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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 ϕ n = n n If f x ˆ a ϕ . ( ) is a function that we can expand in powers of x , show that ϕ is an eigenfunction of f ( A ˆ ) with eigenvalue f a ( n ) : n f A ( ˆ ) ϕ n a ϕ n = f ( ) n 2. Displacement operator. Just as U t t ˆ ( ) , 0 = exp iH t ˆ ( t 0 ) h is the time-evolution operator which displaces ψ ( r t , ) in time, ˆ ( , 0 ) = exp ( ip ˆ ( r r 0 ) D r r h ) is the spatial displacement operator that moves ψ in space. a) Defining D λ = exp ( ip h ) , for a one-dimensional displacement, show that ˆ ( ) ˆ λ D xD ˆ ˆ = x + λ where λ is a displacement vector. The relationship you’ve seen in 5.73 2 2 ( ) 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 n n + i λ ⎞ ˆ ˆ 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 ψ 0 , show that the wavefunction ψ 0 λ = D ˆ ( λ ) ψ 0 is the same as the wavefunction of the state 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 between two displaced harmonic oscillators. Specifically, calculate the Franck Condon factor for overlap of a harmonic oscillator with eigenstates with a ψ n displaced harmonic oscillator in the ground state . That is:
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5.74, Problem Set #1 Page 2 2 2 I = ψ ψ 0 λ = D ˆ λ ( ) n ψ n ψ 0 Evaluate this by expressing D in terms of raising and lowering operators. You ˆ ˆ ˆ ˆ 1 ˆ ˆ A B A B 2 ˆ
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