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A metastable state of excitation (e.g., a metastable energy level of an orbiting electron) is one that lasts longer than typical (e.g., lasts seconds, minutes, or hours rather than a tiny fraction of a second). Examples regarding atoms:
Such atomic metastable states are often associated with forbidden lines: the state can be so delicate that any disturbance is likely to cause them to relax, and the state is unlikely to live out an undisturbed life due to frequent interaction with other molecules. In a sufficient vacuum, such interaction is less likely and the spectral line is more likely to form.
Metastability is actually a more general concept, i.e., that small disturbances result in negative feedback, but disturbances on some larger scale would eventually receive positive feedback, i.e., lead to instability. In quantum mechanics, the potential "disturbance" can be a quantum-mechanical probability of some change, and metastability results in much longer half-lives (which is a way of expressing what happens to electrons in metastable orbits). The structure of an atomic nucleus (nuclear isomer) can be in a metastable state: an arrangement of the nucleons which is not at the minimal energy level, resulting in radioactivity with a long half-life (but in the case of nuclei, the "long half-life" may well still be a tiny fraction of a second).