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Transformation Theory

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Mathematical Methods in Solid State and Superfluid Theory

Abstract

In these lectures we focus attention on the use of simple explicit canonical transformations to analyze typical Hamiltonians that occur in nonrelativistic physics. In its broadest aspects transformation theory and the “diagonalizing” of Hamiltonians include essentially all of quantum theory. It is therefore not possible to avoid repeating material presented in modern texts on quantum theory. We limit our discussion by ignoring for the most part those unitary transformations that are generated by operators that are exact constants of the motion. The constants of the motion permit one to classify the multiplets corresponding to a given Hamiltonian. The study of the implications of the exact invariance of Hamiltonians under groups of transformations belongs to the subject of standard group theory as applied to quantum mechanics and numerous presentations exist.

Work supported by U.S. Air Force, Office of Scientific Research.

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Bibliography

  • M. Born and P. Jordan, Elementare Quantenmechanik (Springer-Verlag, Berlin, 1930).

    Book  MATH  Google Scholar 

  • P. A. M. Dirac, The Principles of Quantum Mechanics (Oxford University Press, London, 1958), 4th edition.

    MATH  Google Scholar 

  • J. Frenkel, Wave Mechanics, Advanced General Theory (Dover Publications, Inc., New York, 1950).

    Google Scholar 

  • K. Gottfried, Quantum Mechanics (W. A. Benjamin, Inc., New York, 1966), Vol. i. E. M. Henley and W. Thirring, Elementary Quantum Field Theory (McGraw-Hill Book Company, Inc., New York, 1962).

    Google Scholar 

  • F. A. Kaempffer, Concepts in Quantum Mechanics (Academic Press, Inc., New York, 1965).

    MATH  Google Scholar 

  • A. Katz, Classical Mechanics, Quantum Mechanics, Field Theory (Academic Press, Inc., New York, 1965).

    Google Scholar 

  • B. Kursunoglu, Modern Quantum Theory (W. H. Freeman and Company, San Francisco, 1962).

    MATH  Google Scholar 

  • A. Messiah, Quantum Mechanics (John Wiley and Sons, Inc., New York, 1962), Vols, I, II.

    MATH  Google Scholar 

  • P. Roman, Advanced Quantum Theory (Addison-Wesley Publishing Company, Reading, Massachusetts, 1965).

    MATH  Google Scholar 

  • J. Schwinger, in Brandeis University 1960 Summer Institute in Theoretical Physics Lecture Notes (Brandeis University, 1960).

    Google Scholar 

1.7. References for Section 1

  1. V. I. Kogan and V. M. Galitskiy, Problems in Quantum Mechanics (Prentice-Hall Inc., Englewood Cliffs, New Jersey, 1963).

    MATH  Google Scholar 

  2. H. Tung-Li, Thesis, Brandeis University (1967).

    Google Scholar 

  3. B. Kursunoglu, Modern Quantum Theory (W. H. Freeman and Co., San Francisco, 1962).

    MATH  Google Scholar 

  4. E. P. Gross, in Physics of Many Body Systems, E. Meeron, Editor (Gordon and Breach Science Publishers, Inc., New York, 1967); J. Math. Phys., 4, 195 (1963).

    Google Scholar 

2.4. References for Section 2

  1. W. Magnus, Comm. Pure and Appl. Math., 7, 649 (1954).

    Article  MATH  MathSciNet  Google Scholar 

  2. S. Tani, Progr. Theoret. Phys. (Kyoto) 11, 190 (1954).

    Article  ADS  MATH  MathSciNet  Google Scholar 

  3. P. Carruthers and M. M. Nffito, Amer. J. Phys. 33, 537 (1965).

    Article  ADS  MathSciNet  Google Scholar 

  4. R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill Book Company, Inc., New York, 1965).

    MATH  Google Scholar 

  5. R. W. Fuller, S. M. Harris and E. L. Slaggie, Amer. J. Phys. 31, 431 (1963).

    Article  ADS  MATH  MathSciNet  Google Scholar 

  6. K. Husimi, Progr. The ret. Phys. (Kyoto) 9, 381 (1953).

    ADS  MathSciNet  Google Scholar 

  7. L. M. Scarfone, Amer. J. Phys. 32, 158 (1964).

    Article  ADS  MATH  MathSciNet  Google Scholar 

  8. Selected Problems in Quantum Mechanics, D. ter Haar, Editor (Academic Press Inc., New York, 1964).

    Google Scholar 

  9. M. Gell-Mann and F. Low, Phys. Rev. 84, 350 (1951).

    Article  ADS  MATH  MathSciNet  Google Scholar 

  10. S. S. Schweber, An Introduction to Relativistic Quantum Field Theory (Row, Peterson and Company, Evanson, Illinois, 1961).

    Google Scholar 

  11. S. S. Schweber, An Introduction to Relativistic Quantum Field Theory (Row, Peterson and Company, Evanson, Illinois, 1961).

