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Uncertainty rule

  1. Aug 30, 2009 #1
    Dear PF members,
    I want to know some accurate informations regarding the time-energy uncertainty principle.
    From several websites i got that [tex]\Delta[/tex]E[tex]\Delta[/tex]t[tex]\geq[/tex][tex]\hbar[/tex]/2 (for e.g., hyperphysics, wiki, etc.).
    But in some books they use [tex]\Delta[/tex]E[tex]\Delta[/tex]t[tex]\geq[/tex][tex]\hbar[/tex].
    Can anyone clear this why it is like that...Also is there any small derivation for that?

  2. jcsd
  3. Aug 31, 2009 #2


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    The uncertainty is of order hbar. The 1/2 is the absolute minimum for a Gaussian distribution in time and energy, which is not usually the case for energy and time.
    Some books just don't bother with factors like 1/'2 when giving order of magnitude lower limits.
  4. Aug 31, 2009 #3
    There is no time energy uncertainty relation like that at all! See e.g. here:


    Pages 6, 7 and 8.
  5. Aug 31, 2009 #4
  6. Aug 31, 2009 #5


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    In quantum mechanics, energy eigenstates have a time dependence of the form [tex] \exp(i\omega t) [/tex]. Since all solutions to the dynamical equation (Schrodinger equation) are superpositions of energy eigenstates (on spacetime), the time dependence of an amplitude will be generally of the form

    [tex] A(t) = \int_{-\infty}^{\infty} \tilde{A}(\omega) e^{i\omega t} d\omega [/tex]

    where [tex] \tilde{A} [/tex] is the Fourier transform of A(t). If A(t) is mostly finite only in a region of size Δt, then by familiar properties of the Fourier transform, [tex] \tilde{A}(\omega) [/tex] will be finite in region of size Δω ~ 1/Δt, or (using [tex] E = \hbar \omega [/tex])

    ΔE Δt ~ h

    The precise constant of proportionality depends on the definition of 'Δ', i.e. what we mean by "mostly finite only in a region of size Δt".
  7. Aug 31, 2009 #6


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    Regardless of formalism, the natural width of a spectral line is related to the lifetime of the state by [tex]\Delta E\Delta t\sim\hbar[/tex].
  8. Sep 7, 2009 #7
    Hi Dx,
    thanks for your reply..Now i understand..abour delta.
    Clem..the link that you send are good..But one should write properly and precisely ...since hbar is very small..
  9. Sep 7, 2009 #8
    Yes, I agree. The problem is that this is not a universal result. In general, there is no energy time uncertainty relation of this simple form.
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