Normalized Wave Functions for Hydrogen Atom s orbitals

Quantum numbers n ℓ mℓ 1 0 0 2 0 0 3 0 0 Radial Wave Functions R(r) for Hydrogen Atom Quantum numbers n ℓ R(r) 1 0 2 0 3 0 Angular Wave Functions ΘΦ(Өφ) for Hydrogen Atom Quantum numbers ℓ mℓ 0 0 a0 = (4 π ε0 ħ n2) / (me e2 Z)

0

3 2 Ζ ra

1s0

1 Zψ = eaπ

−

0

3 2 Ζ r2a

2s0 0

1 Z Ζψ = 2 r ea a4 2π

−

−

0

3 2 Ζ2 r3a2

3s 20 0 0

1 Z 18Ζ 2Ζψ = 27 r + r ea a a81 3π

−

−

0

0

3 2 Ζ raZ2 e

a

−

0

3 2 Ζ r2a

o 0

1 Z Ζ 2 r ea a2 2

−

−

0

3 2 Ζ2 r3a2

2o 0 0

2 Z 18Ζ 2Ζ 27 r + r ea a a81 3

−

−

12 π

Radial Plots of the 1s Orbital

Angular Plot Angular Probability Plot

Electron Density (Contour) Plot 1s 2s 3s 4s 5s 6s

Normalized Wave Functions for Hydrogen Atom p orbitals

Quantum numbers n ℓ mℓ 2 1 0 2 1 ±1 3 1 0 3 1 ±1 Radial Wave Functions R(r) for Hydrogen Atom Quantum numbers n ℓ R(r) 2 1 3 1 Angular Wave Functions ΘΦ(Өφ) for Hydrogen Atom Quantum numbers ℓ mℓ 1 0 ±1 a0 = (4 π ε0 ħ n2) / (me e2 Z)

0

z

3 2 Z r2a

2p0 0

1 Z Zψ = re cosθa a4 2π

−

φ

0

x y

3 2 Z r2a

2p ,2p0 0

1 Z Zψ = re sinθ cosa a4 2π

−

0

z

3 2 Z2 r3a2

3p 20 0 0

2 Z Z Zψ = r r e cosθa a a81 π

− 6−

φ

0

x y

3 2 Z2 r3a2

3p ,p 20 0 0

2 Z Z Zψ = r r e sinθ cosa a a81 π

− 6−

o

3 2 Z r2a

o o

1 Z Z r ea a2 6

−

0

3 2 Z2 r3a2

2o 0 0

4 Z Z Z r r ea a a81 6

− 6−

1 3 cos θ2 π

( )φ1 3 sin θ cos2 π

Radial Plots of the 2p Orbital

Radial Plots of the 3p Orbital

Angular Plot

Angular Probability Plot

Electron Density (Contour) Plot 2p 3p 4p 5p 6p

Normalized Wave Functions for Hydrogen Atom d orbitals

Quantum numbers n ℓ mℓ 3 2 0 3 2 ±1 3 2 ±2 Radial Wave Functions R(r) for Hydrogen Atom Quantum numbers n ℓ R(r) 3 2 Angular Wave Functions ΘΦ(Өφ) for Hydrogen Atom Quantum numbers ℓ mℓ 2 0 ±1 ±2 a0 = (4 π ε0 ħ n2) / (me e2 Z)

( )

0

2z

3 2 Z2 r3a2 2

3d 20 0

1 Z Zψ = r e 3cos θ 1a a81 6π

−

−

0

xz yz

3 2 Ze2 r3a2

3d ,3d 20 0

2 Z Zψ = r e sinθ cosθ cosφa a81 π

−

−

0

xy 2 2x y

3 2 Z2 r3a2 2

3d ,3d 20 0

1 Z Zψ = r e sin θ cos2φa a81 2π

−

o

3 2 Z23a2

2o 0

4 Z Z r ea a81 30

−

( )21 5 3cos θ 14 π

−

1 15 cos θ sin θ cos 2 π

φ

21 15 sin θ cos 24 π

φ

Radial Plots of the 3d Orbital

Angular Plot

Angular Probability Plot

Electron Density (Contour) Plot

3 dx2 – y2 3 dx2 – y2 3 dz2