A NOVEL APPROACH TO SCHRÖDINGER'S WAVE EQUATION: UTILIZING METRIC TENSOR IN SPHERICAL COORDINATES
DOI:
https://doi.org/10.33003/fjs-2024-0805-2688Keywords:
Schrodinger wave equation, Laplacian operator, Quantum systems, Howusu metric tensorAbstract
In quantum mechanics, the Schrödinger equation is fundamental for describing particle wave functions, traditionally within flat spacetime, ignoring gravitational effects. This paper introduces the Howusu Metric Tensor to extend the Schrödinger equation into spherical coordinates, accommodating gravitational fields that are regular and continuous with a reciprocal decrease at infinity. This leads to the derivation of the Riemannian Schrödinger equation, offering insights into quantum behavior in curved spacetime. Building on previous work integrating quantum mechanics with general relativity and Finsler geometry, our approach addresses the limitations in capturing gravitational subtleties. By incorporating the Howusu Metric Tensor, our model accounts for gravitational potential in spherical coordinates, providing a more precise description of quantum phenomena under gravity. The resulting Riemannian Schrödinger equation reveals new quantum behavior influenced by gravitational forces, opening new research possibilities in cosmology and astrophysics, where quantum-gravitational interactions are key. This study demonstrates the advantages of using the Howusu Metric Tensor over previous models, highlighting its potential to unify quantum mechanics with gravitational effects more coherently and comprehensively.
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Onyenegecha, C. P., & Ikot, A. N. (2017). Analytical solution of the Schrödinger equation for the ring-shaped multi-parameter exponential type potential. Transaction of the Nigerian Association of Mathematical Physics, 3, 223-2.
Ord, G. (1983). Fractal space-time. Journal of Physics A: Mathematical and General, 16, 18-69.
Ord, G. (2003). Entwined paths, difference equations, and the Dirac equation. Physical Review A, 67, 0121XX3.
Prigogine, I., Rössler, O., & El Naschie, M. S. (1995). Quantum mechanics, diffusion and chaotic fractals. Pergamon. ISBN: 0 08 04227 3.
Rund, H. (1959). The differential geometry of Finsler spaces. Springer.
Tavernelli, I. (2016). Annals of Physics, 371, 239.
Weinberg, S. (1972). Gravitation and cosmology: Principles and applications of the general theory of relativity. John Wiley & Sons.
Wheeler, J. (1990). Physical Review D, 41, 431–441.
Bes, D. R. (2012). Quantum mechanics: A modern and concise introductory course (3rd ed.). Springer.
Blau, M., Frank, D., & Weiss, S. (2006). *Classical and Quantum Gravity, 23, 3993. https://arxiv.org/abs/hep-th/0603109
Bohm, D. (1951). Quantum mechanics. Routledge.
Bracken, P. (2003). International Journal of Theoretical Physics, 42, 775.
Bracken, P. (2008). Pacific Journal of Applied Mathematics, 1, 77.
Chern, S. S., Chen, W. H., & Lam, K. S. (1999). *Lectures on differential geometry. World Scientific.
Chern, S. S., & Shen, Z. (2005). *Riemann-Finsler geometry. World Scientific.
Claudel, C. M., Virbhadra, K. S., & Ellis, G. F. R. (2001). Journal of Mathematical Physics, 42, 818. https://arxiv.org/abs/gr-qc/0005050
Cruz, Y., Cruz, S., Negro, J., & Nieto, L. M. (2007). Classical and quantum position-dependent mass harmonic oscillators. Physics Letters A, 369, 400-406.
Daniel R. Bes. (2012). Quantum mechanics: A modern and concise introductory course (3rd ed.). Springer.
Eisberg, R. M. (1961). Fundamentals of modern physics. John Wiley & Sons.
El Naschie, M. S. (1993). On the universal behavior and statistical mechanics of multidimensional triadic Cantor sets. SAMS, 11, 217-225.
El Naschie, M. S. (1998). Superstrings, knots and non-commutative geometry in E-infinity space. *International Journal of Theoretical Physics, 37(12), 212-234.
El Naschie, M. S. (2001). Quantum collapse of wave interference pattern in the two-slit experiment: A set theoretical resolution. Nonlinear Science Letters A, 2(1), 1-9.
El Naschie, M. S. (2004a). A review of E-infinity theory and the mass spectrum of high energy particle physics. Chaos, Solitons & Fractals, 19, 209-236.
El Naschie, M. S. (2004b). The concept of E-infinity: An elementary introduction to the Cantorian-fractal theory of quantum physics. Chaos, Solitons & Fractals, 22, 495-511.
