Equation related to the radioactive decay being random

Question

How is time decay randomness calculated? it's similar that on variables like position?

If we describe a system with $\phi\left(x,t\right)$ as wave function, $\left|\phi\left(x,t\right)\right|^2$ as probability. From probability density we can calculate probability of a given position. We can answer questions like "what's the probability of measuring a particle in $x_0$ (spatial coordinate)?". So uncertainty is about $x$.

In a radioactive decay, uncertainty is in time of occurrence, it's not about position, that's this question about, what are the equation related to this probability computation? Is there a "time operator"?

It's easy to confuse randomness in position with time, because given any fixed $x_0$ (or a neighborhood zone), we will have to wait random time to get a hit there, of course, but that's not this question about!, that's spatial distribution of probability, the "when" of a radioactive decay event doesn't depend on "where" it hits.

Answer 1

The exponential decay formalism typically describes persistence of populations, so it is a probability distribution of cumulative outcome, so persistence probability, not transition instant. In a naive 2-state system, it could amount, schematically, to something of the sort, $$ |\langle \psi| e^{-iHt/\hbar}|\psi\rangle|^2 \sim e^{-t/\tau}/\tau. $$

Of course, in realistic situations, there are more final states involved, and more elaborate formulas computable via Fermi's golden rule, itself a consequence of time-dependent perturbation theory, but it is not clear to me these methods are what you are asking about, specifically.

For one particle, the same distribution specifies the distribution of decay times, with τ as the mean time of the particle's demise, the lifetime.

If the rate is small (rare events), it is a random, (homogeneous) Poisson process; are you asking about the QM determination of the salient parameters (like the rate 1/τ)? It underlies the above exponential decay law. Typically, you may find the distribution of times between decays (Fig. 3) in the entire population.

• In QM, time is a parameter, not a dynamical variable, and as such there is no "operator" for it. The Hamiltonian operator effects time translations.

Answer 2

"Time decay randomness" is calculated by calculating the transition probability. Radioactive decay, like any quantum transition, is a stochastic process. We can calculate with what probability it occurs, but we we do not know when this probability will materialise.

This stochastic behaviour is the essence of quantum mechanics.