From the Hamilton-Jacobi equation to the Schrödinger equation and vice versa, without additional terms and approximations

Authors

  • J.D. Bulnes Dep. Ciencias Exatas e Tecnologia, Universidade Federal do Amapa, Rod. J. Kubitschek, 68903-419, Macapa, AP, Brazil
  • M.A.I. Travassos Department of Scientific Research, Innovation and Training of Scientific and Pedagogical Staff, University of Economics and Pedagogy, Karshi, 180100, Uzbekistan
  • D.A. Juraev Department of Mathematics, Anand International College of Engineering, Near Kanota, Agra Road, Jaipur-303012, Rajasthan, India
  • J. Lopez-Bonilla ESIME-Zacatenco, Instituto Politecnico Nacional, Edif. 4, 1er. Piso, Col. Lindavista CP 07738, CDMX, Mexico

DOI:

https://doi.org/10.3126/hp.v11i1.66159

Abstract

In this article, we will answer a question posed in the book Classical Mechanics by H. Goldstein: ``"Is the Hamilton-Jacobi equation the short wavelength limit of the Schrödinger equation?" But, before that, we will identify an essential element that will take us from the Hamilton-Jacobi equation to the dynamic equation of non-relativistic quantum mechanics for a function ψ through an exact procedure. This element is the linear independence of the functions ψ and ψ* (their complex conjugate). Their independence is demonstrated for physical systems where the acting physical potential does not explicitly depend on time. Proceeding in reverse, from the Schrödinger equation, we obtain the Hamilton-Jacobi equation, exactly, without additional terms.

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Published

2024-05-24

How to Cite

Bulnes, J., Travassos, M., Juraev, D., & Lopez-Bonilla, J. (2024). From the Hamilton-Jacobi equation to the Schrödinger equation and vice versa, without additional terms and approximations. Himalayan Physics, 11(1). https://doi.org/10.3126/hp.v11i1.66159

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Research Articles