Angular Momentum r p = L = r i r + + r - - PowerPoint PPT Presentation

Angular Momentum

• L =

r × p =

i

r × ∇

• ∇ = ˆ

r ∂

∂r + ˆ

θ 1

r ∂ ∂θ + ˆ

φ

1 r sin θ ∂ ∂φ

ˆ θ = cos θ cos φ ˆ x + cos θ sin φ ˆ y − sin θ ˆ z ˆ φ = − sin φ ˆ x + cos φ ˆ y

• L

=

• i
• r × ˆ

r ∂

∂r +

r × ˆ θ 1

r ∂ ∂θ +

r × ˆ φ

1 r sin θ ∂ ∂φ

• =
• i
• ˆ

φ ∂

∂θ − ˆ

θ

1 sin θ ∂ ∂φ

• =
• i
• (− sin φ ˆ

x + cos φ ˆ y) ∂

∂θ

− (cos θ cos φ ˆ x + cos θ sin φ ˆ y − sin θ ˆ z)

1 sin θ ∂ ∂φ

• Lx

=

• i
• − sin φ ∂

∂θ − cos θ sin θ cos φ ∂ ∂φ

• Ly

=

• i
• cos φ ∂

∂θ − cos θ sin θ sin φ ∂ ∂φ

• Lz

=

• i

∂ ∂φ

Angular Momentum

L± = Lx ± i Ly =

• i
• (− sin φ ± i cos φ) ∂

∂θ − (cos φ ± i sin φ) cot θ ∂ ∂φ

• =

± e±iφ

∂ ∂θ ± i cot θ ∂ ∂φ

• L+L−

= 2 eiφ

∂ ∂θ + i cot θ ∂ ∂φ

−e−iφ

∂ ∂θ − i cot θ ∂ ∂φ

• =

−2 eiφ e−iφ

∂2 ∂θ2 − i −1 sin2 θ ∂ ∂φ − i cot θ ∂2 ∂θ∂φ

• i cot θ (−i) e−iφ

∂ ∂θ − i cot θ ∂ ∂φ

• i cot θ e−iφ

∂2 ∂θ∂φ − i cot θ ∂2 ∂φ2

• =

−2

∂2 ∂θ2 + cot θ ∂ ∂θ + cot2 θ ∂2 ∂φ2 + i ∂ ∂φ

• L2

= L+L− − i[Ly, Lx] + L2

z = L+L− + L2 z − Lz

= −2

∂2 ∂θ2 + cot θ ∂ ∂θ + cot2 θ ∂2 ∂φ2 + i ∂ ∂φ + ∂2 ∂φ2 + 1 i ∂ ∂φ

• =

−2

1 sin θ ∂ ∂θ

• sin θ ∂

∂θ

• +

1 sin2 θ ∂2 ∂φ2

Angular Momentum

L2 = −2

1 sin θ ∂ ∂θ

• sin θ ∂

∂θ

• +

1 sin2 θ ∂2 ∂φ2

• eigenfunctions Y m

ℓ (θ, φ)

no Y m

ℓ

for half-integer ℓ

• 1

2µr2

• −2 d

dr

• r2 d

dr

• + L2

+ V (r)

• ψ = E ψ

Krad = p2

2µ,

Krot = I2

2I ,

I = µr2

Diatomic Molecule

m1 r1 = m2 r2 (m1 + m2) r1 = m2 (r1 + r2), (m1 + m2) r2 = m1 (r1 + r2) I = m1r2

1 + m2r2 2 = m1 m2

2(r1+r2)2

(m1+m2)2 + m2 m2

1(r1+r2)2

(m1+m2)2

=

m1m2(r1+r2)2 m1+m2

= µr2 Hψ = L2

2I ψ = Eψ

ψ = Y m

ℓ (θ, φ)

E = 2 ℓ(ℓ+1)

2I

O2 Diatomic Molecule

Erot =

2 2 MOr2 = 22 c2 16Mpc2r2

≈

(6.58×10−16eV s)2 (3×108m/s)2 8×938×106eV (10−10m)2

≈

40×10−16eV 2 8×108eV 10−20

≈ 5 × 10−3 eV fvib = 5 × 1013Hz Evib = hfvib = 2π 6.58 × 10−16eV s × 5 × 1013 1/s = 0.2 eV Evib ≫ Erot

Diatomic Molecule