Effect of a Moving Mirror on the Free Fall of a Quantum Particle in a Homogeneous Gravitational Field
Abstract
:1. Introduction
2. Schrödinger Equation with Time-Dependent Boundary Conditions
3. Quantum Bouncer
3.1. Fixed Mirror
3.2. Moving Mirror
4. Results
4.1. Initial Basis State of the Moving Mirror Bouncer
4.2. Initial Eigenstate of the Fixed Mirror Bouncer
4.3. Initial Gaussian Wavepacket
5. Discussion
5.1. Dynamical Effects
5.2. Non-Locality
5.3. Experimental Observation
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
- Sakurai, J.J. Modern Quantum Mechanics; Pearson: Harlow, UK, 1994. [Google Scholar]
- Landau, L.D.; Lisfshitz, E.M. Quantum Mechanics; Pergamon: Oxford, UK, 1977. [Google Scholar]
- Gea-Banacloche, J. A quantum bouncing ball. Am. J. Phys. 1999, 67, 776–782. [Google Scholar]
- Doncheski, M.A.; Robinett, R.W. Expectation value analysis of wave packet solutions for the quantum bouncer: Short-term classical and long-term revival behaviors. Am. J. Phys. 2001, 69, 1084–1090. [Google Scholar] [CrossRef] [Green Version]
- Goodings, D.A.; Szeredi, T. The quantum bouncer by the path integral method. Am. J. Phys. 1991, 59, 924–930. [Google Scholar] [CrossRef]
- Loh, Y.L.; Gan, C.K. Path-integral treatment of quantum bouncers. J. Phys. A Math. Theor. 2021, 54, 405302. [Google Scholar] [CrossRef]
- Nesvizhevsky, V.V.; Börner, H.G.; Petukhov, A.K.; Abele, H.; Baeßler, S.; Rueß, F.J.; Stöferle, T.; Westphal, A.; Gagarski, A.M.; Petrov, G.A.; et al. Quantum states of neutrons in the Earth’s gravitational field. Nature 2002, 415, 297. [Google Scholar] [CrossRef] [PubMed]
- Suda, M.; Faber, M.; Bosina, J.; Jenke, T.; Käding, C.; Micko, J.; Pitschmann, M.; Abele, H. Spectra of neutron wave functions in Earth’s gravitational field. Z. Naturforschung. A 2022, 77, 875. [Google Scholar] [CrossRef]
- Di Martino, S.; Facchi, P. Quantum systems with time-dependent boundaries. Int. J. Geom. Methods Mod. Phys. 2015, 12, 1560003. [Google Scholar] [CrossRef] [Green Version]
- Mostafazadeh, A. Perturbative calculation of the adiabatic geometric phase and particle in a well with moving walls. J. Phys. A 1999, 32, 8325. [Google Scholar] [CrossRef]
- Greenberger, D.M. A new non-local effect in quantum mechanics. Physica B+C 1988, 151, 374–377. [Google Scholar]
- Matzkin, A. Single particle nonlocality, geometric phases and time-dependent boundary conditions. J. Phys. A 2018, 51, 095303. [Google Scholar] [CrossRef] [Green Version]
- Abele, H.; Jenke, T.; Leeb, H.; Schmiedmayer, J. Ramsey’s method of separated oscillating fields and its application to gravitationally induced quantum phase shifts. Phys. Rev. D 2010, 81, 065019. [Google Scholar] [CrossRef] [Green Version]
- Duffin, C.; Dijkstra, A.G. Controlling a Quantum System via its Boundary Conditions. Eur. Phys. J. D 2019, 73, 221. [Google Scholar] [CrossRef]
- Makowski, A.J. Two classes of exactly solvable quantum models with moving boundaries. J. Phys. A 1992, 25, 3419. [Google Scholar] [CrossRef]
- Glasser, M.L.; Mateo, J.; Negro, J.; Nieto, L.M. Quantum infinite square well with an oscillating wall. Chaos Solitons Fract. 2009, 41, 2067–2074. [Google Scholar] [CrossRef]
