Heisenberg - Uncertainty principle - lifetime of a particle

  • #1
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Main Question or Discussion Point

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
 

Answers and Replies

  • #2
hilbert2
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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.
 
  • #3
jfizzix
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The energy-time uncertainty relation:

[itex]\Delta E \Delta t \geq\frac{\hbar}{2}[/itex]

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

[itex]\Delta t \equiv \frac{\Delta L}{\Big|\langle\frac{d \hat{L}}{dt}\rangle\Big|}[/itex]

where [itex]\langle \cdot \rangle[/itex] 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 [itex]\tau_{1/2}[/itex] of a given quantum state:

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

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 [itex]\Delta H[/itex]

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