I Particle lifetime (half-life) question

[aaronll]

If I have a particle with a average lifetime of 15min, if I take 10 particles confined in a box, after 15 min there will be 5 particles.
After 15min 2.5 particles and so on... , but so, at the end there will be the last particle that decades.
That particle lived far longer than 15min, but is the same kind of other particles.
So why some particles lives less and some more? Are related to when they are "born" ( I know this word is bad)

And for particle with an enormous lifetime, e.g. 10^20 years, I can take a large number of particles, like 10^20, and see if there is a decay in some time interval,for example 1 years, if there is not i can say that almost its lifetime is 10^20 years.
But how those particle doesn't decay, for lucky, when I experiment on them?

thank you

 

[Vanadium 50]

If I have a particle with a average lifetime of 15min, if I take 10 particles confined in a box, after 15 min there will be 5 particles.

No. That's a half life, not an average lifetime.

So why some particles lives less and some more?

It's completely probabilistic.

 

[aaronll]

No. That's a half life, not an average lifetime.It's completely probabilistic.

Yes, I meant half life ( excuse me ).

So it's probabilistic, but... why? It's like when we calculate expectation value of an observable according to probabilistic interpretation of wave function? So we "found" some value with some probability to found if we measure it, but why we found a particular value it's a "mystery".
it's the same things?
To me thinking that a particle has a probabilistic lifetime it's very strange.Last question: if we take a particles, randomly (we cannot say when it was "born") so statistically we can observe it for a time equal to its average lifetime? We can observe some that disappear instantaneously, and some later than lifetime interval, but on average is the lifetime, is correct?

thanks

 

[Vanadium 50]

To me thinking that a particle has a probabilistic lifetime it's very strange.

It is strange. But that's how nature behaves.

Also, nuclei don't "wear out" the probability of a nucleus to decay is independent of how old it is.

 

[mfb]

Last question: if we take a particles, randomly (we cannot say when it was "born") so statistically we can observe it for a time equal to its average lifetime? We can observe some that disappear instantaneously, and some later than lifetime interval, but on average is the lifetime, is correct?

Right. And it doesn't matter when it was born, as long as it exists when you start the measurement.

 

[jtbell]

If I have a particle with a average lifetime of 15min, if I take 10 particles confined in a box, after 15 min there will be 5 particles.

As noted above, you really mean the half-life, $t_{1/2}$, not the average lifetime.

Also, nuclei don't "wear out" the probability of a nucleus to decay is independent of how old it is.

In general, a half-life corresponds to a decay constant (probability of decay, per unit time) of $$\lambda = \frac {-\ln 0.5} {t_{1/2}}$$ For the given example, $$\lambda = \frac {0.693} {15~\rm{min}} = 0.0462~\rm{min^{-1}}$$ If you pick any particle, it has a probability of 0.0462 = 4.62% of decaying within the next minute, regardless of how long it has lived already.

 

[vanhees71]

If a particle is certainly present at $t=0$ then the probability that it decays within the time interval $[0,t]$ is
$$P_{\text{decay}}=1-\exp(-\lambda t).$$

 

[jtbell]

Oops, I shouldn't have blindly assumed the first-order approximation.

The exact calculation above gives P = 0.0451 = 4.51%.