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 $$\Delta$$E$$\Delta$$t$$\geq$$$$\hbar$$/2 (for e.g., hyperphysics, wiki, etc.).
But in some books they use $$\Delta$$E$$\Delta$$t$$\geq$$$$\hbar$$.
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 (Schrödinger 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.