Preface  pp. xi-xiv

Preface

By Nanny Fröman and Per Olof Fröman

Image View Previous Chapter Next Chapter



Only a few quantal problems are exactly soluble by analytical methods. In practice one has almost always to resort to an appropriate approximation method or to numerical integration of the Schrödinger equation. Among the great variety of approximate methods used in quantum mechanics, one may discern three different main types: perturbation methods, variational methods and asymptotic methods. The topic of the present book falls into this last category.

The merit of an asymptotic treatment is that it not only provides a convenient means for approximate calculation of details, but often it also has a fundamental physical significance. In our present context, the asymptotic treatment of the Schrödinger equation yields in its zeroth order the classical Hamilton–Jacobi equation, in its first order it yields the semi-classical approximation and in its higher orders it yields, in many important situations, a full account of quantal details, the physical quantities being sometimes obtained with an accuracy hardly obtainable by any other means.

The phase-integral method for the solution of physical problems used in this book is the result of a systematic work by the present authors and their coworkers over about four decades. The book is written with physical needs in mind. It contains three chapters. In Chapter 1, we give a historical survey. The purpose of Chapter 2 is to introduce the reader to the phase-integral method. The form of the wave function is discussed in Section 2.1, and the phase-integral approximation generated from an unspecified base function is presented in Section 2.2.

