m l (θ, φ) = (2κ) 3/2 A nl ρ l e ρ/2 F nl (ρ)y m l (θ, φ) F (ρ) = = a i ρ i (i + l + 1) λ a i+1 = (i + 1)(i + 2l + 2) a i a 0 = 1 i=0 E n = E ρ = 2κ κ = 2µ E F nl (ρ) = L 2l+1 n l 1 (ρ) A nl = 2 λ = Ze2 µ 2 E (n l 1)! 2n[(n + l)!] 3 Jey Gilfoyle The Hydogen Atom 5 / 18

16 Hydogen Eigenvalues (Enegy Levels) 5 Continuum States 0 E n = µ(e2 ) n 2 ev = 13.6 n 2 Enegy (ev) -5 Discete States Jey Gilfoyle The Hydogen Atom 6 / 18

17 Hydogen Bound State Eigenfunctions ψ Enlm (, θ, φ) = R nl ()Yl m (θ, φ) ( kmax ) = A nl ρ l e ρ b k ρ k Yl m (θ, φ) k=0 Jey Gilfoyle The Hydogen Atom 7 / 18

18 Hydogen Bound State Eigenfunctions ψ Enlm (, θ, φ) = R nl ()Yl m (θ, φ) ( kmax ) = A nl ρ l e ρ b k ρ k Yl m (θ, φ) k=0 ψ Enlm = 2(k + l + 1) λe2 b k+1 = (k + 1)(k + 2l + 2) b k b 0 = 1 2µW 2µ E n = W ρ = κ κ = λ = 2 2 W ( 2 na 0 a0 = 2 me 2 ) 3 ( ) l ( ) (n l 1)! 2n[(n + l)!] 3 e /na 0 2 (n + l)! L 2l+1 2 n l 1 Yl m (θ, φ) na 0 na 0 Jey Gilfoyle The Hydogen Atom 7 / 18

19 Recall the Solid Angle Jey Gilfoyle The Hydogen Atom 8 / 18

20 Spheical Diffeential Volume Element Jey Gilfoyle The Hydogen Atom 9 / 18

21 Hydogen Eigenfunctions Hydogen Pobability Density (n=4) Red - l=0 P Jey Gilfoyle The Hydogen Atom 10 / 18

22 Hydogen Eigenfunctions Hydogen Pobability Density (n=4) Red - l=0 Blue - l=1 P Jey Gilfoyle The Hydogen Atom 11 / 18

23 Hydogen Eigenfunctions Hydogen Pobability Density (n=4) Red - l=0 Blue - l=1 Geen - l=2 P Jey Gilfoyle The Hydogen Atom 12 / 18

24 Hydogen Eigenfunctions Hydogen Pobability Density (n=4) P Red - l=0 Blue - l=1 Geen - l=2 Gay - l= Jey Gilfoyle The Hydogen Atom 13 / 18

25 Do the peaks line up? Red : n=1, Blue: n= Pobability Density (angstoms) Jey Gilfoyle The Hydogen Atom 14 / 18

26 Old Obitals Jey Gilfoyle The Hydogen Atom 15 / 18

27 Old Obitals - New Obitals Jey Gilfoyle The Hydogen Atom 15 / 18

28 Old Obitals - New Obitals How ae these plots elated to what we know? Jey Gilfoyle The Hydogen Atom 15 / 18

29 Moe Hydogen Eigenfunctions Jey Gilfoyle The Hydogen Atom 16 / 18

30 Hydogen Eigenvalues 13.6 ev E n = n 2 Quantitative compaison fo Balme seies hydogen in units of σ. Line My Results (Å) NIST Results (Å) Nomalized Pecent Diffeence Diffeence α 6.64 ± β 4.85 ± γ 4.39 ± α : n = 3 n = 2 β : n = 4 n = 2 γ : n = 5 n = 2 Jey Gilfoyle The Hydogen Atom 17 / 18

31 Some Plots Jey Gilfoyle The Hydogen Atom 18 / 18