02s_mtxsol

02s_mtxsol - MIT OpenCourseWare http://ocw.mit.edu 5.80...

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Unformatted text preview: 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 . Lecture #2 Supplement Contents A Matrix Solution of Harmonic Oscillator Problem . . . . . . . . . . . . . . . . . . . . . . 1 B Derivation of Heisenberg Equation of Motion . . . . . . . . . . . . . . . . . . . . . . . . 4 C Matrix Elements of any Function of X and P . . . . . . . . . . . . . . . . . . . . . . . . 5 A Matrix Solution of Harmonic Oscillator Problem We wish to obtain all possible information about the eigenstates of a harmonic oscillator without ever solving for the actual eigenfunctions. The energy levels and the expectation values of any positive integer power of x = r r e and p = m dx will be obtained. dt The first step is always to write down the Hamiltonian operator, which for the harmonic oscillator is: p 2 kx 2 H = + (1) 2 m 2 In order to construct the matrix for H we need to know the matrix elements of p 2 and x 2 in some convenient basis set. Because we are lazy (and clever) we would like to choose a basis set which results in a diagonal H matrix. We know such a basis set must exist (because any Hermitian matrix can be diagonalized), so we choose that basis set and try to obtain the p and x matrices in that basis without initially knowing the properties of those basis functions. We know: A. H is in diagonal form (choice of basis); B. [ x , p ] = xp px = i (a fundamental postulate of quantum mechanics; C....
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02s_mtxsol - MIT OpenCourseWare http://ocw.mit.edu 5.80...

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