Abstract
We discuss through the examples of the oscillator with centrifugal potential and the Coulomb problem in dilated form, the general procedure for finding the representation in quantum mechanics of non-linear and non-bijective canonical transformations. The ambiguity group associated with the canonical transformations and the irreducible representations of this group, which lead to the concept of ambiguity spin, suggest also that some features of the spectra of quantum operators are already present in the classical picture.
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References
P. Kramer, M. Moshinsky and T.H. Seligman, J. Math. Phys. 19 (1978) 683.
M. Moshinsky and T.H. Seligman, Ann. Phys. (N.Y.) (1978) (in press).
P.A. Mello and M. Moshinsky, J. Math. Phys. 16 (1975) 2017
H. Goldstein, "Classical Mechanics" (Addison-Wesley, Reading 1957)
R.S. Burington and C.C. Torrance, "Higher Mathematics", (Mc Graw-Hill Inc., New York 1939) p. 123
M. Moshinsky, T.H. Seligman and K.B. Wolf, J. Math. Phys. 13 (1972) 901
I.S. Gradstein and I.M. Ryzik, "Tables of integrals, series and products", (Academic Press, New York, 1965) p. 81; pp. 1057
M. Moshinsky, "Canonical transformations and their representation in quantum mechanics" in "Differential Geometric Methods in Mathematical Physics, Bonn Conference 1975, Lecture Notes in Mathematical Physics 570", Editors K. Bleuler and A. Reetz, (Springer, Heidelberg, 1977).
P.A.M. Dirac, "The principles of quantum mechanics", (Clarendon Press, Oxford, First Edition 1930, Third Edition 1947, pp. 103–107).
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Moshinsky, M., Seligman, T.H. (1978). Ambiguities in canonical transformations of classical systems and the spectra of quantum observables. In: Bleuler, K., Reetz, A., Petry, H.R. (eds) Differential Geometrical Methods in Mathematical Physics II. Lecture Notes in Mathematics, vol 676. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0063685
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DOI: https://doi.org/10.1007/BFb0063685
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