The free fall in three Physics theories
DOI:
https://doi.org/10.31349/RevMexFisE.21.010214Keywords:
Asymptotic Quantum Mechanics, Correspondence Principle, Probability Density, Relativistic Form of Newton's Second Law, Weak Equivalence PrincipleAbstract
This paper explores the interplay between Classical Mechanics, Relativistic Mechanics, and Quantum Mechanics through an analysis of the free fall phenomenon. We investigate the probability density functions and corresponding plots in each theory, alongside calculating the expected values of position and momentum. By observing the behavior of these results as they approach the classical limit, we confirm the hypothesis that these theories can be connected through their probability density functions. Furthermore, we discuss the validity of the correspondence principle in Quantum Mechanics, while also examining, in a non-rigorous manner, the validity of the weak equivalence principle within each of the aforementioned theories.
References
E. Chacón, Apuntes del Curso de Mecánica Cuántica de Marcos Moshinsky, 1st ed. (Las Prensas de Ciencias, CDMX, 2008).
N. Bohr, The Correspondence Principle (North Holland, Amsterdam, 2013).
R. Robinett, Quantum and classical probability distributions for position and momentum, Am. J. Phys. 63 (1995) 823,
https://doi.org/10.1119/1.17807
A. Kolmogorov, Foundations of the Theory of Probability, 2nd ed. (Dover Publications, Mineola, NY, 2013), p. 37, https://www.york.ac.uk/depts/maths/histstat/kolmogorov_foundations.pdf.
N. Wheeler, Classical/Quantum Motion in a Uniform Gravitational Field,
R. Resnick, D. Haliday, and K. Krane, Physics Vol. 1, 4th ed. (Wiley, Hoboken, NJ, 1991), p. 28.
G. Galilei, Dialogues Concerning Two New Sciences (The Macmilan Company, NY, 1914).
P. Roll, R. Krotkov, and R. Dicke, The equivalence of inertial and passive gravitational mass, Ann. Phys. 26 (1964) 442, https://doi.org/10.1016/0003-4916(64)90259-3
I. Ciufolini and J. Wheeler, Gravitation and Inertia (Princeton University, NJ, 1995), pp. 117–119.
P. Brax, Satellite Confirms the Principle of Falling, Phys. 15 (2022) 94, https://doi.org/10.1103/Physics.15.94
C. Möller, The Theory of Relativity, 3rd ed. (Clarendon Press, Oxford, 1952), p. 75.
N. Lemos, Analytical Mechanics, 2nd ed. (Cambridge University Press, Cambridge, 2018), p. 200.
G. Naber, The Geometry of Minkowski Space Time: An Introduction to the Mathematics of the Special Theory of Relativity, 2nd ed. (Springer, New York, NY, 2011), pp. 1–6.
J. Cañas, La Caida Libre Cuántica, Thesis Degree (2019), Universidad Juárez Autónoma de Tabasco.
J. Cañas, J. Bernal, and A. Martín-Ruiz, Exact classical limit of the quantum bouncer, Eur. Phys. J. 137 (2022) 1310, https://doi.org/10.1140/epjp/s13360-022-03529-2
N. Zetilli, Quantum Mechanics: Concepts and Applications, 2nd ed. (Wiley, Hoboken, NJ, 2009), p.179, http://www.mmmut.ac.in/News_content/02110tpnews_11232020.pdf.
P. Langhoff, Schrödinger Particle in a Gravitational Well, Am. J. Phys. 39 (1971) 954, https://doi.org/10.1119/1.1986333
R. Gibbs, The quantum bouncer, Am. J. Phys. 43 (1975) 25, https://doi.org/10.1119/1.10024
M. Vallée, O. Soares, Airy functions and aplications to physics (Imperial College Press, Covent Garden, L, 2004).
J. Bernal, A. Martín-Ruiz, and J. García Melgarejo, A simple mathematical formulation of the correspondence principle, J. Mod. Phys. 4 (2013) 108, http://dx.doi.org/10.4236/jmp.2013.41017
A. Martín-Ruiz, et al., The Classical Limit of the Quantum Kepler Problem, J. Mod. Phys. 4 (2013) 818, http://dx.doi.org/10.4236/jmp.2013.41017
J. Albright, Integrals of products of Airy functions, J. Phys. A: Math. Gen. 10 (1977) 485, https://doi.org/10.1119/1.10024
N. Temme, Error Functions, Dawson’s and Fresnel Integrals (Cambridge University Press, Cambridge, 2010).
C. Cohen Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics Vol.1, 1st ed. (Wiley, Hoboken, NJ, 1991), p. 229.
P. Ehrenfest, Bemerkung uber die angen ̈ aherte g ̈ ultigkeit der klassischen mechanik innerhalb der quantenmechanik, Z. Phys. 45 (1927) 455, https://doi.org/10.1007/BF01329203
S. Singh, S. Suman, and V. Singh, Quantum–classical correspondence for a particle in a homogeneous field, Eur. Phys. J. 37 (2016) 6, https://doi.org/10.1088/0143-0807/37/6/065405
N. Piskunov, Differential and Integral Calculus (Mir, Moscow, Msk, 1969).
J. Burden, R.L. Faires, Numerical Analysis, 9th ed. (Cengage Learning, Boston, MA, 2010), pp. 206–207.
J. R. Sousa, Generalized Harmonic Numbers, https://arxiv.org/abs/1810.07877 (2018), 10.48550/ARXIV.1810.07877.
S. Ramanujan, On the sum of the square roots of the first n natural numbers, J. Indian Math. Soc. 7 (1915) 173, http://ramanujan.sirinudi.org/Volumes/published/ram09.pdf
T. Apostol, An Elementary View of Euler’s Summation Formula, Am. Math. Mon. 106 (1999) 409, https://doi.org/10.2307/2589145
M. Anthony, The Towering Zeta Function, APM. 6 (2016) 351, http://dx.doi.org/10.4236/apm.2016.65026
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2024 Edgar Luis Luna Hernández, Jorge Alejandro Bernal Arroyo, Luis Enrique Ramón Pedrero
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
Authors retain copyright and grant the Revista Mexicana de Física E right of first publication with the work simultaneously licensed under a CC BY-NC-ND 4.0 that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.