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TimeEnergy Uncertainty Problem

  1. Nov 12, 2009 #1
    1. The problem statement, all variables and given/known data

    In laser physics a common approximation for the time-energy uncertainty is given by.

    [tex]\Delta \omega \cdot \tau \geq \approx 2 \pi[/tex]

    The problem is to use the Energy-Time Uncertainty relationship to derive a more exact answer than this.

    2. Relevant equations

    [tex]\sqrt{\left\langle E^2 \right\rangle \left\langle t^2 \right\rangle} \geq \frac{\hbar}{2}[/tex]

    It should be noted that we are supposed to use FWHM for the uncertainties, so the following relationship is supplied.

    [tex]\Delta_{FWHM} x = 2\sqrt{2 ln(2)}\sqrt{\left\langle x^2 \right\rangle}[/tex]

    3. The attempt at a solution

    [tex]\sqrt{\left\langle E^2 \right\rangle \left\langle t^2 \right\rangle} = \frac{\hbar}{2}[/tex]

    I substitute

    [tex]E = \hbar \omega[/tex]

    Giving

    [tex]\sqrt{\left\langle \hbar^2 \omega^2 \right\rangle \left\langle t^2 \right\rangle} = \frac{\hbar}{2}[/tex]

    I divide with [tex]\hbar[/tex] on both sides

    [tex]\sqrt{\left\langle \omega^2 \right\rangle \left\langle t^2 \right\rangle} = \frac{1}{2}[/tex]

    We then substitute

    [tex]\tau = 2\sqrt{2 ln(2)} \sqrt{\left\langle t^2 \right\rangle}[/tex]

    And

    [tex]\Delta \omega = 2\sqrt{2 ln(2)} \sqrt{\left\langle \omega^2 \right\rangle}[/tex]

    Giving

    [tex]\Delta \omega \cdot \tau = 4 ln(2)[/tex]

    I don't know where I've done anything wrong, but I would appreciate some help a lot :3
     
  2. jcsd
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