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arXiv · 2403.15555

Galilean relativity and wave-particle duality imply the Schr\"odinger equation

Gustavo Rigolin
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Abstract

We show that the Schr\"odinger equation can be derived assuming the Galilean covariance of a generic wave equation and the validity of the de Broglie's wave-particle duality hypothesis. We also obtain from this set of assumptions the transformation law for the wave function under a Galilean boost and prove that complex wave functions are unavoidable for a consistent description of a physical system. The extension to the relativistic domain of the above analysis is also provided. We show that Lorentz covariance and wave-particle duality are consistent with two different transformation laws for the wave function under a Lorentz boost. This leads to two different wave equations, namely, the Klein-Gordon equation and the Lorentz covariant Schr\"odinger equation.

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Gustavo Rigolin. 2024-03-22. Galilean relativity and wave-particle duality imply the Schr\"odinger equation. https://doi.org/10.1103/physreva.109.052209

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