The main problem of quantum mechanics is to
elucidate why the probability density is the modulus square of wave function.
For the purpose of solving this problem, we explored the possibility of
deducing the fundamental equation of quantum mechanics by starting with the
probability density. To do so, it is necessary to formulate a new theory of
quantum mechanics distinguished from the previous ones. Our investigation shows
that it is possible to construct quantum mechanics in phase space as an
alternative autonomous formulation and such a possibility enables us to study
quantum mechanics by starting with the probability density rather than the wave
function. This direction of research is contrary to configuration-space formulation
of quantum mechanics starting with the wave function. Our work leads to a full
understanding of the wave function as the both mathematically and physically
sufficient representation of quantum-mechanical state which supplements
information on quantum state given solely by the probability density with phase
information on quantum state. The final result of our work is that quantum
mechanics in phase space satisfactorily elucidates the relation between the
wave function and the probability density by using the consistent procedure
starting with the probability density, thus corroborating the ontological
interpretation of the wave function and withdrawing a main assumption of
quantum mechanics.
References
[1]
Auletta, G. (2001) Foundations and Interpretation of Quantum Mechanics. World Scientific Publishing, Singapore.
[2]
Born, M. (1926) Zur Quantenmechanik der Stoß vorgänge. Zeitschrift für Physik, 37, 863-867. https://doi.org/10.1007/BF01397477
[3]
Stoica, O.C. (2022) Born Rule: Quantum Probability as Classical Probability. ArXiv: 2209.08621.
[4]
Taoppenden, P. (2022) Pilot-Wave Theory without Nonlocality. Foundations of Physics, 52, Article No. 107. https://doi.org/10.1007/s10701-022-00627-0
[5]
Drezet, A. (2021) Making Sense of Born’s Rule with the Many-Minds Interpretation. Quantum Studies: Mathematics and Foundations, 8, 315-336.
[6]
Sun, C.P. (2017) On Interpretations of Quantum Mechanics. Physics, 46, 481-498. (In Chinese).
[7]
Einstein, A., Podolsky, B. and Rosen, N. (1935) Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? Physical Review, 47, 777-780. https://doi.org/10.1103/PhysRev.47.777
[8]
Long, G.L., Qin, W., Yang, Z. and Li, J.-L. (2017) Realistic Interpretation of Quantum Mechanics and Encounter-Delayed-Choice Experiment. Science China Physics, Mechanics & Astronomy, 61, Article No. 030311. https://doi.org/10.1007/s11433-017-9122-2
[9]
Long, G.L. (2006) General Quantum Interference Principle and Duality Computer. Communications in Theoretical Physics, 45, 825-844. https://doi.org/10.1088/0253-6102/45/5/013
[10]
Long, G.L. (2017) What Is the Wave Function in Quantum Mechanics? Science Bulletin, 62, 1355-1356. https://doi.org/10.1016/j.scib.2017.10.004
[11]
Esfeld, M., Lazarovici, D., Hubert, M. and Dürr, D. (2014) The Ontology of Bohmian Mechanics. The British Journal for the Philosophy of Science, 65, 773-796. https://doi.org/10.1093/bjps/axt019
[12]
Suárez, M. (2015) Bohmian Dispositions. Synthese, 192, 3203-3228. https://doi.org/10.1007/s11229-015-0741-1
[13]
Bhaumik, M.L. (2022) Is the Quantum State Real in the Hilbert Space Formulation. ArXiv: 2210.13973.
[14]
Myrvold, W.C. (2018) ψ-Ontology Result without the Cartesian Product Assumption. Physical Review A, 97, Article ID: 052109. https://doi.org/10.1103/PhysRevA.97.052109
[15]
Hermens, R. (2021) How Real Are Quantum States in ψ-Ontic Model. Foundations of Physics, 51, Article No. 38. https://doi.org/10.1007/s10701-021-00448-7
[16]
Bolotin, A. (2021) Wave-Particle Duality and the Objectiveness of “True’’ and “False’’. Foundations of Physics, 51, Article No. 78. https://doi.org/10.1007/s10701-021-00478-1
[17]
Callender, C. (2015) One World, One Beable. Synthese, 192, 3153-3177. https://doi.org/10.1007/s11229-014-0582-3
[18]
Miller, E. (2014) Quantum Entanglement, Bohmian Mechanics, and Humean Supervenience. Australasian Journal of Philosophy, 92, 567-583. https://doi.org/10.1080/00048402.2013.832786
[19]
Diósi, L. (2022) Schrödinger-Newton Equation with Spontaneous Wave Function Collapse. ArXiv: 2210.15057.
