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Time-energy uncertainty and derivative of an operator

  1. Jan 2, 2015 #1
    1. The problem statement, all variables and given/known data
    I would appreciate feedback on the following two problems:
    (1) For a given operator A with no explicit time dependence I am asked to show that d/dt(eAt)=A(eAt)
    (2) A free wave packet of width Δx is traveling at a constant velocity v0=p0/m. I am asked to estimate the time-energy uncertainty relation.

    2. Relevant equations


    3. The attempt at a solution
    (1) Is this as straightforward as it appears to be? Probably isn't, right? Is this to be solved using Ehrenfest's theorem? Or how?
    (2) I can easily show that ΔEΔt ~ ħ but is this what I am plainly asked, or am I rather asked to arrive at a more exact result (i.e. find the proportionality coefficient)?
     
  2. jcsd
  3. Jan 2, 2015 #2

    jfizzix

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    To take the time derivative of [itex]e^{At}[/itex] with respect to [itex]t[/itex], you can take the power series expansion of [itex]e^{At}[/itex], differentiate term by term, and re-exponentiate, since what you get left should be [itex]A[/itex] times another exponential.

    For the second problem, it looks like they're asking you to calculate the uncertainty product [itex]\Delta E\Delta t[/itex] from the given information. Then you can see for yourself how much larger than [itex]\frac{\hbar}{2}[/itex] it is.
     
  4. Jan 2, 2015 #3
    For the second problem, how may I go about it then? Is it necessary to write the general wave function for the wave packet and then find the variances?
     
  5. Jan 2, 2015 #4

    jfizzix

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    I would think so, if you know what the wavefunction of the wave packet is. I can't say off the top of my head what that would be, though.
     
  6. Jan 2, 2015 #5
    The question, as formulated, does not provide any additional information other than that in #1. Any ideas?
     
  7. Jan 2, 2015 #6

    jfizzix

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    I'm guessing that there is a standard formula for a free wave packet of given [itex]\Delta x[/itex] and [itex]p_{0}[/itex], to which the problem is referring.

    Failing that, the best I can think of is to find a limit to the momentum uncertainty with [itex]\Delta p\geq \frac{\hbar}{2 \Delta x}[/itex].

    Then, knowing [itex]p_{0}[/itex] and a bound for [itex]\Delta p[/itex], you can use that [itex]E=\frac{p^{2}}{2m}[/itex] and the propagation of uncertainty

    [itex](\Delta E)^{2}\approx (\frac{\partial E}{\partial p})^{2}(\Delta p)^{2}[/itex]

    Getting an estimate for the time uncertainty is a bit trickier (since time is not an observable), but you can say that

    [itex]\Delta t = \frac{\Delta x}{(\frac{d <x>}{dt})}[/itex]

    That's about all I can say on the matter, though.
     
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