- Matzkin, A.; Mousavi, S.V.; Waegell, M. Nonlocality and local causality in the Schrödinger equation with time-dependent boundary conditions. Phys. Lett. A 2018, 382, 3347–3354. [Google Scholar] [CrossRef] [Green Version]
- Jenke, T.; Geltenbort, P.; Lemmel, H.; Abele, H. Realization of a gravity-resonance-spectroscopy technique. Nat. Phys. 2011, 7, 468–472. [Google Scholar] [CrossRef]
- Cronenberg, G.; Brax, P.; Filter, H.; Geltenbort, P.; Jenke, T.; Pignol, G.; Pitschmann, M.; Thalhammer, M.; Abele, H. Acoustic Rabi oscillations between gravitational quantum states and impact on symmetron dark energy. Nat. Phys. 2018, 14, 1022–1026. [Google Scholar] [CrossRef] [Green Version]
- Waegell, M.; Matzkin, A. Nonlocal Interferences Induced by the Phase of the Wavefunction for a Particle in a Cavity with Moving Boundaries. Quantum Rep. 2020, 2, 514–528. [Google Scholar] [CrossRef]
- Scheininger, C.; Kleber, M. Quantum to classical correspondence for the Fermi-acceleration model. Phys. D Nonlinear Phenom. 1991, 50, 391–404. [Google Scholar] [CrossRef]
- Doescher, W.; Rice, H.H. Infinite Square-Well Potential with a Moving Wall. Am. J. Phys. 1969, 37, 1246–1249. [Google Scholar] [CrossRef]
- Makowski, A.J. On the solvability of the bouncer model. J. Phys. A Math. Gen. 1996, 29, 6003. [Google Scholar] [CrossRef]
- Cervero, J.P.; Polo, P.P. The one dimensional SchrZXLdinger equation: Symmetries, solutions and Feynman propagators. Eur. J. Phys. 2016, 37, 055401. [Google Scholar] [CrossRef]
- Colin, S.; Matzkin, A. Non-locality and time-dependent boundary conditions: A Klein-Gordon perspective. Europhys. Lett. 2020, 130, 50003. [Google Scholar] [CrossRef]
- Rauch, H.; Werner, S.A. Neutron Interferometry; Clarendon: Oxford, UK, 2000. [Google Scholar]
- Mousavi, S.V.; Majumdar, A.S.; Home, D. Effect of quantum statistics on the gravitational weak equivalence principle. Class. Quantum Grav. 2015, 32, 215014. [Google Scholar] [CrossRef] [Green Version]
- Emelyanov, V.A. On free fall of quantum matter. Eur. Phys. J. C 2022, 82, 318. [Google Scholar] [CrossRef]
- Emelyanov, V.A. Non-universality of free fall in quantum theory. arXiv 2022, arXiv:2204.03279. [Google Scholar]
- Perez, P.G.; Banerjee, D.; Biraben, F.; Brook-Roberge, D.; Charlton, M.; Clade, P.; Comini, P.; Crivelli, P.; Dalkarov, O.D.; Debu, P.; et al. The GBAR antimatter gravity experiment. Hyperfine Interact 2015, 233, 21–27. [Google Scholar] [CrossRef]
- Doser, M.; Amsler, C.; Belov, A.; Bonomi, G.; Bräunig, P.; Bremer, J.; Brusa, R.; Burkhart, G.; Cabaret, L.; Canali, C.; et al. Exploring the WEP with a pulsed cold beam of antihydrogen. Class. Quantum Grav. 2012, 29, 184009. [Google Scholar] [CrossRef]
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Allam, J.; Matzkin, A. Effect of a Moving Mirror on the Free Fall of a Quantum Particle in a Homogeneous Gravitational Field. Quantum Rep. 2023, 5, 1-11. https://doi.org/10.3390/quantum5010001
Allam J, Matzkin A. Effect of a Moving Mirror on the Free Fall of a Quantum Particle in a Homogeneous Gravitational Field. Quantum Reports. 2023; 5(1):1-11. https://doi.org/10.3390/quantum5010001
Chicago/Turabian StyleAllam, Jawad, and Alex Matzkin. 2023. "Effect of a Moving Mirror on the Free Fall of a Quantum Particle in a Homogeneous Gravitational Field" Quantum Reports 5, no. 1: 1-11. https://doi.org/10.3390/quantum5010001