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Unformatted text preview: PHY6938 Proficiency Exam Spring 2002 April 5, 2002 Modern Physics and Quantum Mechanics 1. Consider the Schr odinger equation for the linear harmonic oscillator, h 2 2 m d 2 dx 2 + 1 2 m 2 x 2 = E , where m is the mass of the particle and is the angular frequency. The wavefunction and energy of the ground and first excited state are given by GS ( x ) = s exp( 2 x 2 / 2) , E GS = 1 2 h E 1 ( x ) = s 2 2 x exp( 2 x 2 / 2) , E E 1 = 3 2 h , respectively, where = q m/ h . A perturbation term H = m 2 xx is added to the Hamiltonian of the harmonic oscillator. The following integral may be useful: Z  dx x 2 exp( x 2 ) = / 2 (a) Calculate the transition matrix element h GS  H  E 1 i . h GS  H  E 1 i = m 2 x 2 s 2 Z  dx x 2 exp h 2 x 2 i = [change variable, x = x ] = m 2 x s 2 1 Z  dx x 2 exp h x 2 i = m 2 x s 2 1 2 = m 2 x 1 2...
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This note was uploaded on 03/11/2012 for the course PHY 3900 taught by Professor Staff during the Fall '1 term at FSU.
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