Interferometric discrepancy between the non-relativistic solution to the Klein-Gordon and Schrödinger wave equations due to their dissimilar phase velocities
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866911524032872448 |
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| author | Kowalski, Frank Victor |
| author_facet | Kowalski, Frank Victor |
| contents | Adding a constant energy offset leaves classical dynamics unchanged. In quantum mechanics it changes the phase velocity of the wavefunction. The inclusion of the constant rest energy in the Klein-Gordon formulation leads to significantly higher phase velocities compared with the Schrödinger equation. The Schrödinger equation predicts an attenuation of the wavefunction along one of the paths in a Sagnac interferometer when a beamsplitter's trajectory along that path includes a segment where its speed exceeds the phase velocity of a free particle. Such an attenuation does not occur for electromagnetic waves nor for eigenstates of momentum in the Klein-Gordon equation since the speed of the beamsplitter cannot then exceed the phase velocity of the wave. This attenuation reduces the amplitude without introducing a phase shift, preserving the overall structure of the transmitted wave group. While a Klein-Gordon wave group undergoes three traversals of the beamsplitter that moved, it experiences attenuation equivalent to only a single pass, whereas the Schrödinger equation predicts the expected attenuation for three passes. This discrepancy highlights a fundamental incompatibility between the Klein-Gordon equation and the Schrödinger equation in a regime where their predictions should converge. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_08007 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Interferometric discrepancy between the non-relativistic solution to the Klein-Gordon and Schrödinger wave equations due to their dissimilar phase velocities Kowalski, Frank Victor Quantum Physics Adding a constant energy offset leaves classical dynamics unchanged. In quantum mechanics it changes the phase velocity of the wavefunction. The inclusion of the constant rest energy in the Klein-Gordon formulation leads to significantly higher phase velocities compared with the Schrödinger equation. The Schrödinger equation predicts an attenuation of the wavefunction along one of the paths in a Sagnac interferometer when a beamsplitter's trajectory along that path includes a segment where its speed exceeds the phase velocity of a free particle. Such an attenuation does not occur for electromagnetic waves nor for eigenstates of momentum in the Klein-Gordon equation since the speed of the beamsplitter cannot then exceed the phase velocity of the wave. This attenuation reduces the amplitude without introducing a phase shift, preserving the overall structure of the transmitted wave group. While a Klein-Gordon wave group undergoes three traversals of the beamsplitter that moved, it experiences attenuation equivalent to only a single pass, whereas the Schrödinger equation predicts the expected attenuation for three passes. This discrepancy highlights a fundamental incompatibility between the Klein-Gordon equation and the Schrödinger equation in a regime where their predictions should converge. |
| title | Interferometric discrepancy between the non-relativistic solution to the Klein-Gordon and Schrödinger wave equations due to their dissimilar phase velocities |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2601.08007 |