What is this? Jerry Gilfoyle The Hydrogen Atom 1 / 18 What is - - PowerPoint PPT Presentation

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What is this? Jerry Gilfoyle The Hydrogen Atom 1 / 18 What is - - PowerPoint PPT Presentation

Feb 07, 2024 48 likes •358 views

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

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SLIDE 1

What is this?

Jerry Gilfoyle The Hydrogen Atom 1 / 18

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SLIDE 2

What is this?

The Hydrogen Atom

Jerry Gilfoyle The Hydrogen Atom 1 / 18

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SLIDE 3

What is this?

The Hydrogen Atom

Jerry Gilfoyle The Hydrogen Atom 1 / 18

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SLIDE 4

What is this?

The Hydrogen Atom

1 λ = RH

  • 1

n2

f

− 1 n2

i

  • RH - Rydberg constant

Jerry Gilfoyle The Hydrogen Atom 1 / 18

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SLIDE 5

Hydrogen Eigenvalues

En = −13.6 eV n2 Quantitative comparison for Balmer series hydrogen in units of σ. Line My Results (˚ A) NIST Results (˚ A) Normalized Percent Difference Difference α 6.64 ± 0.09 × 103 6.56280 × 103 0.95 1.2 β 4.85 ± 0.15 × 103 4.86133 × 103 0.11

  • 0.2

γ 4.39 ± 0.06 × 103 4.34047 × 103 0.9 1.2 α : n = 3 → n = 2 β : n = 4 → n = 2 γ : n = 5 → n = 2

Jerry Gilfoyle The Hydrogen Atom 2 / 18

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SLIDE 6

n = 8, l = 3, m = 1

Jerry Gilfoyle The Hydrogen Atom 3 / 18

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SLIDE 7

How do we build the quantum model?

1 What is the mechanical energy? Jerry Gilfoyle The Hydrogen Atom 4 / 18

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SLIDE 8

How do we build the quantum model?

1 What is the mechanical energy?

E = p2 2µ − e2 r = p2

r

2µ + L2 2µr2 − e2 r µ = mpme mp + me ≈ me

Jerry Gilfoyle The Hydrogen Atom 4 / 18

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SLIDE 9

How do we build the quantum model?

1 What is the mechanical energy?

E = p2 2µ − e2 r = p2

r

2µ + L2 2µr2 − e2 r µ = mpme mp + me ≈ me

2 What is the Schroedinger equation? Jerry Gilfoyle The Hydrogen Atom 4 / 18

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SLIDE 10

How do we build the quantum model?

1 What is the mechanical energy?

E = p2 2µ − e2 r = p2

r

2µ + L2 2µr2 − e2 r µ = mpme mp + me ≈ me

2 What is the Schroedinger equation?

− 2 2µ∇2ϕs( r) − e2 r ϕs( r) = Eϕs( r)

− 2 2µ 1 r2 ∂ ∂r r2 ∂ ∂r + 1 r2 sin θ ∂ ∂θ sin θ ∂ ∂θ + 1 r2 sin2 θ ∂2 ∂2φ

  • ϕs(

r)− e2 r ϕs( r) = Eϕs( r)

Jerry Gilfoyle The Hydrogen Atom 4 / 18

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SLIDE 11

How do we build the quantum model?

1 What is the mechanical energy?

E = p2 2µ − e2 r = p2

r

2µ + L2 2µr2 − e2 r µ = mpme mp + me ≈ me

2 What is the Schroedinger equation?

− 2 2µ∇2ϕs( r) − e2 r ϕs( r) = Eϕs( r)

− 2 2µ 1 r2 ∂ ∂r r2 ∂ ∂r + 1 r2 sin θ ∂ ∂θ sin θ ∂ ∂θ + 1 r2 sin2 θ ∂2 ∂2φ

  • ϕs(

r)− e2 r ϕs( r) = Eϕs( r)

3 What do we know about the solution? Jerry Gilfoyle The Hydrogen Atom 4 / 18

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SLIDE 12

How do we build the quantum model?

1 What is the mechanical energy?

E = p2 2µ − e2 r = p2

r

2µ + L2 2µr2 − e2 r µ = mpme mp + me ≈ me

2 What is the Schroedinger equation?

− 2 2µ∇2ϕs( r) − e2 r ϕs( r) = Eϕs( r)

− 2 2µ 1 r2 ∂ ∂r r2 ∂ ∂r + 1 r2 sin θ ∂ ∂θ sin θ ∂ ∂θ + 1 r2 sin2 θ ∂2 ∂2φ

  • ϕs(

r)− e2 r ϕs( r) = Eϕs( r)

3 What do we know about the solution?

ϕs( r) = R(r)Θ(θ)Φ(φ) = R(r)Y m

l (θ, φ)

Jerry Gilfoyle The Hydrogen Atom 4 / 18

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SLIDE 13

How do we build the quantum model?

1 What is the mechanical energy?

E = p2 2µ − e2 r = p2

r

2µ + L2 2µr2 − e2 r µ = mpme mp + me ≈ me

2 What is the Schroedinger equation?

− 2 2µ∇2ϕs( r) − e2 r ϕs( r) = Eϕs( r)

− 2 2µ 1 r2 ∂ ∂r r2 ∂ ∂r + 1 r2 sin θ ∂ ∂θ sin θ ∂ ∂θ + 1 r2 sin2 θ ∂2 ∂2φ

  • ϕs(

r)− e2 r ϕs( r) = Eϕs( r)

3 What do we know about the solution?

ϕs( r) = R(r)Θ(θ)Φ(φ) = R(r)Y m

l (θ, φ)

GO SOLVE IT!