El Naschie, M. S. (2004c). Quantum gravity from descriptive set theory. Chaos, Solitons & Fractals, 19, 1339-1344.
El Naschie, M. S. (2006). Elementary prerequisites for E-infinity (Recommended background readings in nonlinear dynamics, geometry, and topology). Chaos, Solitons & Fractals, 30, 579-605.
Exirifard, Q., & Karimi, E. (2022). Schrödinger equation in a general curved spacetime geometry. International Journal of Modern Physics D, 31*(3). https://doi.org/10.1142/S0218271822500183
Hawking, S. W., & Ellis, G. F. R. (1973). The large-scale structure of space-time. Cambridge University Press.
He, J. H. (2011). Quantum golden mean entanglement test as the signature of the fractality of micro space-time. Nonlinear Science Letters B, 1(2), 45-50.
Howusu, S. X. K. (2009). The metric tensors for gravitational fields and the mathematical principles of Riemannian theoretical physics. Jos University Press.
Hu, J., & Yu, H. (2021). *European Physical Journal C, 81, 470. https://arxiv.org/abs/2105.10642
Koffa, D. J., Omonile, J. F., Oladimeji, E. O., Edogbanya, H. O., Eghaghe, O. S., Obaje, V. O., & Taofiq, I. T. (2023). A unique generalization of Einstein field equation: Pathway for continuous generation of gravitational waves. Communication in Physical Sciences, 10(1), 122-129.
Landau, L. D., & Lifshitz, E. M. (1977). Quantum mechanics. Pergamon.
Mauldin, R. D., & Williams, S. C. (1986). Random recursive construction. *Transactions of the American Mathematical Society, 295, 325-346.
Marek-Crnjac, L. (2012). Quantum gravity in Cantorian space-time. In R. Sobreiro (Ed.), Quantum Gravity (pp. 87-88). InTech. ISBN: 978-953-51-0089-8.
Marek Crnjac, L. (2009). A Feynman path-like integral method for deriving the four dimensionalities of space-time from the first principle. Chaos, Solitons & Fractals, 41, 2471-2473.
Manasse, F. K., & Misner, C. W. (1963). Journal of Mathematical Physics, 4, 735.
Messiah, A. (1999). *Quantum mechanics* (Vol. I & II). Dover.
Misner, C. W., Thorne, K. S., & Wheeler, J. A. (1973). *Gravitation. W. H. Freeman.
Nyam, G. G. (2017). The generalized Riemannian Schrödinger wave equation for hydrogen atom. IOSR Journal of Applied Physics, 9(4), 32-34. https://doi.org/10.9790/4861-0904033234
Obaje, V. O. (2023). A comparative study of the Schwarzschild metric tensor and the Howusu metric tensor using the radial distance parameter as a measuring index. Journal of Physics: Theoretical and Application, 7(2), 214-222.
Onyenegecha, C. P., & Ikot, A. N. (2017). Analytical solution of the Schrödinger equation for the ring-shaped multi-parameter exponential type potential. Transaction of the Nigerian Association of Mathematical Physics, 3, 223-2.
Ord, G. (1983). Fractal space-time. Journal of Physics A: Mathematical and General, 16, 18-69.
Ord, G. (2003). Entwined paths, difference equations, and the Dirac equation. Physical Review A, 67, 0121XX3.
Prigogine, I., Rössler, O., & El Naschie, M. S. (1995). Quantum mechanics, diffusion and chaotic fractals. Pergamon. ISBN: 0 08 04227 3.
Rund, H. (1959). The differential geometry of Finsler spaces. Springer.
Tavernelli, I. (2016). Annals of Physics, 371, 239.
Weinberg, S. (1972). Gravitation and cosmology: Principles and applications of the general theory of relativity. John Wiley & Sons.
Wheeler, J. (1990). Physical Review D, 41, 431–441.
Published
2024-09-18
How to Cite
Obaje, V. O., Koffa, D. J., Nyam, G. G., Ukewulonu, K. U., Onate, C. A., Ochala, I., Emeje, K. O., Olademeji, E. O., Aliyu, N. S., Obaje, W. M., Egbunu, F. U., Ibrahim, R. A., Oshatuyi, I. I., Ohiani, A. O., & Egwuje, F. D. (2024). A NOVEL APPROACH TO SCHRÖDINGER’S WAVE EQUATION: UTILIZING METRIC TENSOR IN SPHERICAL COORDINATES. FUDMA JOURNAL OF SCIENCES, 8(5), 89 - 93. https://doi.org/10.33003/fjs-2024-0805-2688
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Research Articles
FUDMA Journal of Sciences
