Abstract
The quantum mechanical relationships between time-dependent oscillators, Hamilton-Jacobi theory and an invariant operator are clarified by making reference to a system with a generalized oscillator. We introduce a linear transformation in position and momentum, and show that the correspondence between classical and quantum transformations is exactly one-to-one. We found that classical canonical transformations are constructed from quantum unitary transformations as long as we are concerned with linear transformations. We also show the relationship between the invariant operator and a linear transformation.
Highlights
Canonical transformations are a highlight in classical mechanics
The quantum mechanical relationships between time-dependent oscillators, HamiltonJacobi theory and an invariant operator are clarified by making reference to a system with a generalized oscillator
We found that classical canonical transformations are constructed from quantum unitary transformations as long as we are concerned with linear transformations
Summary
Canonical transformations are a highlight in classical mechanics. They give solutions to classical mechanical systems, and an insight into the quantization of them. The idea of canonical transformations has so far not been fully utilized in quantum systems This issue was raised by Dirac [1] [2] [3] just after the birth of quantum mechanics. There, he only discussed the case of a time-independent canonical transformation. There has been renewed interest coming in this field in the context of Hamilton-Jacobi theory [4] [5] and action-angle variables [6] While these articles were focused on time-independent transformations, time-dependent ones were discussed [7]. The introduction of an invariant operator to construct solutions for time-dependent Hamiltonian systems has been proposed [8] [9] [10]
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
