01_090308_580

01_090308_580 - MIT OpenCourseWare http:/ocw.mit.edu 5.80...

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MIT OpenCourseWare http://ocw.mit.edu 5.80 Small-Molecule Spectroscopy and Dynamics Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
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5.80 Lecture #1 Fall, 2008 Page 1 Lecture #1: Matrices are Useful in Spectroscopic Theory For next time, read handout on the Van Vleck transformation and look at the notes on coupled harmonic oscillators. Outline * Correspondences bra ⎛ ⎞ row column i = ⎜ ⎟ N × 1 column vector ⎝ ⎠ ψ i ket ψ * i i = 1 × N row vector operator O O N × N square matrix * Solution of Schrödinger Equation corresponds to solving a determinantal secular equation for the {E k } ψ ; * The {E k } are the H kk in the special “diagonal” representation of H * The { ψ k } are obtained from the basis states { φ k } as columns of the unitary matrix T from the similarity transformation T H φ T = H ψ that “diagonalizes” H φ ; * Matrix representation of arbitrary f( Q ). Usual procedure to obtain a fit model. e.g. Born- expressed Van Vleck a finite matrix Oppenheimer in terms of transformation expressed in combine and normal structure terms of linearly mode parameters introduces microscopic dependent separations e.g. f ij force many small exact H a a complete exact H truncate H molecular constants Fit model * finite simplified matrix OR * algebraic formulas expressed in terms of effective molecular constants Dunham constants ω e , ω e x e Approximate H perturbation theory differential set of basis operator functions an infinite matrix model of variables constants terms parameters not necessarily the same even if they have same name
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5.80 Lecture #1 Fall, 2008 Page 2 Matrices are useful because: * they display all necessary information; * can be “read” and simplified by perturbation theory “order sorting” via E i o H ij E j ; o * labor saving tricks for avoiding the evaluation of unnecessary integrals, such as formulas for all matrix elements of Q and P in the Harmonic Oscillator basis set.
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01_090308_580 - MIT OpenCourseWare http:/ocw.mit.edu 5.80...

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