Distinguish between a metastable state and a normally allowed energy state within an atom. Discuss the role of metastable states in the operation of laser systems.

Normally allowed energy states within an atom are the stable states in which the electrons remain for a relatively long time. In these states, electrons are arranged in the lowest possible energy levels, with each energy level filled in accordance with the rules of quantum mechanics. Metastable states, on the other hand, are excited energy states that lie within the continuous range of energies that are partially allowed due to electron-electron interactions and other phenomena. These states have lifetimes that are significantly longer than typical excited states, yet shorter than ground state lifetimes. This means that electrons in metastable states will eventually decay to lower energy levels, but the process takes much longer than for normal excited states.

In a laser system, the key to generating coherent beams of light is to achieve what is called a "population inversion." This means that a larger number of atoms or molecules are in a higher excited state than in ground or lower energy states. Achieving population inversion is crucial for stimulated emission to occur, in which an incoming photon can interact with an excited atom and cause it to release another photon of identical properties (wavelength, phase, direction). Metastable states play a vital role in achieving this population inversion. In a three-level laser system, the energy of the pumping source (usually light or electric current) is used to promote electrons to an excited state, which then quickly decay to a metastable state. Because the lifetime of this metastable state is relatively long, there is time for several atoms to accumulate in this state before they decay back to the ground state through stimulated emission. This creates the population inversion necessary for lasing action to occur. In summary, metastable states are essential in enabling the required population inversion for laser systems to function effectively, leading to the generation of coherent light output.