- #1

tryingtolearn1

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- Homework Statement:
- Muons decay time distribution

- Relevant Equations:
- ##N(t) = N_0 exp(−\lambda t)## and ##D(t) = \lambda \exp(−\lambda t)##

I know for muons that the the probability that a muon decays in some small time interval ##dt## is ##\lambda dt##, where ##\lambda## is a decay rate. Thus the change in the population of muons is just ##dN/N(t) = −\lambda dt##. Integrating gives ##N(t) = N_0 \exp(−\lambda t)##. This makes sense to me but my book goes on to say the following,

What exactly is that explaining? Don't we need to know what ##\lambda## is before using the ##D(t)## equation? Because trying to find ##\lambda## using ##D(t) = \lambda \exp(−\lambda t)## will give the wrong results.

By decay time distribution D(t), we mean that the time-dependent probability that a muon decays in the time interval between ##t## and ##t + dt## is given by ##D(t)dt##. If we had started with ##N_0## muons, then the fraction ##−dN/N_0## that would on average decay in the time interval between ##t## and ##t + dt## is just given by differentiating the above relation: ##−dN = N_0\lambda \exp(−\lambda t) dt## ##\therefore## ##−dN/ N_0 = \lambda \exp(−\lambda t) dt##. The left-hand side of the last equation is nothing more than the decay probability, so ##D(t) = \lambda \exp(−\lambda t)##.

What exactly is that explaining? Don't we need to know what ##\lambda## is before using the ##D(t)## equation? Because trying to find ##\lambda## using ##D(t) = \lambda \exp(−\lambda t)## will give the wrong results.