The identification of particle velocity with the velocity of a wave group was a crucial assumption that resolved important inconsistencies in the theoretical developments of Louis de Broglie and Erwin Schrödinger. Interestingly, this was one of the few common aspects of their work. In this paper, we present a reconstruction of how group velocity became essential for both de Broglie and Schrödinger, as well as a comparison of important differences in their theories. Pedagogical lessons to be extracted from this episode are also outlined.

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The distinction between phase and group velocity had been pointed out by Hamilton and Stokes in the 19th century, and it was introduced by Rayleigh in the physical context in his Theory of Sound, written in 1877. We thank one of the anonymous reviewers for this comment.
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In this paper, we follow Louis de Broglie's terminology of “rest mass.” However, it should be mentioned that the concept of rest mass has been widely criticized in the literature, since it is more coherent to treat mass as a relativistic invariant. For more details, see
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11.
Equation (7) is another formulation of the usual expression for group velocity as g=ωk, since ω=2πν and k=2πνu. De Broglie's original derivation of this expression in his doctoral thesis is presented in  Appendix A. See also Chapter 48 (Vol. 1) of the Feynman lectures (<https://www.feynmanlectures.caltech.edu/>) for a pedagogical explanation of group velocity.
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This inconsistency in de Broglie's use of wave groups has been pointed out for the first time, to the best of our knowledge, by MacKinnon (Ref. 3), but there seems to be no consensus in the literature. We are aligned with the arguments given by Brown and Martins (Ref. 5) on this matter and strongly recommend the interested reader to consult their paper to get a deeper understanding of the nuances of this problem.
14.
This is not so simple as it may seem. In general, one cannot insert the relativistic expression of momentum in the nonrelativistic formulation of Maupertuis's principle. Since de Broglie's approach is essentially relativistic, he needs the relativistic version of this principle. However, he manages to relate these two versions and justify his use of Eq. (14). This is discussed in detail in  Appendix B.
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It is important to stress that Schrödinger's research program was relativistic from the very beginning (Ref. 16) and was even guided by general relativity (Ref. 17). Despite that, Schrödinger's original derivation in the 1926 papers was completely nonrelativistic. Although he had first obtained a relativistic equation (as it was found in his notebooks), the main motivation for not publishing it was due to the fact that it could not fully account for the Hydrogen spectrum (Ref. 16).
22.
It should be noted that Schrödinger knew the concept of group velocity even before his acquaintance with the works of de Broglie (E. Schrödinger, “Zur akustik der atmosphäre,” Phys. Z. 18, 445–453 (1917)). The importance of de Broglie's work was drawn to Schrödinger's attention by A. Einstein. Schrödinger even used the name de Broglie–Einstein wave theory (E. Schrödinger, “Zur Einsteinschen Gastheorie.” Phys. Z. 27, 95–101 (1926)). We thank one of the anonymous reviewers for this comment.
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Schrödinger derives his time-dependent equation by expressing, once again, the time dependence of the wave function as a complex exponential ψe2πiEt/h and differentiating it once with respect to time ψ̇=2πiEhψ, which enables him to eliminate the parameter E from his time-independent equation. By substituting Eψ=h2πiψ̇ in Eq. (30), the more familiar time-dependent Schrödinger equation is obtained (Ref. 8, p. 22).
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The interested reader can find an excellent explanation of Schrödinger's “micro-macro” paper in Chapter 14.6
of
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