Checkpoint: Transition Rates

This checkpoint ties together the chain of ideas in the module: matrix element → golden-rule rate → exponential decay → lifetime. These are the quantities you actually measure for a decaying state.

From rate to exponential decay

Fermi's golden rule gives a constant transition rate $W$ out of the initial state. When the only thing that can happen is decay, the population obeys a simple first-order rate equation,

\frac{dN}{dt} = -W\,N \quad\Longrightarrow\quad N(t) = N_0\,e^{-Wt} .

The constancy of $W$ is exactly what makes the decay exponential — the hallmark of irreversible decay into a continuum, in contrast to the oscillatory behavior of a two-level system with no continuum (Rabi flopping).

Lifetime and survival probability

The mean lifetime is the reciprocal of the rate,

\tau = \frac{1}{W} ,

and the probability that a given system has not yet decayed by time $t$ is the survival probability

P_{\text{survive}}(t) = e^{-Wt} = e^{-t/\tau} .

After exactly one lifetime, $t=\tau$ , this is $e^{-1}\approx 0.368$ — about 37% of the population remains, independent of the specific numbers. This universal ratio is a good sanity check on any decay calculation.

The natural linewidth

A finite lifetime $\tau$ implies, by the time–energy uncertainty relation, a finite spread in the emitted energy: $\Gamma = \hbar W = \hbar/\tau$ is the natural linewidth of the transition. Short-lived states (large $W$ ) emit broad spectral lines; long-lived states emit sharp ones. This is the same time–frequency reciprocity we met in the resonance lineshape.

Try it

An excited state decays into a continuum with $|V_{fi}|=0.05$ and $\rho(E_f)=80.0$ (units $\hbar=1$ ). Compute the golden-rule rate $W=(2\pi/\hbar)|V_{fi}|^2\rho$ , the lifetime $\tau=1/W$ , and the survival probability after one lifetime $P_{\text{survive}}(\tau)=e^{-W\tau}$ . return that survival probability. (You should get $e^{-1}$ .)

Run your code to see the quantum state.