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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: H = K + V ( x, y, z ) =- h 2 2 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, , ) x = rsincos y = rsinsin z = rcos 2 = 1 r 2 r r 2 r + 1 r 2 sin sin + 1 r 2 sin 2 2 2 1 Thus, in spherical polar coordinates, H ( r, , ) ( r, , ) = E ( r, , ) be- comes: "- h 2 2 1 r 2 r r 2 r + 1 r 2 sin sin + 1 r 2 sin 2 2 2 !# ( r, , ) = E ( r, , ) For the...
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