Galilean relativity and wave-particle duality imply the Schrödinger equation

• URL: http://arxiv.org/abs/2403.15555v2
• Date: Fri, 10 May 2024 17:58:22 GMT
• Title: Galilean relativity and wave-particle duality imply the Schrödinger equation
• Authors: Gustavo Rigolin,
• Abstract summary: We show that complex wave functions are unavoidable for a consistent description of a physical system. This leads to two different wave equations, namely, the Klein-Gordon equation and the Lorentz covariant Schr"odinger equation.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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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