Heisenberg - Uncertainty principle - lifetime of a particle

[fab13]

I have seen that the more a particle has a high energy, i.e $E$, the more its lifetime is short, respecting so the uncertainty principle.

But by the definition of this uncertainty principle :

$E\,\Delta t \geq \dfrac{\hbar}{2}$, I can write :

$\Delta t \geq \dfrac{\hbar}{2E}$, then $\Delta t$ has a lower limit and not an upper limit.

If this was a upper limit, this would mean that $\Delta t$, i.e. the apparition time, should be observed in a time interval lower than $\dfrac{\hbar}{2E}$ : for example, if a detector had a time resolution greater than $\dfrac{\hbar}{2E}$, the particle could not be detected, could it ?

So in which case can we write : $\Delta t \leq \dfrac{\hbar}{2E}$ ??

It seems that I have confusions with this principle.
Any clarification is welcome

 

[hilbert2]

I think the correct version of this states that when a particle or system has a short lifetime, then it has a large uncertainty $\Delta E$ in the energy. This can be seen in e.g. the collision cross sections of particle reactions. So it's the $\Delta E$, not $E$ that we're talking about.

This kind of uncertainty relation is not the same as the momentum-position uncertainty, because the time $t$ is not a real observable.

 

[jfizzix]

The energy-time uncertainty relation:

\Delta E \Delta t \geq\frac{\hbar}{2}

is special because \Delta t is actually defined relative to some observable \hat{L} (whichever one you want) as the approximate amount of time it takes the statistics of \hat{L} to drift by one standard deviation \Delta L. More formally:

\Delta t \equiv \frac{\Delta L}{\Big|\langle\frac{d \hat{L}}{dt}\rangle\Big|}

where \langle \cdot \rangle denotes expectation value or mean value.

As far as particle lifetimes go, you can use the corresponding relation derived from this one which gives bounds on the half-life \tau_{1/2} of a given quantum state:

\tau_{1/2}\Delta H \geq \frac{\pi\hbar}{4}

Where particle decays are transitions from one quantum state to another, this relation gives a solid lower bound to the half-life of an unstable particle given the standard deviation of its energy \Delta H

For a well-written, if technical discussion of the energy-time uncertainty relation, see:
https://arxiv.org/pdf/quant-ph/0105049.pdf