Reference Title: References

Reference Type: reference-list

Reference Type: reference-list

Abawi, A. T., Dashen, R. F., and Levine, H., 1997, J Math Phys 38, 1623–1649
Abramowitz, M., and Stegun, I. A., Editors, 1965, Handbook of Mathematical Functions. Applied Mathematics Series 55. Fourth Printing 1965, with corrections, National Bureau of Standards, Washington DC
de Alfaro, V., and Regge, T., 1965, Potential Scattering. North-Holland, Amsterdam
Amaha, A., 1993, Molec Phys 78, 345–356
Bateman Manuscript Project, 1953, Higher Transcendental Functions, Vols I and II, McGraw-Hill Book Company, New York
Bateman Manuscript Project, 1954, Tables of Integral Transforms, Vols I and II, McGraw-Hill Book Company, New York
Bateman Manuscript Project, 1955, Higher Transcendental Functions, Vol III, McGraw-Hill Book Company, New York
Berry, M. V., and Mount, K. E., 1972, Rep Prog Phys 35, 315–397
Bertocchi, L., Fubini, S., and Furlan, G., 1965, Nuovo Cim 35, 599–632
Birkhoff, G. D., 1908, Trans Am Math Soc 9, 219–231
Birkhoff, G. D., 1933, Bull Am Math Soc 39, 681–700
Blumenthal, O., 1912, Archiv Math Phys (Dritte Reihe) 19, 136–174
Brillouin, L., 1926a, C R Acad Sci (Paris) 183, 24–26
Brillouin, L., 1926b, J Phys Radium (Série VI) 7, 353–368
Broer, L. J. F., 1963, Appl Sci Res B10, 110–118
Campbell, J. A., 1972, J Comp Phys 10, 308–315
Campbell, J. A., 1979, J Phys A: Math Gen 12, 1149–1154
Carlini, F., 1817, Ricerche sulla convergenza della serie che serve alla soluzione del problema di Keplero. Appendice all' Effemeridi Astronomiche di Milano per l'Anno 1818, pp. 3–48. (Review: Giornale di Fisica, Chimica, Storia Naturale Medicina ed Arti. X, 458–460, 1817)
Carlini, F., 1850, Astron Nachr 30, 197–254. Translation into German and revision by Jacobi of the paper by Carlini (1817); also published in C. G. J. Jacobi's Gesammelte Werke, edited by K. Weierstrass, Band 7, pp 189–245, Berlin 1891
Cherry, T. M., 1950, Trans Am Math Soc 68, 224–257
Child, M. S., 1974, Molecular Collision Theory. Academic Press, London. Reprinted with corrections 1984
Choi, S., and Ross, J., 1962, Proc Nat Acad Sci 48, 803–806
Cole, M. W., and Good, Jr, R. H., 1978, Phys Rev A18, 1085–1088
Connor, J. N. L., 1968, Mol Phys 15, 37–46
Copson, E. T., 1965, Asymptotic Expansions. Cambridge University Press, Cambridge
Dagens, L., 1969, J Physique (Paris) 30, 593–597
Dammert, Ö., and Fröman, P. O., 1980, J Math Phys 21, 1683–1687
Debye, P., 1909, Math Ann 67, 535–558
Delves, L. M., 1963, Nucl Phys 41, 497–503
Dingle, R. B., 1965, Proc Phys Soc 86, 1366–1368
Dingle, R. B., 1973, Asymptotic Expansions: Their Derivation and Interpretation. Academic Press, London
Erdélyi, A., 1956, Proc Int Congr Math 1954, III, 92–101
Erdélyi, A., 1960, J Math Phys 1, 16–26
Flügge, S., 1974, Practical Quantum Mechanics, Vol. I. Springer-Verlag, New York
Ford, K. W., Hill, D. L., Wakano, M., and Wheeler, J. A., 1959, Ann Phys (NY) 7, 239–258
Fowler, R. H., Gallop, E. G., Lock, C. N. H., and Richmond, W. H., 1921, Phil Trans Roy Soc London A221, 295–387
Fröman, N., 1966a, Ark Fys 31, 381–408
Fröman, N., 1966b, Ark Fys 31, 445–451
Fröman, N., 1966c, Ark Fys 32, 79–97
Fröman, N., 1966d, Ark Fys 32, 541–548
Fröman, N., 1970, Ann Phys (NY) 61, 451–464
Fröman, N., 1974, Phys Lett 48A, 137–139
Fröman, N., 1978a, Phys Rev A17, 493–504
Fröman, N., 1978b, J Math Phys 19, 1141–1146
Fröman, N., 1980, Semiclassical and Higher-Order Approximations: Properties. Solution of Connection Problems. Proceedings of the NATO Advanced Study Institute, Cambridge, England, in September 1979, on Semiclassical Methods in Molecular Scattering and Spectroscopy, pp. 1–44, edited by M. S. Child. NATO Advanced Study Institute Series C-Mathematical and Physical Sciences, Volume 53. Reidel, Dordrecht
