Vector Spaces and Linear Transformations

  • Ralph E. Christoffersen
Part of the Springer Advanced Texts in Chemistry book series (SATC)


Having seen that it was necessary to construct a new theory to describe the behavior of sub-microscopic particles, it should not be surprising that mathematical techniques were introduced concurrently to aid in the development of the new theory.


Vector Space Linear Operator Basis Vector Linear Transformation Product Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 2.
    See, for example, Paul R. Halmos, Finite-Dimensional Vector Spaces, Van Nostrand, New York, 1958, pp. 3–4.Google Scholar
  2. 17.
    See, for example, Advanced Calculus, by A. E. Taylor, Ginn and Co., NY, 1955, Chapter 16.Google Scholar
  3. 25.
    G. F. Roach, Green’s Functions, van Nostrand Reinhold, New York, 1970.Google Scholar
  4. 29.
    For a good discussion of the different convergence criteria, see F. W. Byron and R. W. Fuller, Mathematics of Classical and Quantum Physics, Vol. I, Addison-Wesley Publishers, Reading, MA, 1969.Google Scholar
  5. 30.
    A more general mathematical treatment of this problem shows that both 3C and F belong to an abstract Hilbert space. This shows their basic similarity since both the functions and sequences belong to the same abstract space. For a detailed discussion of this point, see W. Schmeidler, Linear Operators in Hilbert Space,Academic Press, New York, 1965.Google Scholar
  6. 31.
    For more details, the reader is referred to treatises such as R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. I, Interscience Publishers, New York, 1953, pp. 69–81.Google Scholar
  7. 32.
    For additional discussion see, for example, W. Kaplan, Advanced Calculus, Addison-Wesley, Cambridge, MA, 1952, p. 434.Google Scholar
  8. 33.
    Tables of Integral Transforms Vols. I-II, edited by A. Erdelyi, McGraw-Hill Book Co., New York, 1954.Google Scholar
  9. 34.
    This “function” was introduced by P. A. M. Dirac, Quantum Mechanics,Oxford University Press, 1958, pp. 58–62.Google Scholar
  10. E. Meibacher, Quantum Mechanics, 2nd ed., Wiley, New York, 1970, pp. 82–85.Google Scholar
  11. 35.
    Proper proofs of these properties can be given using Distribution Theory (cf. A. Messiah, Quantum Mechanics, Vol. 1, North-Holland Publishing Co., Amsterdam, 1965, Appendix A). The “proofs” given here are strictly formal proofs.Google Scholar
  12. 38.
    For additional discussion, see G. Arfken, Mathematical Methods for Physicists, Academic Press, New York, 1968, Sect. 17. 6.Google Scholar
  13. 40.
    More general mappings of the form a: V→V’ are discussed in most advanced books in linear algebra, see, e.g., K. Hoffman and R. Kunze, Linear Algebra, Prentice-Hall, Englewood Cliffs, NJ, 1961.Google Scholar
  14. 41.
    See, for example, J. Indritz, Methods in Analysis, Macmillan, New York, 1963, p. 22.Google Scholar
  15. 44.
    For a development of the properties and uses of these operators, see, for example, P. O. Löwdin, Phys. Rev., 97, 1509 (1955)CrossRefGoogle Scholar
  16. P. O. Löwdin, Rev. Mod. Phys., 34, 520 (1962)CrossRefGoogle Scholar
  17. P. O. Löwdin, J. Math Phys., 3, 969 (1962).CrossRefGoogle Scholar
  18. J. V. Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, Princeton, 1955.Google Scholar
  19. 45.
    P. O. Löwdin, Rev. Modern Phys., 34, 520 (1962).CrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1989

Authors and Affiliations

  • Ralph E. Christoffersen
    • 1
  1. 1.The Upjohn CompanyKalamazooUSA

Personalised recommendations