# Heisenberg - Uncertainty principle - lifetime of a particle

• I
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

Gold Member
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.

Gold Member
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

Demystifier and vanhees71