Fröman, N., and Dammert, Ö., 1970, Nucl Phys A147, 627–649
Fröman, N., and Fröman, P. O., 1965, JWKB Approximation, Contributions to the Theory. North-Holland, Amsterdam. Russian translation: MIR, Moscow 1967
Fröman, N., and Fröman, P. O., 1970, Nucl Phys A147, 606–626
Fröman, N., and Fröman, P. O., 1972, Phys Rev A6, 2064–2067
Fröman, N., and Fröman, P. O., 1974a, Ann Phys (NY) 83, 103–107. (Review: Zentralblatt für Mathematik und ihre Grenzgebiete, Mathematics Abstracts 279, 190–191, 1974.)
Fröman, N., and Fröman, P. O., 1974b, Nuovo Cim 20B, 121–132
Fröman, N., and Fröman, P. O., 1977, J Math Phys 18, 903–906
Fröman, N., and Fröman, P. O., 1978, J Math Phys 19, 1823–1829
Fröman, N., and Fröman, P. O., 1981, J Physique (Paris) 42, 1491–1504
Fröman, N., and Fröman, P. O., 1984, Phase-Integral Calculation with Very High Accuracy of the Stark Effect in a Hydrogen Atom. Colloque du 10 à 15 septembre 1984, CIRM (Luminy), sur Méthodes Semi-Classiques en Mécanique Quantique, p. 45, edited by D. Helffin and D. Robert. Publications de l'Université de Nantes, Institut de Mathématiques et d'Informatiques, 2 rue de la Houssinière, 44072 Nantes CEDEX, France
Fröman, N., and Fröman, P. O., 1985a, On the History of the So-called WKB-Method from 1817 to 1926. Proceedings of the Niels Bohr Centennial Conference, Copenhagen 25–28 March 1985 on Semiclassical Descriptions of Atomic and Nuclear Collisions, pp 1–7, edited by J. Bang and J. de Boer. North-Holland, Amsterdam
Fröman, N., and Fröman, P. O., 1985b, Ann Phys (NY) 163, 215–226
Fröman, N., and Fröman, P. O., 1989, Int J Quant Chem 35, 751–760
Fröman, N., and Fröman, P. O., 1996, Phase-Integral Method Allowing Nearlying Transition Points, With adjoined papers by A. Dzieciol, N. Fröman, P. O. Fröman, A. Hökback, S. Linnæus, B. Lundborg, and E. Walles. Springer Tracts in Natural Philosophy Vol 40, edited by C. Truesdell. Springer-Verlag, New York
Fröman, N., and Fröman, P. O., 1998, J Math Phys 39, 4417–4429
Fröman, N., and Myhrman, U., 1970, Ark Fys 40, 497–508
Fröman, N., Fröman, P. O., and Karlsson, F., 1979, Molec Phys 38, 749–767
Fröman, N., Fröman, P. O., and Larsson, K., 1994, Phil Trans R Soc Lond A347, 1–22
Fröman, N., Fröman, P. O., and Lundborg, B., 1988a, Math Proc Camb Phil Soc 104, 153–179
Fröman, N., Fröman, P. O., and Lundborg, B., 1988b, Math Proc Camb Phil Soc 104, 181–191
Fröman, N., Fröman, P. O., and Lundborg, B., 1996, adjoined paper (Chapter 5) in Fröman and Fröman (1996)
Fröman, N., Fröman, P. O., Andersson, N., and Hökback, A., 1992, Phys Rev D45, 2609–2616
Fröman, N., Fröman, P. O., Myhrman, U., and Paulsson, R., 1972, Ann Phys (NY) 74, 314–323
Fröman, P. O., 1957, Mat Fys Skr Dan Vid Selsk 1, No. 3
Fröman, P. O., 1974, Ann Phys (NY) 88, 621–630
Fröman, P. O., 2000, J Math Phys 41, 7952–7963
Fröman, P. O., Karlsson, F., and Yngve, S., 1986, J Math Phys 27, 2738–2747
Fröman, P. O., Yngve, S., and Fröman, N., 1987, J Math Phys 28, 1813–1826
Fröman, P. O., Hökback, A., Walles, E., and Yngve, S., 1985, Ann Phys (NY) 163, 252–264
Fulling, S. A., 1983, SIAM J Math Anal 14, 780–795
Furry, W. H., 1947, Phys Rev 71, 360–371
Gans, R., 1915, Ann Physik (Vierte Folge) 47, 709–736
Glaser, W., and Braun, G., 1954, Act Phys Austriaca 9, 41–74
Glaser, W., and Braun, G., 1955, Act Phys Austriaca 9, 267–296
Green, G., 1837, Trans Camb Phil Soc 6, 457–462
Gustafson, S.-å., and Lindahl, S., 1977, J Comp Phys 24, 81–95
ter Haar, D., Editor, 1964, Selected Problems in Quantum Mechanics. Revised and augmented second edition of Gol'dman, Krivchenkov, Kogan and Galitskii, Problems in Quantum Mechanics. Academic Press, New York
