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pra_fin - •  Show that if an Hamiltonian H is...

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Unformatted text preview: •  Show that if an Hamiltonian H is separable into two terms, one involving only coordinates q1 and the other involving only coordinates q2 [H(q1,q2) = H1 (q1) + H2(q2)], then the eigenfuncCons of H are products of the eigenfuncCons of H1 and H2 and the eigenvalues of H are the sums of eigenvalues of H1 and H2. •  Can the Hamiltonian of a molecular system composed of nuclei and electrons be wriFen in a fully separable form (i.e. as a sum of a nuclear and electronic part)? •  Explain what the Born ­Oppenheimer approximaCon is and consider the molecule, in the following group, for which you think the BO approximaCon is most appropriate: HCl, HI and KCl ( jusCfy your choice). The infrared spectrum of such a molecule has an intense line at 278 cm ­1. Calculate the force constant and the vibraConal frequency of the molecule. Use the harmonic approximaCon. KCl (heaviest): From the allowed energies of the HO and knowing the reduced mass of the molecule one may compute the force constant: ν= (1/2π) sqrt(k/µ) •  Compute the average value of the square of the posiCon operator in one dimension for the ground state of a harmonic oscillator. Use the expression to evaluate the root mean square amplitude of vibraCon in the molecule chosen in 3. <x2> = h/[4π sqrt (µk)] •  Explain what the following types of spectroscopic techniques are and which physical quanCty they may be used to probe: Infrared, Raman, NMR, UV ­visible, laser spectroscopy. •  Explain what the Fermi golden rule is and which approximaCons are involved in its derivaCon. Write an expression for the rate of transiCon between an iniCal and an excited state of a molecule in the presence of electromagneCc radiaCon (consider electric field only for simplicity). Discuss selecCon rules. ...
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