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BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access September 24, 2010

The Maslov correction in the semiclassical Feynman integral

Peter. Horváthy
From the journal Open Physics

Abstract

The Maslov correction to the wave function is the jump of $$ \left( { - \frac{\pi } {2}} \right) $$ in the phase when the system passes through a caustic. This can be explained by studying the second variation and the geometry of paths, as conveniently seen in Feynman’s path integral framework. The results can be extended to any system using the semiclassical approximation. The 1-dimensional harmonic oscillator is used to illustrate the different derivations reviewed here.

[1] M. Gouy, Comptes rendus hebdomadaires des séances de l’Académie des Sciences 110, 1251 (1890) Search in Google Scholar

[2] J.B. Keller, Ann. Phys. 4, 180 (1958) http://dx.doi.org/10.1016/0003-4916(58)90032-010.1016/0003-4916(58)90032-0Search in Google Scholar

[3] V.P. Maslov, Asymptotic methods in the calculus of perturbations (MGU, Moscow, 1965) (in Russian) Search in Google Scholar

[4] V.I. Arnold, Funktional’nyi Analiz i Ego Prilozheniya 1, 1 (1967) (in Russian) http://dx.doi.org/10.1007/BF0107586110.1007/BF01075861Search in Google Scholar

[5] J.-M. Souriau, Lec. Notes Phys. 50, 117 (1976) http://dx.doi.org/10.1007/3-540-07789-8_1310.1007/3-540-07789-8_13Search in Google Scholar

[6] V. Marino, L. Gualandari, Indice di Maslov. Lezioni del Prof. Souriau, Pubblicazioni del’Istituto di Matematica Applicata N. 191. Università di Roma (1977) (in Italian) Search in Google Scholar

[7] R.P. Feynman, A.R. Hibbs, Quantum Mechanics and path integrals (McGraw-Hill, New York, 1965) Search in Google Scholar

[8] P.A. Horváthy, Int. J. Theor. Phys. 18, 245 (1979) http://dx.doi.org/10.1007/BF0067176110.1007/BF00671761Search in Google Scholar

[9] L.S. Schulman, In: A.M. Arthurs (Ed.),, Functional integration and its applications (Clarendon Pross, Oxford, 1975) 144 Search in Google Scholar

[10] B.K. Cheng, Int. J. Theor. Phys. 23, 1099 (1984) http://dx.doi.org/10.1007/BF0221342210.1007/BF02213422Search in Google Scholar

[11] J.Q. Liang, G. Morandi, Phys. Lett. A 160, 9 (1991) http://dx.doi.org/10.1016/0375-9601(91)90197-G10.1016/0375-9601(91)90197-GSearch in Google Scholar

[12] P.A. Horváthy, L. Úry, Acta Phys. Hung. 42, 251 (1977) http://dx.doi.org/10.1007/BF0315749310.1007/BF03157493Search in Google Scholar

[13] P.A. Horváthy, Phys. Lett. A 76, 11 (1980) http://dx.doi.org/10.1016/0375-9601(80)90133-410.1016/0375-9601(80)90133-4Search in Google Scholar

[14] P.A. Horváthy, Lect. Notes Math. 836, 67 (1980) http://dx.doi.org/10.1007/BFb008972710.1007/BFb0089727Search in Google Scholar

[15] B.K. Cheng, Phys. Scripta 29, 351 (1984) http://dx.doi.org/10.1088/0031-8949/29/4/01210.1088/0031-8949/29/4/012Search in Google Scholar

[16] B.K. Cheng, Phys. Rev. A 30, 1491 (1984) http://dx.doi.org/10.1103/PhysRevA.30.149110.1103/PhysRevA.30.1491Search in Google Scholar

[17] G. Sagnac, Comptes rendus hebdomadaires des séances de l’Académie des Sciences 157, 708 (1913) Search in Google Scholar

[18] F. Hasselbach, M. Niklas, Phys. Rev. A 48, 143 (1993) http://dx.doi.org/10.1103/PhysRevA.48.14310.1103/PhysRevA.48.143Search in Google Scholar

[19] R. Anderson, H.R. Bilger, G.E. Stedman, Am. J. Phys. 62, 975 (1994) http://dx.doi.org/10.1119/1.1765610.1119/1.17656Search in Google Scholar

[20] F. Hasselbach, Rep. Prog. Phys. 73, 016101 (2010) http://dx.doi.org/10.1088/0034-4885/73/1/01610110.1088/0034-4885/73/1/016101Search in Google Scholar

