MIT5_74s09_pset01

MIT5_74s09_pset01 - MIT OpenCourseWare http:/ocw.mit.edu...

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MIT OpenCourseWare http://ocw.mit.edu 5.74 Introductory Quantum Mechanics II Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
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5.74, Problem Set #1 Spring 2009 Due Date: February 19, 2009 1. Functions of operators. Let the eigenfunctions and eigenvalues of an operator A ˆ be ϕ n and a n respectively: A n = n n If fx ˆ a . ( ) is a function that we can expand in powers of x , show that is an eigenfunction of f ( A ˆ ) with eigenvalue fa ( n ) : n fA ( ˆ ) n a n = f () n 2. Displacement operator. Just as Utt ˆ () , 0 = e x p iHt ˆ ( − t 0 ) h is the time-evolution operator which displaces ψ ( r t , ) in time, ˆ ( , 0 ) = e x p ( i p ˆ ( r r 0 ) h ) is the spatial displacement operator that moves ψ in space. a) Defining D λ = exp (− i p h ) , for a one-dimensional displacement, show that ˆ () ˆ λ DxD = x + λ where λ is a displacement vector. The relationship you’ve seen in 5.73 22 ( ) A exp ( λ ˆ ) A G,A i 2 λ ! ˆ ⎦⎦ + exp iG ˆ λ ˆ iG = ˆ + λ i ˆ ˆ + G, G,A ˆ ˆ ⎤⎤ K nn + i λ ⎞ ˆˆ ˆ K ˆ ˆ K G, G, G G,A K + n! ⎦⎦ will be useful here. b) For the ground eigenstate of the one-dimensional harmonic oscillator 0 , show ψ 0 λ D ( λ ψ 0 ψ 0 , only shifted by λ . c) In spectroscopy the Franck-Condon factor, I , quantifies the overlap of vibronic levels in ground and excited electronic states. Let’s calculate this for overlap Specifically, calculate the Franck Condon factor for overlap of a harmonic oscillator with eigenstates with a ψ
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5.74, Problem Set #1 Page 2 2 2 I = ψψ 0 λ = D ˆ λ () n ψ n 0 Evaluate this by expressing D in terms of raising and lowering operators. You
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MIT5_74s09_pset01 - MIT OpenCourseWare http:/ocw.mit.edu...

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