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sp02mod_sol

# sp02mod_sol - PHY6938 Prociency Exam Spring 2002 April 5...

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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 ψ dx 2 + 1 2 2 x 2 ψ = Eψ , where m is the mass of the particle and ω is the angular frequency. The wavefunction and energy of the ground and first excited state are given by ψ GS ( x ) = s α π exp( - α 2 x 2 / 2) , E GS = 1 2 ¯ ψ E 1 ( x ) = s α 2 π 2 αx exp( - α 2 x 2 / 2) , E E 1 = 3 2 ¯ hω , respectively, where α = q mω/ ¯ h . A perturbation term H 0 = - 2 xx 0 is added to the Hamiltonian of the harmonic oscillator. The following integral may be useful: Z -∞ dx x 2 exp( - x 2 ) = π/ 2 (a) Calculate the transition matrix element h ψ GS | H 0 | ψ E 1 i . h ψ GS | H 0 | ψ E 1 i = - 2 x 0 α 2 s 2 π Z -∞ dx x 2 exp h - α 2 x 2 i = [change variable, x 0 = αx ] = - 2 x 0 s 2 π 1 α Z -∞ dx 0 x 0 2 exp h - x 0 2 i = - 2 x 0 s 2 π 1 α π 2 = - 2 x 0 1 α 2 = - 2 x 0 ¯ h 2 !

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sp02mod_sol - PHY6938 Prociency Exam Spring 2002 April 5...

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