[20]
Wigner, E. (1932) On the Quantum Correction for Thermodynamic Equilibrium. Physical Review, 40, 749-759. https://doi.org/10.1103/PhysRev.40.749
[21]
Groenewold, H. (1946) On the Principles of Elementary Quantum Mechanics. Physica, 12, 405-460. https://doi.org/10.1016/S0031-8914(46)80059-4
[22]
Zachos, C.K., Fairlie, D.B. and Curtright, T.L. (2005) Quantum Mechanics in Phase Space. In: World Scientific Series in 20th Century Physics, Vol. 34, World Scientific Publishing, Singapore. https://doi.org/10.1142/5287
[23]
Feynman, R.P. (1950) Mathematical Formulation of the Quantum Theory of Electromagnetic Interaction. Physical Review, 80, 440-457. https://doi.org/10.1103/PhysRev.80.440
[24]
Neuman, J.V. (1931) Die Eindeutigkeit der Schrödingerschen Operatoren. Mathematische Annalen, 104, 570-578. https://doi.org/10.1007/BF01457956
[25]
Greiner, W. (2001) Quantum Mechanics: An Introduction. Springer, Berlin. https://doi.org/10.1007/978-3-642-56826-8
Moyal, J. (1949) Quantum Mechanics as a Statistical Theory. Mathematical Proceedings of the Cambridge Philosophical Society, 45, 99-124. https://doi.org/10.1017/S0305004100000487
[28]
Baker, G. (1958) Formulation of Quantum Mechanics Based on the Quasi-Probability Distribution Induced on Phase Space. Physical Review, 109, 2198-2206. https://doi.org/10.1103/PhysRev.109.2198
[29]
Takabayasi, T. (1952) On the Formulation of Quantum Mechanics Associated with Classical Pictures. Progress of Theoretical Physics, 8, 143-182. https://doi.org/10.1143/ptp/8.2.143
[30]
Takabayasi, T. (1954) The Formulation of Quantum Mechanics in Terms of an Ensemble in Phase Space. Progress of Theoretical Physics, 11, 341-373. https://doi.org/10.1143/PTP.11.341
[31]
Goldstein, S. (1998) Quantum Theory without Observers. Physics Today, 51, 38. https://doi.org/10.1063/1.882241
[32]
Goldstein, S., Tumulka, R. and Zanghì, N. (2011) Bohmian Trajectories as the Foundation of Quantum Mechanics. In: Chattaraj, P., Ed., Quantum Trajectories, CRC Press, Boca Raton, FL, 1-15.
[33]
Home, D. and Whitaker, A. (2007) Einstein’s Struggles with Quantum Theory. Springer, Berlin.
Wyatt, R.E. (2005) Quantum Dynamics with Trajectories. Springer, Berlin.
[36]
Bohm, D. (1952) A Suggested Interpretation of the Quantum Theory in Terms of “Hidden” Variables. I. Physical Review, 85, 166-179. https://doi.org/10.1103/PhysRev.85.166
[37]
Bohm, D. (1952) A Suggested Interpretation of the Quantum Theory in Terms of “Hidden” Variables. II. Physical Review, 85, 180-193. https://doi.org/10.1103/PhysRev.85.180
[38]
Madelung, V.E. (1926) Quantentheorie in Hydrodynamischer Form. Zeitschrift für Physik, 40, 322-326. https://doi.org/10.1007/BF01400372
[39]
Dürr, D. and Teufel, S. (2009) Bohmian Mechanics. Springer, Berlin. https://doi.org/10.1007/b99978
[40]
Sanz, á.S. and Miret-Artés, S. (2011) An Account of Quantum Interference from a Hydrodynamical Perspective. In: Chattaraj, P.K., Ed., Quantum Trajectories, CRC Press, Boca Raton, 197-221.
[41]
Hughes, K.H., Burghardt, I. and Chattaraj, P.K. (2011) A Hybrid Hydrodynamic-Liouvillian Approach to Non-Markovian Dynamics. In: Chattaraj, P.K., Ed., Quantum Trajectories, CRC Press, Boca Raton, 163-181.
[42]
Gasser, I. and Markowich, P.A. (1997) Quantum Hydrodynamics, Wigner Transforms, the Classical Limit. Asymptotic Analysis, 14, 97-116. https://doi.org/10.3233/ASY-1997-14201
[43]
Sbitnev, V.I. (2012) Bohmian Trajectories and the Path Integral Paradigm-Complexified Lagrangian Mechanics. In: Pahlavani, M.R., Eds., Theoretical Concepts of Quantum Mechanics, IntechOpen, London, 313-340. https://doi.org/10.5772/33064
[44]
Meier, C., Beswick, J.A. and Yefsah, T. (2011) Mixed Quantum/Classical Dynamics: Bohmian and DVR Stochastic Trajectories. In: Chattaraj, P.K., Ed., Quantum Trajectories, CRC Press, Boca Raton, 149-161.
Martens, C.C., Donoso, A. and Zheng, Y. (2011) Quantum Trajectories in Phase Space, In: Chattaraj, P.K., Ed., Quantum Trajectories, CRC Press, Boca Raton, 95-111.
[47]
Bohm, D. and Hiley, B.J. (1993) The Undivided Universe. Routledge, London.
