Line Profiles

The previous discussion suggests that spectral lines would be infinitely narrow and sharp. In reality, however, they are somewhat broadened. We will now consider briefly the factors affecting the shape of a spectral line, called a line profile. An exact treatment would take us too deep into quantum mechanics, so we cannot go into the details here.

According to quantum mechanics everything cannot be measured accurately at the same time. For example, even in principle, there is no way to determine the x coordinate and the momentum px in the direction of the x axis with arbitrary precision simultaneously. These quantities have small uncertainties Ax and A px, such that

A similar relation holds for other directions, too. Time and energy are also connected by an uncertainty relation,

The natural width of spectral lines is a consequence of this Heisenberg uncertainty principle.

If the average lifetime of an excitation state is T, the energy corresponding to the transition can only be determined with an accuracy of AE = h/T = h/(2nT). From (5.1) it follows that Av = AE/h.In fact, the uncertainty of the energy depends on the lifetimes of both the initial and final states. The natural width of a line is defined as

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