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Unformatted text preview: 1 Self Evaluation #3 (Chem113A W07, Thu. 03/22/07 11:30am2:30pm+15mins, 100+10pts, max=100pts) Name: _______ By writing down my name, I confirm that I strictly obey the academic ethic code when taking this exam. (i) Six questions + one extra credit question. Please budget your time. You may want to start with the parts you are more familiar with. Formula sheets are attached at the end. Need more formula? Please raise your hand. (ii) It is very important to show calculation or algebraic details. (iii) Academic ethics need to be strictly obeyed. No exceptions and no kidding. Theme of SE#3. We blend in additional advanced, uptodate materials into Chem113A W07 to pursue a solid understanding to some abstract, deep, and striking concepts and to make these concepts useful to our future careers. Richard Feynman: I think I can safely say that nobody understands quantum mechanics. After our 10 weeks of intensive, diligent stud together, lets see how much we understand quantum chemistry now. [1] Objective #1 Basic Material: Molecular Spectroscopy (total 14 pts) [1](a) Eigenfunctions and eigenvalues of the Hamiltonian operator of a particle on a sphere (4 pts) Denote the angular momentum operator by L . The eigenfunctions and eigenvalues for the operators L 2 are: L 2 Y lm ( , ) = l ( l +1) 2 Y lm ( , ), where l =0,1,2, and m =0,1,2,..., , l . In classical mechanics, the energy for a rotating particle (e.g., an imagined relativemotion particle representing the relative motion of a rotating diatomic molecule) is E = L 2 /(2 I ), where moment of inertia I = R 2 . Please derive the eigenfunctions and eigenvalues for the Hamiltonian operator of a rotating particle with moment of inertia I . 2 [1](b) Rotational spectra bond length (5 pts) Based on the results in [1](a) and [1](b), we can obtain valuable information about the rotating diatomic molecule from the spectroscopic data. For example, in 1 H 19 F, the frequency of light absorbed in a change from the l =3 to the l =4 rotational state is measured to be 2.5 *10 12 s1 . Please calculate the bond length of 1 H 19 F. [1](c) Vibrational spectra force constant (5 pts) The quantized energy levels for the vibrational motion of a diatomic molecule can be found in the formula sheets. 1 H 19 F absorbes a photon with wavenumber 4138.32 cm1 to go from the vibrational ground state (n=0) to the 1 st vibrational excited state (n=1). Please calculate the force constant for the chemical bond of 1 H 19 F. (Hint: wavenumber = 1/ ) 3 [2] Objective #1 Basic Material: Hydrogen Atom (total 16 pts) [2](a) Timeindependent Schrodinger equation (TISE) for Hatom (4 pts) The TISE for Hatoms electron in the spherical coordinate system {r, , } is shown below: The angular momentum operator, L 2 , in the spherical coordinate system {r, , } can be expressed as: Please show by algebra that the above TISE for Hatom can be simplified to: Where V(r)= ....
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This note was uploaded on 10/12/2009 for the course CHEM 113A taught by Professor Lin during the Winter '07 term at UCLA.
 Winter '07
 Lin
 Quantum Chemistry

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