The Analytical Function Defined in the Hamilton-Jacobi and Schrödinger Approach and the Classical Schrödinger Equation
DOI:
https://doi.org/10.22105/kmisj.v1i2.63Keywords:
Schrödinger ansatz, Hamilton-Jacobi equation, Analytic function, Laplace equation, Classical schrödinger equationAbstract
In this paper, we show that a specific extension of the Schrödinger ansatz for
two free particles, under the requirement that this function (ansatz) be analytical, is compatible with the expected physical result for a quantum description at the quantum-classical boundary; that is, its total erasure in the transition to a description compatible with classical physics. Using the Cauchy-Riemann relation, the Laplace equation, and the Hamilton-Jacobi equation, we have shown that this function verifies a classical equation arising from the time-dependent Schrödinger equation, since in the assumed context the time variable can be taken as a parameter since it is irrelevant in the process of approximation to the quantum-classical boundary.
References
A.I. Prilepko, D.G. Orlovsky, I.A. Vasin, Methods for solving inverse problems in mathematical physics, Marcel Dekker, Inc., New York, (2000).
R.M. Santilli, Foundations of Theoretical Mechanics I. The Inverse Problem in Newtonian Mechanics, Springer Science, (1978).
E. Schrödinger, Collected Papers on Wave Mechanics, Chelsea, New York, (1978).
L.D. Landau, E.M. Lifshitz, Quantum Mechanics, Addison-Wesley, Reading, (1965).
B. Bhalgamiya, M.A. Novotny, Exact solutions for the inverse problem of the time-independent Schrödinger equation, Am. J. Phys., 92:12, 975–979 (2024).
D.A. Juraev, E.E. Elsayed, J.D. Bulnes, P. Agarwal, R.K. Saeed, History of ill-posed problems and their application to solve various mathematical problems, Engineering Applications, 2:3, 279–290 (2023).
A. Rangel-Merino, R. Linares y Miranda, J. López-Bonilla, Genetic Algorithms and the Inverse Problem of Electromagnetic Radiation, Journal of the Institute of Engineering, 8:1, 231236 (2011).
J.D. Bulnes, M. Cavalcante, M.A.I. Travassos, J.L. López-Bonilla, Una simulación matemática del efecto Zeeman de un campo magnético, Journal de Ciencia e Ingeniería, 15:2, 24–32 (2023).
Y. Fayziev, Q. Buvaev, D. Juraev, N. Nuralieva, Sh. Sadullaeva, The inverse problem for determining the source function in the equation with the Riemann-Liouville fractional derivative, Global and Stochastic Analysis, 9:2, 4352 (2022).
D.A. Juraev, A.A. Tagiyeva, J.D. Bulnes and B. Drumea, On the regularization of the Cauchy problem for matrix factorizations of the Helmholtz equation in Rm, Karshi Multidisciplinary International Scientific Journal, 1:2, 51–68 (2024).
D.A. Juraev, N.M. Mammadzada and M. Israr, On the Application of Ill-posed problems of equations of mathematical physics, Karshi Multidisciplinary International Scientific Journal, 1:2, 43–50 (2024).
I.E. Niyozov, D.A. Juraev, R.F. Efendiev, M.J. Jalalov, The Cauchy Problem for the System of Elasticity, Journal of Contemporary Applied Mathematics, 14:2, 92–107 (2024).
A. Small and K.S. Lam, Simple derivations of the Hamilton-Jacobi equation and the eikonal equation without the use of canonical transformations, Am. J. Phys., 79:6, 678–681 (2011).
P.R. Sarma, Direct derivation of Schrödinger equation from Hamilton-Jacobi equation using uncertainty principle, Rom. Journ. Phys., 56:10, 1053-1056 (2010).
L . Kocis, Ehrenfest Theorem for the Hamilton-Jacobi Equation, Acta Physica Polonica A, 102:6, 709–716 (2002).
D. Derbes, Feynman’s derivation of the Schrödinger equation, Am. J. Phys., 64:7, 881–884 (1996).
H. Goldstein, Classical Mechanics, Addison Wesley, (1980).
J.D. Bulnes, M.A.I. Travassos, D.A. Juraev, J. López-Bonilla, From the Hamilton-Jacobi equation to the Schrödinger equation and vice versa, without additional terms and approximations, Himalayan Physics, 11, 21–27 (2024).
D.A. Juraev, N.M. Mammadzada, J.D. Bulnes, S.K. Gupta, G.A. Aghayeva, V.R. Ibrahimov, Regularization of the Cauchy problem for matrix factorizations of the Helmholtz equation in an unbounded domain, Mathematics and Systems Science, 2:2, 1–16 (2024).
D.A. Juraev, A.A. Tagiyeva, J.D. Bulnes and G.X.-G. Yue, On the solution of the Ill-posed Cauchy problem for elliptic systems of the first order, Karshi Multidisciplinary International Scientific Journal, 1:1, 17–26 (2024).
D.A. Juraev, P. Agarwal, A. Shokri, E.E. Elsayed, Integral formula for matrix factorizations of Helmholtz equation, Recent trends in fractional calculus and its applications, advanced studies in complex systems, Elsevier, pp. 123-146 (2024).
