Time-Energy Uncertainty Principle: Info & Derivation

[Rajini]

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

Thanks.

 

[Meir Achuz]

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.

 

[Count Iblis]

There is no time energy uncertainty relation like that at all! See e.g. here:

http://arxiv.org/abs/quant-ph/0609163

Pages 6, 7 and 8.

 

[Count Iblis]

And here:

http://prola.aps.org/abstract/PR/v122/i5/p1649_1

 

[dx]

In quantum mechanics, energy eigenstates have a time dependence of the form \exp(i\omega t). 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

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

where \tilde{A} 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, \tilde{A}(\omega) will be finite in region of size Δω ~ 1/Δt, or (using E = \hbar \omega)

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".

 

[Meir Achuz]

There is no time energy uncertainty relation like that at all! See e.g. here:

http://arxiv.org/abs/quant-ph/0609163

Pages 6, 7 and 8.

Regardless of formalism, the natural width of a spectral line is related to the lifetime of the state by \Delta E\Delta t\sim\hbar.

 

[Rajini]

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..
Thanks

 

[Count Iblis]

Regardless of formalism, the natural width of a spectral line is related to the lifetime of the state by \Delta E\Delta t\sim\hbar.

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.