    Google Scholar 

3.3. References for Section 3, 3.3.1 Operator Techniques

  1. K. Aizu, J. Math. Phys. 4, 762 (1963).

    Article  ADS  MATH  MathSciNet  Google Scholar 

  2. F. A. Berezin, The Method of Second Quantization (Academic Press, Inc., New York, 1966).

    MATH  Google Scholar 

  3. R. P. Feynman, Phys. Rev. 84, 108 (1951).

    Article  ADS  MATH  MathSciNet  Google Scholar 

  4. K. O. Friedrichs, Mathematical Aspects of the Quantum Theory of Fields (Interscience Publishers, Inc., New York, 1953).

    MATH  Google Scholar 

  5. K. Kumar, J. Math. Phys. 6, 1928 (1965); Perturbation Theory and the Nuclear Many Body Problem (North-Holland Publishing Co., Amsterdam, 1962).

    Article  ADS  MATH  Google Scholar 

  6. W. H. Louisell, Radiation and Noise in Quantum Electronics (McGraw-Hill Book Company, Inc., New York, 1964).

    Google Scholar 

  7. R. M. Wilcox, J. Math. Phys. 8, 962 (1967).

    Article  ADS  MATH  MathSciNet  Google Scholar 

3.3.2 Evolution Operator

  1. B. S. DeWitt, Phys. Rev. 100, 905 (1955).

    Article  ADS  MATH  MathSciNet  Google Scholar 

  2. M. Gellman and M. L. Goldberger, Phys. Rev. 91, 398 (1953).

    Article  ADS  Google Scholar 

  3. M. L. Goldberger, Phys. Rev. 84, 929 (1951).

    Article  ADS  MATH  MathSciNet  Google Scholar 

  4. W. Heitler, The Quantum Theory of Radiation (Oxford University Press, London 1954), 3rd edition.

    MATH  Google Scholar 

  5. N. M. Hugenholtz, Physica 23, 533 (1957).

    Article  ADS  MATH  MathSciNet  Google Scholar 

  6. B. A. Lippman and J. Schwinger, Phys. Rev. 79, 469 (1950).

    Article  ADS  Google Scholar 

  7. A. Messiah, Quantum Mechanics (John Wiley and Sons, Inc., New York, 1962), Vols, I, II.

    MATH  Google Scholar 

  8. P. Roman, Advanced Quantum Theory (Addison-Wesley Publishing Company, Reading, Massachusetts, 1965).

    MATH  Google Scholar 

  9. M. Schonberg, Nuovo Cimento Ser. 9, 8, 243, 403, 651, 817 (1951).

    MathSciNet  Google Scholar 

  10. L. Van Hove, Physica 21, 901 (1955), 22, 343 (1956).

    Article  ADS  MATH  MathSciNet  Google Scholar 

  11. A. Visconti, Journal de Physique et le Radium 14, 591 (1953); Theorie Quantique des Champs II (Gauthiers Villars et Cie, Paris, 1965).

    Article  MATH  MathSciNet  Google Scholar 

4.7. References for Section 4

  1. Polarons and Excitons, Scottish Universities’ Summer School 1962, Editors C. G. Kuper and G. D. Whitfield (Oliver and Boyd, Edinburgh and London, 1963).

    Google Scholar 

  2. R. Jost, Phys. Rev. 72, 815 (1947).

    Article  ADS  MATH  Google Scholar 

  3. T. D. Lee, F. Low and D. Pines, Phys. Rev. 90, 297 (1953).

    Article  ADS  MATH  MathSciNet  Google Scholar 

  4. T. Kato, T. Kobayashi and M. Namiki, Progr. Theoret. Phys. (Kyoto) Suppl. No. 15, 3 (1960).

    Google Scholar 

  5. O. W. Greenberg and S. S. Schweber, Nuovo Cimento Ser. 10, 8, 378 (1958).

    Google Scholar 

  6. S. S. Schweber, An Introduction to Relativistic Quantum Field Theory (Row, Peterson and’Company, Evanston, Illinois, 1961).

