Level Width

the uncertainty in the energy of a quantum-mechanical system having discrete energy levels ℰ_k in a state that is not strictly stationary. The system may be an atom, a molecule, or an atomic nucleus.

The level width Δℰ_k characterizes the broadening of an energy level and the energy spread in the level. It depends on the mean life of a system in a given energy state—that is, on the lifetime τ_k of the system in the level—and, in accordance with the uncertainty relation for energy and time, is given by the expression Δℰ_k ≈ ℏ/τ_k, where ℏ is Planck’s constant. For a stationary state of a system, τ_k = ∞ and Δℰ_k = 0. The lifetime τ_k and, consequently, the level width result from the possibility of quantum transitions of the system to states of lower energy.

For a free system, such as a single atom, spontaneous radiative transitions from level ℰ_k to lower-lying levels ℰ_i (ℰ_i < ℰ_k) determine the radiation width, or the natural width, of the level: (Δℰ_k)_rad ≈ ℏA_k, where A_k = ∑_iA_ki is the total probability of spontaneous emission from level ℰ_k; A_ki is Einstein’s coefficient for spontaneous emission. The broadening of an energy level may also be caused by spontaneous nonradiative transitions, for example, by the alpha decay of a radioactive atomic nucleus. The width of an energy level of an atom is very small in comparison with the energy of the level.

In other cases, the level width may become comparable to the level spacing. For example, the probability of quantum transitions in excited nuclei is determined by the emission of neutrons and is very high.

Any interactions that increase the probability of a transition of a system to other states result in additional broadening of the system’s energy levels. The Stark broadening of the energy levels of an atom or ion in a plasma as a result of collisions with ions or electrons may be cited as an example. In the general case, the total level width is proportional to the sum of the probabilities of all possible transitions—both spontaneous transitions and transitions induced by various interactions—from a given energy level.

The level width determines the line width of a spectral line.