Jerry Gilfoyle The Hydrogen Atom 4 / 18

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SLIDE 14

Hydrogen Bound State Eigenfunctions

ϕnlm(r, θ, φ) = Rnl(r)Y m

l (θ, φ)

= (2κ)3/2 Anlρle−ρ/2Fnl(ρ)Y m

l (θ, φ)

Jerry Gilfoyle The Hydrogen Atom 5 / 18

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SLIDE 15

Hydrogen Bound State Eigenfunctions

ϕnlm(r, θ, φ) = Rnl(r)Y m

l (θ, φ)

= (2κ)3/2 Anlρle−ρ/2Fnl(ρ)Y m

l (θ, φ)

F(ρ) =

∞

  • i=0

= aiρi ⇒ ai+1 = (i + l + 1) − λ (i + 1)(i + 2l + 2)ai a0 = 1 En = −|E| ρ = 2κr κ =

  • 2µ|E|

2 λ = Ze2

  • µ

2|E| Fnl(ρ) = L2l+1

n−l−1(ρ)

Anl =

  • (n − l − 1)!

2n[(n + l)!]3

Jerry Gilfoyle The Hydrogen Atom 5 / 18

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SLIDE 16

Hydrogen Eigenvalues (Energy Levels)

En = −µ(e2)2 22n2 = −13.6 eV n2

Discrete States Continuum States 2 4 6 8

  • 15
  • 10
  • 5

5 Energy (eV) Jerry Gilfoyle The Hydrogen Atom 6 / 18

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SLIDE 17

Hydrogen Bound State Eigenfunctions

ψEnlm(r, θ, φ) = Rnl(r)Y m

l (θ, φ)

= Anlρle−ρ kmax

  • k=0

bkρk

  • Y m

l (θ, φ) Jerry Gilfoyle The Hydrogen Atom 7 / 18

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SLIDE 18

Hydrogen Bound State Eigenfunctions

ψEnlm(r, θ, φ) = Rnl(r)Y m

l (θ, φ)

= Anlρle−ρ kmax

  • k=0

bkρk

  • Y m

l (θ, φ)

bk+1 = 2(k + l + 1) − λe2 (k + 1)(k + 2l + 2)bk b0 = 1 En = −W ρ = κr κ =

  • 2µW

2 λ =

  • 2µ

2W a0 = 2 me2 ψEnlm = 2 na0 3 (n − l − 1)! 2n[(n + l)!]3 e−r/na0 2r na0 l (n + l)! L2l+1

n−l−1

2r na0

  • Y m

l (θ, φ) Jerry Gilfoyle The Hydrogen Atom 7 / 18

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SLIDE 19

Recall the Solid Angle

Jerry Gilfoyle The Hydrogen Atom 8 / 18

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SLIDE 20

Spherical Differential Volume Element

Jerry Gilfoyle The Hydrogen Atom 9 / 18

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SLIDE 21

Hydrogen Eigenfunctions

Red - l=0 10 20 30 40 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 r P Hydrogen Probability Density (n=4)

Jerry Gilfoyle The Hydrogen Atom 10 / 18

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SLIDE 22

Hydrogen Eigenfunctions

Red - l=0 Blue - l=1 10 20 30 40 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 r P Hydrogen Probability Density (n=4)

Jerry Gilfoyle The Hydrogen Atom 11 / 18

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SLIDE 23

Hydrogen Eigenfunctions

Red - l=0 Blue - l=1 Green - l=2 10 20 30 40 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 r P Hydrogen Probability Density (n=4)

Jerry Gilfoyle The Hydrogen Atom 12 / 18

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SLIDE 24

Hydrogen Eigenfunctions

Red - l=0 Blue - l=1 Green - l=2 Gray - l=3 10 20 30 40 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 r P Hydrogen Probability Density (n=4)

Jerry Gilfoyle The Hydrogen Atom 13 / 18

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SLIDE 25

Do the peaks line up?

0.0 0.2 0.4 0.6 0.8 1.0 0.000 0.002 0.004 0.006 0.008 0.010 0.012 r (angstroms) Probability Density Red : n=1, Blue: n=4

Jerry Gilfoyle The Hydrogen Atom 14 / 18

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SLIDE 26

Old Orbitals

Jerry Gilfoyle The Hydrogen Atom 15 / 18

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SLIDE 27

Old Orbitals - New Orbitals

Jerry Gilfoyle The Hydrogen Atom 15 / 18

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SLIDE 28

Old Orbitals - New Orbitals

How are these plots related to what we know?

Jerry Gilfoyle The Hydrogen Atom 15 / 18

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SLIDE 29

More Hydrogen Eigenfunctions

Jerry Gilfoyle The Hydrogen Atom 16 / 18

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SLIDE 30

Hydrogen Eigenvalues

En = −13.6 eV n2 Quantitative comparison for Balmer series hydrogen in units of σ. Line My Results (˚ A) NIST Results (˚ A) Normalized Percent Difference Difference α 6.64 ± 0.09 × 103 6.56280 × 103 0.95 1.2 β 4.85 ± 0.15 × 103 4.86133 × 103 0.11

  • 0.2

γ 4.39 ± 0.06 × 103 4.34047 × 103 0.9 1.2 α : n = 3 → n = 2 β : n = 4 → n = 2 γ : n = 5 → n = 2

Jerry Gilfoyle The Hydrogen Atom 17 / 18

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SLIDE 31

Some Plots

Jerry Gilfoyle The Hydrogen Atom 18 / 18