The h index has been comprehensively developed and diffusely applied following its initial proposal in 2005 by Jorge Hirsch. However, while the numbers of papers and authors measured continue to grow, it is not known whether such increases in quantity mean that the research itself is advancing. Accordingly, the present study sets out to establish an analysis method, based on the concept of entropy, with which to elucidate the evolution of the h index. The results suggest that there are four stages, as follows: the emerging stage, the preliminary growth stage, the real growth stage, and the maturity stage. Meanwhile, the research of the h index is found to be in flux, and it is concluded that further development of the metric will require profound theoretical achievements as well as embedded innovation, or else the evolution of the h index will decline following a shortened maturity stage.