Abstract
We rewrite the Klein–Gordon equation in an arbitrary space-time transforming it into a generalized Schrödinger equation. Then, we take the weak field limit and show that this equation has certain differences with the traditional Schrödinger equation plus a gravitational field. Thus, this procedure shows that the Schrödinger equation derived in a covariant manner is different from the traditional one. We study the KG equation in a Newtonian space-time to describe the behavior of a scalar particle in an inertial system. This particle is immersed in a gravitational field with the new Schrödinger equation. We study particular physical systems given examples for which we find their energy levels, effective potential and the wave function of the systems. The results contain the gravitational effects due to the curvature of space-time. Finally, we discuss the possibility of the experimental verification of these effects in a laboratory using non-inertial reference frames.
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Acknowledgements
This work was partially supported by CONACyT México under grants CB-2011 No. 166212, CB-2014-01 No. 240512, Project No. 304001, 269652 and Fronteras Project 281; The authors are gratefully for the computing time granted by LANCAD and CONACYT in the Supercomputer Hybrid Cluster “Xiuhcoatl” at GENERAL COORDINATION OF INFORMATION AND COMMUNICATIONS TECHNOLOGIES (CGSTIC) of CINVESTAV. URL: https://clusterhibrido.cinvestav.mx/ and Abacus clusters at Cinvestav, IPN; I0101/131/07 C-234/07 of the Instituto Avanzado de Cosmología (IAC) collaboration (http://www.iac.edu.mx). O.G. acknowledges financial support from CONACyT doctoral fellowship and appreaciates Angelica C. Aguirre Castañón for her valuable review and support. Works of T.M. are partially supported by Conacyt through the Fondo Sectorial de Investigación para la Educación, grant CB-2014-1, No. 240512. The data that support the findings of this study are available from the corresponding author upon reasonable request.
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Gallegos, O., Matos, T. Weak gravitational quantum effects in boson particles. Gen Relativ Gravit 53, 50 (2021). https://doi.org/10.1007/s10714-021-02810-6
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DOI: https://doi.org/10.1007/s10714-021-02810-6