Construction and Properties of Canonical Transforms

José M. Cerveró, Rodriguez-Gonzalez, A.

In this Paper we present an approach to Quantum Mechanical Canonical Transformations. Our main result is that Time Dependent Quantum Canonical Transformations can always be cast in the form of Squeezing Operators. We revise the main properties of these operators in regard to its Lie group properties, how two of them can be combined to yield another operator of the same class and how can also be decomposed and fragmented. In the second part of the paper we show how this procedure works extremely well for the Time Dependent Quantum Harmonic Oscillator. The issue of the systematic construction of Quantum Canonical Transformations is also discussed along the lines of Dirac, Wigner and Schwinger ideas and to the more recent work by Lee. The main conclusion is that the Classical Phase Space Transformation can be maintained in the operator formalism but the construction of the Quantum Canonical Transformation is not clearly related to the Classical Generating Function of a Classical Canonical Transformation. We propose the road of Squeezing Operators rather than the old one attached to Quantum Operators constructed under the guideline of the exponential of the Classical Generating Function.

Kurt Bernardo Wolf

Tan Si Do

In order to obtain with simplicity the properties of linear canonical transformations (LCTs), it is introduced the notion of dual couple of operators having commutator identical to unity and the property that all relations between them are valuable for any other dual couple. It follows that from the translation operator exp (a\({\partial}\)x) which transforms the dual couple (\({\partial}\)x, \(\hat{X}\) ) into (\({\partial}\)x, \(\hat{X}\) + al) one obtains the dilatation operator exp (aBA) which transforms the dual couple (A, B) into (e -al A, e -al B) then the operator exp (aA2) exp (aB2) and so all, leading to the construction of general linear and linear canonical transformations in phase spaces. Moreover, are also obtained the LCT transforms of functions. By this way different kinds of LCTs such as Fast Fourier, Fourier, Laplace, Xin Ma and Rhodes, Baker-Campbell-Haussdorf, Bargman transforms are found again. We also obtain a clear relationship between linear and linear canonical transforms from the formula representing the aforementioned integral realisation. Numerous LCT examples are provided to highlight the method's ease of use.

Nikolai N. Bogolubov, Anatoliy K. Prykarpatsky

The analytic aspects of the Bogoliubov generating functional equations and their transformation properties within the Bogoliubov canonical transformation method are studied. The classical Bogoliubov idea [2] to use the Wigner density operator transformation for studying the non equilibrium distribution functions within the Bogoliubov canonical transformation method is developed, a new analytic non-stationary solution to the classical N.N. Bogoliubov evolution functional equation is constructed.

時珪 柳