    Google Scholar 

  7. S. F. Edwards and R. E. Peierls, Proc. Roy. Soc. (London) A224, 24 (1954).

    Article  ADS  MATH  MathSciNet  Google Scholar 

  8. L. D. Landau and S. I. Pekar, Zh. Eksperim. i Teor. Fiz. 18, 419 (1948).

    Google Scholar 

  9. S. I. Pekar, Untersuchungen über die Elektronentheorie der Kristalle (Akademie-Verlag, Berlin, 1954).

    MATH  Google Scholar 

  10. E. P. Gross, Annals of Physics 8, 78 (1959), 19, 234 (1962). U.S. Yukon, Thesis, Brandeis University (1967).

    Article  ADS  MATH  MathSciNet  Google Scholar 

5.4. References to Section 5

  1. F. Bloch and A. Nordsieck, Phys. Rev. 52, 54 (1937).

    Article  ADS  Google Scholar 

  2. E. P. Gross, Annals of Physics 8, 78 (1959).

    Article  ADS  MATH  MathSciNet  Google Scholar 

  3. T. D. Lee and D. Pines, Phys. Ret. 92, 883 (1953).

    Article  ADS  MATH  MathSciNet  Google Scholar 

  4. S. I. Tomonaga, Progr. Theoret. Phys. (Kyoto) Suppl. No. 2 (1955).

    Google Scholar 

  5. E. P. Gross, Phys. Rev. 100, 1571 (1955).

    Article  ADS  MATH  Google Scholar 

  6. T. D. Schultz, Solid State and Molecular Theory Group M.I.T., Tech. Report 9 (August 1956).

    Google Scholar 

  7. G. Wentzel, Helv. Phys. Acta 15, 111 (1942).

    Google Scholar 

  8. A. Klein and B. H. McCormick, Phys. Rev. 98, 1428 (1955).

    Article  ADS  MATH  MathSciNet  Google Scholar 

  9. E. M. Henley and W. Thirring, Elementary Quantum Field Theory (McGraw-Hill Book Company, Inc., New York, 1962).

    MATH  Google Scholar 

  10. A. V. Tulub, Zh. Eksperim. i Teor. Fiz. 41, 1828 (1961), Soviet Physics—JETP 14, 1301 (1962). U.S. Yukon, Thesis, Brandeis University (1967).

    Google Scholar 

  11. S. Trubatch, Thesis, Brandeis University (1967).

    Google Scholar 

  12. G. Hohler, Nuovo Cimento Ser. 10, 2, 691 (1955).

    Google Scholar 

  13. V. Buymistrov and S. I. Pekar, Zh. Eksperim. i Teor. Fiz. 32, 1193 (1957),

    Google Scholar 

  14. V. Buymistrov and S. I. Pekar, Soviet Physics—JETP 5, 970 (1957).

    Google Scholar 

6.5. References to Section 6

  1. F. M. Eger and E. P. Gross, Annals of Physics 24, 63 (1963);

    Article  ADS  MathSciNet  Google Scholar 

  2. F. M. Eger and E. P. Gross, J. Math. Phys. 6, 891 (1965).

    Article  ADS  MathSciNet  Google Scholar 

  3. F. M. Eger, Thesis, Brandeis University (1963).

    Google Scholar 

  4. F. M. Eger and E. P. Gross, Nuovo Cimento Ser. 10, 34, 1225 (1964).

    Article  MathSciNet  Google Scholar 

  5. P. Jordan, Z.f. Physik 37, 383 (1926),

    Article  ADS  MATH  Google Scholar 

  6. P. Jordan, Z.f. Physik 38, 513 (1926).

    Article  ADS  Google Scholar 

  7. S. Nakai, Progr. Theoret. Phys. (Kyoto) 13, 380 (1955).

    Article  ADS  MATH  MathSciNet  Google Scholar 

  8. B. Podolsky, Phys. Rev. 32, 812 (1928).

    Article  ADS  MATH  Google Scholar 

  9. B. S. DeWitt, Phys. Rev. 85, 653 (1952);

    Article  ADS  MathSciNet  Google Scholar 

  10. B. S. DeWitt, Rev. Mod. Phys. 29, 377 (1957).

    Article  ADS  MATH  MathSciNet  Google Scholar 

  11. D. Amit, Thesis, Brandeis University (1966).

    Google Scholar 

  12. N. Witriol, Thesis, Brandeis University (1967).

    Google Scholar 

  13. J. Bell, in The Many-Body Problem, 1st Bergen School of Physics, Editor C. Fronsdahl (W. A. Benjamin, Inc., New York, 1962); and in Proceedings of the Rutherford Jubilee International Conference, Manchester (1961), Editor J. B. Birks (Heywood and Company, London).

    Google Scholar 

  14. F. M. Eger and E. P. Gross, J. Math. Phys. 7, 578 (1966).

    Article  ADS  MathSciNet  Google Scholar 

  15. A. Bul, Physica 7, 869 (1940).

    Article  ADS  Google Scholar 

  16. R. B. Dingle, Phil. Mag. Ser. 7, 40, 573 (1949).

    MATH  Google Scholar 

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Authors and Affiliations

  1. Brandeis University, Waltham, Massachusetts, USA

    Eugene P. Gross

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  1. Eugene P. Gross
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Editor information

R. C. Clark G. H. Derrick

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© 1968 Springer Science+Business Media New York

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Gross, E.P. (1968). Transformation Theory. In: Clark, R.C., Derrick, G.H. (eds) Mathematical Methods in Solid State and Superfluid Theory. Mathematical Methods in Solid State and Superfluid Theory. Springer, Boston, MA. https://doi.org/10.1007/978-1-4899-6435-9_2

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  • DOI: https://doi.org/10.1007/978-1-4899-6435-9_2

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4899-6214-0

  • Online ISBN: 978-1-4899-6435-9

  • eBook Packages: Springer Book Archive

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