D.A. Juraev, P. Agarwal, A. Shokri, E.E. Elsayed, Cauchy problem for matrix factorizations of Helmholtz equation on a plane, Recent trends in fractional calculus and its applications, advanced studies in complex systems, Elsevier, pp. 147-175 (2024).
D.A. Juraev, P. Agarwal, A. Shokri, E.E. Elsayed, The Cauchy problem for matrix factorizations of Helmholtz equation in space, Recent trends in fractional calculus and its applications, advanced studies in complex systems, Elsevier, pp. 177-210 (2024).
D.A. Juraev, Y.S. Gasimov, On the Regularization Cauchy Problem for Matrix Factorizations of the Helmholtz Equation in a Multidimensional Bounded Domain, Azerbaijan Journal of Mathematics, 2:1, 142 (2022).
D.A. Juraev, Solution of the Ill-posed Cauchy problem for matrix factorizations of the Helmholtz equation on the plane, Global and Stochastic Analysis, 8:3, 1–17 (2021).
D.A. Juraev, P. Agarwal, A. Shokri, E.E. Elsayed, J.D. Bulnes, On the solution of the Ill-posed Cauchy problem for elliptic systems of the first order, Stochastic Modelling & Computational Sciences, 3, 1–21, (2023).
J.D. Bulnes, M.A.I. Travassos, D.A. Juraev, G. Dias and J. López-Bonilla. Mathematical simulation of à la Einstein loss of the non-separable character of a function for two particles at the quantum-classical boundary. Open Journal of Mathematical Sciences, 8, 113–127 (2024).
J.D. Bulnes, M.A.I. Travassos, D.A. Juraev, J.E. Ottoni, J. López-Bonilla, Stern-Gerlach Centennial: Parity, gradient effect, and an analogy with the Higgs field, Himalayan Physics, 11, 68–77 (2024).
J.D. Bulnes, An unusual quantum entanglement consistent with Schrödinger’s equation, Global and Stochastic Analysis, 9, 79–87 (2022).
J.D. Bulnes, M.A.I. Travassos, H.E. Caicedo-Ortiz, J. López-Bonilla, Más de un tipo de entrelazamiento en el mundo cuántico y ninguno en el mundo clásico, Journal de Ciencia e Ingeniería, 16:1, 5–12 (2024).
J.D. Bulnes, D.A. Juraev, J.L. López-Bonilla, M.A.I. Travassos, Exact de coupling of a coupled system of two stationary Schrodinger equation, Stochastic Modelling Computational Sciences, 3, 23–28 (2023).
J.D. Bulnes, J.L. López-Bonilla, D.A. Juraev, Klein-Gordon’s equation for magnons without non-ideal effect on spatial separation of spin waves, Stochastic Modelling and Computational Sciences, 3:1, 29–37 (2023).
J.D. Bulnes, J.L. López-Bonilla, Dirac’s linearization applied to the functional, with matrix aspect, for the time of flight of light, Maltepe Journal of Mathematics, 4:2, 38–43 (2022).
J.D. Bulnes, Solving the heat equation by solving an integro-differential equation, Global and Stochastic Analysis, 9:2, 89–97 (2022).
J.D. Bulnes, F.A. Bonk, On the problem of the theoretical identification of a physical entanglement in spins systems in the NMR quantum computation, Lat. Am. J. Phys. Educ., 9:1, 1303.1–1303.3 (2015).
J.D. Bulnes, F.A. Bonk, A case of spurious quantum entanglement originated by a mathematical property with a nonphysical parameter, Latin-American Journal of Physics Education, 8, 4306-1–4306-4 (2014).
J.D. Bulnes, L.A. Peche, Entrelazamiento cuántico espúrio con matrizes seudopuras extendidas 4 por 4, Revista Mexicana de Física, 57, 188–192 (2011).
J.D. Bulnes, Propagadores cuánticos calculados de acuerdo con el postulado de Feynman con caminos aproximados por polinomios, Revista Mexicana de Física, 55, 34–43 (2009).
J.D. Bulnes, F.A. Bonk, R.S. Sarthour, E. Azevedo, J.C.C. Freitas, T.J. Bonagamba, I.S. Oliveira, Quantum information processing through nuclear magnetic resonance, Brazilian Journal of Physics, 35, 617–625 (2005).
J.D. Bulnes, I.S. Oliveira, Construction of exact solutions for the Stern Gerlach effect, Brazilian Journal of Physics, 31:3, 488–495 (2001).
I.V Misyurkeyev, Problems in Mathematical Physics. McGraw-Hill Book Company, New York, (1966). [42] V. Eiderman, An Introduction to Complex Analysis and the Laplace Transform, Taylor & Francis Group, (2022).
I.N. Sneddon, Elements of Partial Differential Equations, McGraw-Hill, New York, (1957).
P.A.M. Dirac, Complex variables in Quantum Mechanics, Proceedings of the Royal Society A, 160:900, 48–59 (1937).
A. Einstein, B. Podolsky, N. Rosen, Can Quantum-Mechanical Description of Physical Reality be considered complete?, Physical Review, 47, 777–780 (1935).
Issue 1