The p-Adic Schrödinger Equation and the Two-slit Experiment in Quantum Mechanics
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866910513464606720 |
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| author | Zúñiga-Galindo, W. A. |
| author_facet | Zúñiga-Galindo, W. A. |
| contents | p-Adic quantum mechanics is constructed from the Dirac-von Neumann axioms identifying quantum states with square-integrable functions on the N-dimensional p-adic space. This choice is equivalent to the hypothesis of the discreteness of the space. The time is assumed to be a real variable. The p-adic quantum mechanics is motivated by the question: what happens with the standard quantum mechanics if the space has a discrete nature? The time evolution of a quantum state is controlled by a nonlocal Schrödinger equation obtained from a p-adic heat equation by a temporal Wick rotation. This p-adic heat equation describes a particle performing a random motion in the N-dimensional p-adic space. The Hamiltonian is a nonlocal operator; thus, the Schrödinger equation describes the evolution of a quantum state under nonlocal interactions. In this framework, the Schrödinger equation admits complex-valued plane wave solutions, which we interpret as p-adic de Broglie waves. These mathematical waves have all wavelength 1/p. In the p-adic framework, the double-slit experiment cannot be explained using the interference of the de Broglie waves. The wavefunctions can be represented as convergent series in the de Broglie waves, but the p-adic de Broglie waves are just mathematical objects. Only the square of the modulus of a wave function has a physical meaning as a time-dependent probability density. These probability densities exhibit interference patterns similar to the ones produced by `quantum waves.' In the p-adic framework, in the double-slit experiment, each particle goes through one slit only. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2308_01283 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The p-Adic Schrödinger Equation and the Two-slit Experiment in Quantum Mechanics Zúñiga-Galindo, W. A. Quantum Physics High Energy Physics - Theory Primary: 81Q35, 81Q65. Secondary: 26E30 p-Adic quantum mechanics is constructed from the Dirac-von Neumann axioms identifying quantum states with square-integrable functions on the N-dimensional p-adic space. This choice is equivalent to the hypothesis of the discreteness of the space. The time is assumed to be a real variable. The p-adic quantum mechanics is motivated by the question: what happens with the standard quantum mechanics if the space has a discrete nature? The time evolution of a quantum state is controlled by a nonlocal Schrödinger equation obtained from a p-adic heat equation by a temporal Wick rotation. This p-adic heat equation describes a particle performing a random motion in the N-dimensional p-adic space. The Hamiltonian is a nonlocal operator; thus, the Schrödinger equation describes the evolution of a quantum state under nonlocal interactions. In this framework, the Schrödinger equation admits complex-valued plane wave solutions, which we interpret as p-adic de Broglie waves. These mathematical waves have all wavelength 1/p. In the p-adic framework, the double-slit experiment cannot be explained using the interference of the de Broglie waves. The wavefunctions can be represented as convergent series in the de Broglie waves, but the p-adic de Broglie waves are just mathematical objects. Only the square of the modulus of a wave function has a physical meaning as a time-dependent probability density. These probability densities exhibit interference patterns similar to the ones produced by `quantum waves.' In the p-adic framework, in the double-slit experiment, each particle goes through one slit only. |
| title | The p-Adic Schrödinger Equation and the Two-slit Experiment in Quantum Mechanics |
| topic | Quantum Physics High Energy Physics - Theory Primary: 81Q35, 81Q65. Secondary: 26E30 |
| url | https://arxiv.org/abs/2308.01283 |