[21] C. DeWitt-Morette, Ann. Phys. 97, 367 (1976) http://dx.doi.org/10.1016/0003-4916(76)90041-510.1016/0003-4916(76)90041-5Search in Google Scholar

[22] S. Levit, U. Smilansky, Ann. Phys. 103, 198 (1977) http://dx.doi.org/10.1016/0003-4916(77)90269-X10.1016/0003-4916(77)90269-XSearch in Google Scholar

[23] M. Morse, Calculus of variations in the large (Transactions of the AMS, Providence, 1934) 10.1090/coll/018Search in Google Scholar

[24] J. Milnor, Morse Theory (Princeton U.P., Princeton, 1963) Search in Google Scholar

[25] W.H. Miller, J. Chem. Phys. 53, 1949 (1970) http://dx.doi.org/10.1063/1.167427510.1063/1.1674275Search in Google Scholar

[26] W.H. Miller, Adv. Chem. Phys. 25, 69 (1974) http://dx.doi.org/10.1002/9780470143773.ch210.1002/9780470143773.ch2Search in Google Scholar

[27] R.A. Marcus, J. Chem. Phys. 54, 3965 (1971) http://dx.doi.org/10.1063/1.167545310.1063/1.1675453Search in Google Scholar

[28] S. Levit, U. Smilansky, D. Pelte, Phys. Lett. B 53, 39 (1974) http://dx.doi.org/10.1016/0370-2693(74)90338-410.1016/0370-2693(74)90338-4Search in Google Scholar

[29] H. Massman, J.O. Rasmussen, Nucl. Phys. A 243, 155 (1975) http://dx.doi.org/10.1016/0375-9474(75)90026-310.1016/0375-9474(75)90026-3Search in Google Scholar

[30] T. Koeling, R.A. Malfliet, Phys. Rep. C 22, 181 (1975) http://dx.doi.org/10.1016/0370-1573(75)90059-910.1016/0370-1573(75)90059-9Search in Google Scholar

[31] K. Horie, H. Miyazaki, I. Tsutsui, Ann. Phys. 273, 267 (1999) http://dx.doi.org/10.1006/aphy.1999.590510.1006/aphy.1999.5905Search in Google Scholar

[32] K. Horie, H. Miyazaki, I. Tsutsui, Ann. Phys. 279, 104 (2000) http://dx.doi.org/10.1006/aphy.1999.597110.1006/aphy.1999.5971Search in Google Scholar

[33] C-I Um, K-H Yeon, J. Korean Phys. Soc. 41, 594 (2002) Search in Google Scholar

[34] C-I Um, K-H Yeon, T.F. George, Phys. Rep. 362, 63 (2002) http://dx.doi.org/10.1016/S0370-1573(01)00077-110.1016/S0370-1573(01)00077-1Search in Google Scholar

[35] H. Kleinert, Path integrals in Quantum Mechanics, 4th Edition (World Scientific, Singapore, 2004) Search in Google Scholar

[36] C. Grosche, F. Steiner, Springer Tr. Mod. Phys. 145, 1 (1998) http://dx.doi.org/10.1007/BFb010952110.1007/BFb0109521Search in Google Scholar

[37] P.Y. Cai, A. Inomata, P. Wang, Phys. Lett. A 91, 331 (1982) http://dx.doi.org/10.1016/0375-9601(82)90425-X10.1016/0375-9601(82)90425-XSearch in Google Scholar

[38] G. Junker, A. Inomata, Phys. Lett. A 110, 195 (1985) http://dx.doi.org/10.1016/0375-9601(85)90122-710.1016/0375-9601(85)90122-7Search in Google Scholar

[39] J.M. Cai, P.Y. Cai, A. Inomata, In: J.Q. Liang, M.L. Wang, S.N. Qiao, D.C. Su (Eds.), ISATQP-Shanxi, 1992, Shanxi, China (Science Press, Beijing, 1993) Search in Google Scholar

[40] U. Niederer, Helv. Phys. Acta 46, 192 (1973) Search in Google Scholar

[41] C. Duval, G. Gibbons, A. Horváthy, Phys. Rev. D 43, 3907 (1991) http://dx.doi.org/10.1103/PhysRevD.43.390710.1103/PhysRevD.43.3907Search in Google Scholar PubMed

[42] C. Duval, P.A. Horváthy, L. Palla, Phys. Rev. D 50, 6658 (1994) http://dx.doi.org/10.1103/PhysRevD.50.665810.1103/PhysRevD.50.6658Search in Google Scholar

Published Online: 2010-9-24
Published in Print: 2011-2-1

© 2010 Versita Warsaw

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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