A particle with a lifetime $\tau$ and rest energy $E_0$ has a wavefunction, in natural units: $$ \psi(t) = \psi_0 e^{-iE_0 t} e^{-t/2\tau}.$$
The modulus squares of its Fourier transform $|\hat{\psi(E)}|^2$ gives the probability density of finding the particle at energy $E$:
$$ |\hat{\psi(E)}|^2 = \frac{\psi_0}{(E_0-E)^2 + \frac{1}{4\tau^2}}. $$
But if the particle has infinite lifetime, i.e. it is stable with $\tau \rightarrow \inf$, why does it still have a finite spread in energy space?