Midterm 7 Solutions

Nb 9 so we see that these quantities satisfy the

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Unformatted text preview: HaL To compute the time evolution of these quantities, we let a = aH0L ‰- w t , where aH0L = 2 — XN \ mw Xx\ = XN \ ‰-Â d . Thus, as a function of time cosHw t + dL X px \ = - 2 m — w XN \ sin Hw t + dL so we see that these quantities satisfy the Ehrenfest relations 1 „ X px \ m w2 „ t „ m „t Xx\ Xx\ = X px \ = Printed by Mathematica for Students Furthermore, we know from (c) DE = a ¬༼ a = N0 ; as N0 becomes large 2 — XN \ mw Xx\ = cosHw t + dL X px \ = - 2 m — w XN \ sin Hw t + dL MidtermExam7Solutions.nb 9 so we see that these quantities satisfy the Ehrenfest relations 1 „ X px \ m w2 „ t „ m „t Xx\ Xx\ = X px \ = Furthermore, we know from (c) DE XE \ DE XE \ a = a Ø 2 + 1 ¬༼ a = N0 ; as N0 becomes large 2 1 N0 As the average excitation number becomes very large, the state becomes highly localized in energy about its expectation value, and its expectation values in position and momentum evolve according to the classical equati...
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