Nonlinear canonical transformations and their representations in quantum mechanics

Journal Article · Wed Oct 01 04:00:00 EDT 1975 · Journal of Mathematical Physics

DOI:https://doi.org/10.1063/1.522433· OSTI ID:4145588

Mello, P. A.

In the last few years an extensive literature has developed on linear canonical transformations and their representation in quantum mechanics. Applications of these results have been made to clustering theory in nuclei, problems of accidental degeneracy, etc. In the present paper we wish to turn our attention to nonlinear canonical transformations. We show that by dealing with appropriate functions f_α (α=1,...,2n) of x _i , p _i (i=1,...,n) rather than with these variables themselves, we can in principle set unambiguously the equations that determine the representation in quantum mechanics of the canonical transformation under study. This result holds when the old and new functions f _α have the same spectrum. We discuss specific examples when this last condition is satisfied: nonlinear canonical transformations in the radial variable that were obtained from projection of linear ones in higher-dimensional spaces; canonical transformations that take us from one Hamiltonian to another with the same spectrum, be this one continuous or discrete; canonical transformations that relate two sets of integrals of motion (which include the Hamiltonians) when we are dealing with phase spaces of dimensions higher than 2, etc. Finally, we discuss briefly, in the concluding section, the possibility of extending our analysis to canonical transformations that do not conserve the spectrum of the relevant operators.

• Research Organization: Universidad de Mexico, Mexico City
• Sponsoring Organization: USDOE
• NSA Number: NSA-33-008609
• OSTI ID: 4145588
• Journal Information: Journal of Mathematical Physics, Journal Name: Journal of Mathematical Physics Journal Issue: 10 Vol. 16; ISSN JMAPAQ; ISSN 0022-2488
• Publisher: American Institute of Physics (AIP)
• Country of Publication: United States
• Language: English

Related Subjects

*QUANTUM MECHANICS-- CANONICAL TRANSFORMATIONS
644002*
N76100* --Physics (Theoretical)--General
NONLINEAR PROBLEMS
QUANTUM OPERATORS