02s_mtxsol

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

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

## This note was uploaded on 11/28/2011 for the course CHEM 5.74 taught by Professor Robertfield during the Spring '04 term at MIT.

### Page1 / 6

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online