Heading, J., 1962, An Introduction to Phase-Integral Methods. Methuen's Monographs on Physical Subjects, London and New York. Russian translation with an appendix by V. P. Maslov concerning the WKB method in the multi-dimensional case: MIR, Moscow 1965
Horn, J., 1899a, Math Ann 52, 271–292
Horn, J., 1899b, Math Ann 52, 340–362
Jacobi, C. G. J., 1849, Astron Nachr 28, 257–270; also published in C. G. J. Jacobi's Gesammelte Werke, edited by K. Weierstrass, Band 7, pp 175–188, Berlin 1891
Jeffreys, H., 1915, Mem Roy Astr Soc 60 (Part VI), 187–217
Jeffreys, H., 1925, Proc Lond Math Soc (Second Series) 23, 428–436
Jeffreys, H., 1956, Proc Camb Phil Soc 52, 61–66
Karlsson, F., and Jedrzejek, C., 1987, J Chem Phys 86, 3532–3538
Kemble, E. C., 1935, Phys Rev 48, 549–561
Kemble, E. C., 1937, The Fundamental Principles of Quantum Mechanics. McGraw-Hill, New York. Reissue: Dover Publications, New York 1958
Kramers, H. A., 1926, Z Phys 39, 828–840
Kraus, L., and Levine, L. M., 1961, Comm Pure Appl Math 14, 49–68
Krieger, J. B., 1969, J Math Phys 10, 1455–1458
Krieger, J. B., and Rosenzweig, C., 1967, Phys Rev 164, 171–173
Lamb, H., 1895, Hydrodynamics. Cambridge University Press, Cambridge
Langer, R. E., 1934, Bull Am Math Soc 40, 545–582
Larsson, K., and Fröman, P. O., 1994, Phil Trans R Soc Lond A347, 23–35
Linnæus, S., and Düring, M., 1985, Ann Phys (NY) 164, 506–515
Liouville, J., 1837, J Math Pure Appl 2, 16–35
Luke, Y. L., 1969, The Special Functions and Their Approximations, Vol. I. Academic Press, New York
Lundborg, B., 1977, Math Proc Camb Phil Soc 81, 463–483
Maslov, V. P., and Fedoriuk, M. V., 1981, Semi-Classical Approximation in Quantum Mechanics. Reidel Publishing Company, Dordrecht
McHugh, J. A. M., 1971, Arch History Exact Sci 7, 277–324
Messiah, A., 1959, Mécanique Quantique, Vol. 1. Dunod, Paris. English translation: Quantum Mechanics, Vol. 1. North-Holland, Amsterdam. Fourth printing 1967
Olver, F. W. J., 1965a, J Res Nat Bur Standards 69B, 271–290
Olver, F. W. J., 1965b, J Res Nat Bur Standards 69B, 291–300
Paulsson, R., 1985, Ann Phys (NY) 163, 245–251
Paulsson, R., and Fröman, N., 1985, Ann Phys (NY) 163, 227–244
Rayleigh, Lord, (J. W. Strutt), 1912, Proc R Soc London A86, 207–226
Rosenzweig, C., and Krieger, J. B., 1968, J Math Phys 9, 849–860
Scheibner, W., 1856a, Astron J 4, 177–182
Scheibner, W., 1856b, Berichte der Kgl Sächs Ges d Wiss zu Leipzig, Math-Phys Classe 8, 40–64
Scheibner, W., 1880a, Math Ann 17, 531–544
Scheibner, W., 1880b, Math Ann 17, 545–560
Schlesinger, L., 1906, C R Acad Sci (Paris) 142, 1031–1033
Schlesinger, L., 1907, Math Ann 63, 277–300
Schlissel, A., 1977, Arch History Exact Sci 16, 307–378
Siebert, E., and Krieger, J. B., 1970, J Math Phys 11, 3111–3115
Silverstone, H. J., 1985, Phys Rev Lett 55, 2523–2526
Silverstone, H. J., and Koch, P. M., 1979, J Phys B: Atom Molec Phys 12, L537–L541
Skorupski, A. A., 1980, Rep Math Phys 17, 161–187
Skorupski, A. A., 1988, J Math Phys 29, 1814–1823
Soop, M., 1965, Ark Fys 30, 217–229
de Sparre, M., 1898, Atti della R Acc dei Lincei (Ser V). Rendiconti. Classe di sc fis, mat e nat 7:2, 111–117
Streszewski, M., and Jedrzejek, C., 1988, Phys Rev A37, 645–648
Thidé, B., 1980, J Math Phys 21, 1408–1415
Wentzel, G., 1926, Z Phys 38, 518–529
Wheeler, J. A., 1976, Studies in Mathematical Physics: Essays in Honor of Valentine Bargmann, pp 351–422, edited by E. H. Lieb, B. Simon and A. S. Wightman. Princeton University Press, Princeton
Yngve, S., 1972, J Math Phys 13, 324–331
Yngve, S., 1986, Phys Rev A33, 96–104
Yngve, S., 1988, J Math Phys 29, 931–936
Yngve, S., and Linnæus, S., 1986, J Phys A: Math Gen 19, 3017–3031
Zwaan, A., 1929, Intensitäten im Ca-Funkenspektrum. Academisch Proefschrift. Joh. Enschedé en Zonen, Harlem. This doctoral dissertation was also published in: Arch Néerlandaises Sci Exactes Naturelles (Série IIIA) 12, 1–76