Lifespan (more precisely: mean lifespan) in physics describes the average "lifespan" of the members of an ensemble of identical objects that are in the same state and isolated from one another.

If there is no state of lower energy available to an object and no energy is supplied to it, it is stable and its lifespan is infinite. However, if the objects can spontaneously change into a state of lower energy (“disintegrate”), their respective lifetimes form a frequency distribution , the arithmetic mean of which is the lifespan. Typically, one speaks of lifetime in connection with unstable particles , radioactive atomic nuclei , as well as with atoms and other systems in an excited state .

A closely related term is the half-life . In the life sciences , the term has life expectancy comparable importance.

## Lifespan and probability of decay

The probability density for a member of the ensemble to disintegrate usually follows an exponential distribution : ${\ displaystyle p}$

${\ displaystyle p (t) = \ lambda e ^ {- \ lambda t}}$

${\ displaystyle \ lambda}$is the decay constant . It is also called the probability of decay , but is a probability per unit of time and is usually given in the unit 1 / second. If several decay channels are possible, the total ( total ) decay constant is the sum of the corresponding individual ( partial ) decay constants.

The lifetime is the reciprocal of the decay constants: ${\ displaystyle \ tau}$

${\ displaystyle \ tau = {\ frac {1} {\ lambda}}}$

It is therefore the time after which the number of particles has dropped to the fraction 1 / e ≈ 0.368.

For elementary particles you get an overview of the different decay channels and decay probabilities in the Review of Particle Physics published by the Particle Data Group or in its short version, the Particle Physics Booklet .

## Partial life

If there are several decay channels, the reciprocal of each of the partial decay constants can be formally specified as “partial lifetime”; this is sometimes done for the sake of clarity. The partial lifetime is a fictitious, unobservable quantity: It would be the lifetime of the system if the decay channel in question were the only possible one. The observable decay always shows - regardless of which of the decay channels is observed - the lifetime that corresponds to the total decay constant.

## Half-life

Sometimes, especially in the radioactivity field, the half-life is used instead of the lifetime , i.e. H. the time after which half of the ensemble is still present. The half-life is calculated from the lifetime or the decay constant with the help of ${\ displaystyle T_ {1/2}}$

${\ displaystyle T_ {1/2} = \ tau \ ln 2 = {\ frac {\ ln 2} {\ lambda}}}$

It is about 69% of the service life. In the case of several decay channels, for the sake of clarity - as with the lifetime - fictitious partial half-lives are sometimes mentioned.

Half-lives and decay channels of radionuclides are z. B. specified in the Karlsruhe nuclide map . Branching ratios and other data can be found in the extensive book Table of Isotopes .

## Connection with quantum theory

Using Heisenberg's uncertainty principle , the following relationship can be found between the uncertainty of any observable and its development over time: ${\ displaystyle A}$

${\ displaystyle \ Delta E \ \ Delta A \ geq {\ frac {1} {2}} \ left | \ langle [H, A] \ rangle \ right | \ geq {\ frac {\ hbar} {2}} \ left | {\ frac {\ mathrm {d}} {\ mathrm {d} t}} \ langle A \ rangle \ right |}$

This results in a connection between the energy uncertainty or decay width of a transition or decay and its lifetime: ${\ displaystyle \ Gamma = 2 \ \ Delta E}$

${\ displaystyle \ Gamma \ \ tau = \ hbar}$

To determine very short lifetimes, the breadth of the energy distribution, for example of emitted photons or a peak in an excitation function , is measured and the lifetime is calculated using this formula. A double standard deviation of about 66 keV results in a service life of 10 −20  seconds.