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Quantum Mechanics 1, Spring 2011 CMI Problem set 9 Due by beginning of class on Monday March 21, 2011 Free particle gaussian wave packet and harmonic oscillator 1. Recall that the gaussian wave packet ψ(x, t = 0) = Aeik0 x e − x2 4a2 , A2 = 1 √ . a 2π (1) has mean momentum hpi = ~k0 at t = 0. Write down ψ̃(k, t = 0) and then obtain ψ̃(k, t) in the energy/momentum basis. h3i 2. Find hpi at t > 0. hpi is most easily calculated in the momentum basis. h4i 3. Calculate h x̂i at time t in the above gaussian wave packet. Since ψ̃(k, t) is known, it is good to work in the momentum basis. So you need to know how x̂ acts in k -space. This was worked ∂ out in problem set 6: x̂ = i ∂k . Hint: In working out the integrals, exploit the fact that integrals of odd functions on even intervals vanish. h9i 4. Do the obtained mean values satisfy Ehrenfest’s principle m ∂hxi ∂t = hpi at all times? h2i √ 5. Find the probability for a particle in the ground state π1/4β e−β x /2 of a harmonic oscillator potential 12 mω2 x2 , to be found outside its classically allowed region. Express this probability as q an integral over dimensionless variables. Does it depend on m or ω? Here β = mω ~ . h5i 6. Find the numerical value of this probability. You may use 1 2 2 R∞ 1 2 dξ e−ξ ≈ 0.14. h2i