This article explains how a simple equation describing the time evolution of a superfluid’s quantum phase can be used as a powerful tool to rapidly deduce exact magnitudes of physical observables in situations involving very complex vortex motion. The equation has been used to simplify the understanding of new phenomena, such as vortex precession. Applications of the equation have also led to solutions of long-standing problems, such as the nucleation of vortices, and to new technology, such as the superfluid gyroscope. The article begins by presenting some basic ideas of superfluidity. This discussion leads to the concept of the quantum phase of the superfluid state and to the prediction that vorticity is quantized. Next a discussion is presented of the phase evolution equation, introduced by Josephson and developed further by Anderson. The utility of the equation is demonstrated by making certain general predictions about the consequences of vortex motion. Two experiments, one in ${}^3\mathrm{He}$ and another in ${}^4\mathrm{He},$ are then described in the context of the phase evolution equation. In the first, the precession frequency of a single vortex filament is easily explained in the context of the equation. In the second, quantized dissipation processes are observed which give detailed information about the creation of quantized vortices. The article concludes by showing how these latter experiments have led to the development of a superfluid sensor of absolute rotation. Although the article focuses on results emerging from the author’s own laboratory, the footnotes lead the reader to some of the parallel ongoing projects, especially in the case of the ${}^4\mathrm{He}$ research at Orsay/Saclay in France, the University of Minnesota, and the University of Trento, Italy.

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