One view of quantitative science is that it is about representing the observable world by equations. Solutions then articulate our understanding and allow us to make predictions. Most equations are too complex to solve without the aid of a computer, but when the problem can be made simple enough that closed form expressions are possible, they greatly aid our understanding. The power of these results is often best appreciated by those performing experiments, or engaged in engineering projects, where clear interpretative models are crucial.
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Figure: Schematic of an inhomogeneously broadened lineshape with "hole burning", where one part (darker shaded) can be saturated. [courtesy Marshall Stoneham; doi: 10.1103/RevModPhys.41.82]