Dale said:
In principle, a radioactive element can still decay even when it is in a stationary state. And in practice, in many atomic clocks any atomic motion degrades the time keeping.
This is a bit misleading. So let's see, how radioactive decay is described within QT. Let's take for simplicity the ##\beta## decay of an atom, which in fact on the most fundamental basis according to our present best knowledge, which is the Standard Model of elementary-particle physics, is the process that a down-quark within a neutron in the nucleus decays to an up-quark, an electron, and an anti-electron-neutrino.
Since thus ##\beta## decay is a process where particles are annihilated (the down-quark) and new particles are created (the up-quark, electron, and anti-electron neutrino).
Now this process is described in perturbation theory, with the weak interaction of the quarks and leptons treated as a small correction to the overwhelmingly much stronger strong nuclear force. So first one considers the nucleus, as if there were only the strong interaction. It's a very complicated bound state of quarks and gluons, i.e., it's a very complicated eigenstate of the QCD Hamiltonian. This implies that the nucleus in this 0th-order approximation, i.e., neglecting the weak interaction completely, were stable, because it's in an eigenstate of the Hamiltonian (of only the strong interaction).
Now, if you consider the full thing, i.e., if you add the weak interaction into the treatment, it's clear that the QCD-energy eigenstate describing the nucleus without taking into account the weak interaction is no longer an eigenstate of the full Hamiltonian, including weak interactions. This means that due to the presence of the weak interaction there's indeed some probability that the ##\beta##-decay occurs, because the weak interaction couples the down quark to the up-quark, electron and anti-electron neutrino. That the corresponding decay can only happen if the down-quark is bound in a neutron and not in a proton is, because the neutron is a bit heavier than the proton, and thus the ##\beta## decay is forbidden, if the down-quark is in a proton, because of energy conservation, but for the down-quark within the neutron there's nothing (in fact no conservation law) which forbids this decay due to the weak interaction, and whatever is an allowed process in a theory, this theory predicts that this process will indeed happen spontaneously and randomly, i.e., we cannot predict, when a given nucleus will ##\beta##-decay, but quantum-field theory predicts the probability rate for such an decay to occur. In this way you can derive the mean lifetime (or equivalently the half-life) of the nucleus, applying the Wigner-Weisskopf approximation, which leads to the well-known exponential-decay law.