04082010 - Quantum Mechanics Vibration and Rotation of...

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Unformatted text preview: Quantum Mechanics: Vibration and Rotation of Molecules 5th April 2010 I. The Rigid Rotor and Q. M. Orbital Angular Momentum Consider a rigid rotating diatomic molecule-- the rigid rotor-- with two masses separated by a distance r o ; the distance is fixed, and the rota- tion occurs in the absence of external potentials. The quantum mechanical description begins with the Hamiltonian: ˆ H = ˆ K + V ( x, y, z ) =- ¯ h 2 2 μ ∇ 2 + 0 ∇ 2 = ∂ 2 ∂x 2 + ∂ 2 ∂y 2 + ∂ 2 ∂z 2 This is simply the kinetic energy operator as we have seen in the past for the particle-in-box and the harmonic oscillator. Now, we can change coordinate systems from Cartesian to polar spherical coordinates. This goes as: Cartesian ( x, y, z ) → sphericalpolar ( r, θ, φ ) x = rsinθcosφ y = rsinθsinφ z = rcosθ ∇ 2 = 1 r 2 ∂ ∂r r 2 ∂ ∂r + 1 r 2 sinθ ∂ ∂θ sinθ ∂ ∂θ + 1 r 2 sin 2 θ ∂ 2 ∂φ 2 1 Thus, in spherical polar coordinates, ˆ H ( r, θ, φ ) ψ ( r, θ, φ ) = Eψ ( r, θ, φ ) be- comes: "- ¯ h 2 2 μ 1 r 2 ∂ ∂r r 2 ∂ ∂r + 1 r 2 sinθ ∂ ∂θ sinθ ∂ ∂θ + 1 r 2 sin 2 θ ∂ 2 ∂φ 2 !# ψ ( r, θ, φ ) = Eψ ( r, θ, φ ) For the...
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04082010 - Quantum Mechanics Vibration and Rotation of...

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