For the understanding of irreversibility at the quantum level, the formation and decay of transient (unstable) states play a fundamental role. If the system is treated within Hermitian quantum mechanics, the resulting energy distribution of the resonance state, whose Fourier transform yields the time-dependent probability of decay,
P(t), is real. The physical constraint of the lower bound in the energy spectrum introduces ``memory,'' and causes nonexponential decay (NED) to set in after $t\ensuremath{\gg}\ensuremath{\tau},$ where \ensuremath{\tau} is the lifetime defined by exponential decay. The closer to threshold the decaying state is, the earlier NED appears. Apart from the constraint of
E> 0, the constraint of
t> 0 must be accounted for at the same time. It results from the discontinuity at
t=0 of the solution of the time-dependent Schr\"odinger equation, which breaks the unitarity associated with the S matrix and gives rise to a complex energy distribution, as a manifestation of the non-Hermitian property of the decaying states. For a narrow isolated resonance state, for which the self-energy is essentially energy-independent, analytic results for
PNED(t) obtained from semiclassical path-integral calculations agree with the quantum-mechanical ones when both physical constraints
E>0 and
t>0 are taken into account. As an example of the difference in the magnitude of the
PNED(t) when using a real and a complex energy distribution, application is made to the decay of the unstable ${\mathrm{He}}_{2}^{2+}{}^{1}{\ensuremath{\Sigma}}_{g}^{+}$ ground molecular state.
Approximately 40 years ago it was realized that the time development of decaying systems might not be precisely exponential. Winter [Phys. Rev. 123, 1503 (1961)] analyzed the simplest nontrivial system---a particle tunneling out of a well formed by a wall and $\ensuremath{\delta}$ function. He calculated the probability current just outside the well and found irregular oscillations on a short-time scale followed by an exponential decrease followed by more oscillations and finally by a decrease as a power of the time. We have reanalyzed this system, concentrating on the survival probability of the particle in the well rather than the probability current, and find a different short-time behavior.
To elucidate the time development of quasi-stationary states, a simple barrier penetration problem has been studied. Both approximate expressions and numerical results for some parameters were obtained for the decay rate. First, irregular oscillations occur for a short time. Second, the exponential region follows. Third, further oscillations occur during which the decay rate dips to negative values, so that the probability of finding the undecayed system increases briefly at several times. Fourth and finally, the decay rate decreases like an inverse power of the time.