[48]
Hartle, J.B. (2004) Bohmian Histories and Decoherent Histories. Physical Review A, 69, Article ID: 042111. https://doi.org/10.1103/PhysRevA.69.042111
[49]
Muga, J.G., Sala, R. and Snider, R.F. (1993) Comparison of Classical and Quantal Evolution of Phase Space Distribution Function. Physica Scripta, 47, 732-739. https://doi.org/10.1088/0031-8949/47/6/007
[50]
Sanz, S.á. and Miret-Artés, S. (2014) A Trajectory Description of Quantum Processes. II. Applications. Springer-Verlag, Berlin. https://doi.org/10.1007/978-3-642-17974-7
[51]
Fairlie, D. (1964) The Formulation of Quantum Mechanics in Terms of Phase Space Function. Mathematical Proceedings of the Cambridge Philosophical Society, 60, 581-586. https://doi.org/10.1017/S0305004100038068
[52]
Royer, A. (1977) Wigner Function as the Expectation Value of a Parity Operator. Physical Review A, 15, 449-450. https://doi.org/10.1103/PhysRevA.15.449
[53]
Dahl, J. (1982) Wigner’s Phase Space Function and Atomic Structure I: The Hydrogen Atom Ground State. Molecular Physics, 47, 1001-1019. https://doi.org/10.1080/00268978200100752
[54]
Curtright, T. and Zachos, C. (2001) Negative Probability and Uncertainty Relation. Modern Physics Letters A, 16, 2381-2385. https://doi.org/10.1142/S021773230100576X
Holland, P. (2011) Quantum Field Dynamics from Trajectories. In: Chattaraj, P.K., Ed., Quantum Trajectories, CRC Press, Boca Raton, 73-85.
[57]
De Nicola, S., Fedele, R., Man’ko, M.A. and Man’ko, V.I. (2007) New Inequalities for Tomograms in the Probability Representation of Quantum States. Theoretical and Mathematical Physics, 152, 1081-1086. https://doi.org/10.1007/s11232-007-0092-2
[58]
Monsa, R.A. and Vaz Jr., J. (2003) Quantum Tomography for Dirac Spinors. Physics Letters A, 315, 418-425. https://doi.org/10.1016/S0375-9601(03)01099-5
[59]
Arkhipov, A.S. and Lozovik, Y.E. (2004) Quantum Tomography as a New Approach to Simulating Quantum Processes. Journal of Experimental and Theoretical Physics, 98, 231-239. https://doi.org/10.1134/1.1675891
[60]
Weyl, H. (1927) Quantenmechanik und Gruppentheorie. Zeitschrift für Physik, 46, 1-46. https://doi.org/10.1007/BF02055756
[61]
Curtright, T., Uematsu, T. and Zachos, C. (2001) Generating All Wigner Function. Journal of Mathematical Physics, 42, 2396-2415. https://doi.org/10.1063/1.1366327
[62]
Feynman, R.P. and Hibbs, A.R. (1965) Quantum Mechanics and Path Integrals. McGraw-Hill, New York.
[63]
Weinberg, S. (1992) Dreams of a Final Theory. Pantheon Books, New York.
[64]
Adler, S.L. (2004) Quantum Theory as an Emergent Phenomenon. Cambridge University Press, Cambridge, UK. https://doi.org/10.1017/CBO9780511535277
[65]
Hooft, G. (2005) Determinism Beneath Quantum Mechanics, In: Elitzur, A.C., Dolev, S. and Kolenda, N., Eds., Quo Vadis Quantum Mechanics? The Frontiers Collection, Springer, Berlin, 99-111. https://doi.org/10.1007/3-540-26669-0_8
[66]
Marshall, W., Simon, C., Penrose, R. and Bouwmeester, D. (2003) Towards Quantum Superposition of a Mirror. Physical Review Letters, 91, Article ID: 130401. https://doi.org/10.1103/PhysRevLett.91.130401
[67]
Penrose, R. (2000) Wavefunction Collapse as a Real Gravitational Effect. In: Fokas, A., Grigourion, A., Kibble, B. and Zegarlinski, B., Eds., Mathematical Physics 2000, Imperial College Press, London, 266-282. https://doi.org/10.1142/9781848160224_0013
[68]
Penrose, R. (2004) The Road to Reality. Jonathan Cape, London.
[69]
Gampel, P. and Gajda, M. (2022) Continuous Simultaneous Measurement of Position and Momentum of a Particle. ArXiv: 2209.01612.
[70]
Busch, P., Grabowski, M. and Lahti, P.J. (1995) Operational Quantum Physics. Springer, Berlin. https://doi.org/10.1007/978-3-540-49239-9
[71]
de Muynck, W.M. (2004) Foundations of Quantum Mechanics, an Empiricist Approach. Kluwer Academic Publishers, New York.
Holland, P.R. (1993) The Quantum Theory of Motion. Cambridge University Press, Cambridge.
[76]
Hartle, J.B. (2005) Quantum Mechanics. In: Gross, D., Sevrin, A. and Henneaux, M., Eds, The Quantum Structure of Space and Time, World Scientific Publishing Company, Singapore, 25-26.
