General Unitary Transformations in Quantum Mechanics

Arlen Anderson, Montréal Pq Canada Ha T

Quantum canonical transformations are defined algebraically outside of a Hilbert space context. This generalizes the quantum canonical transformations of Weyl and Dirac to include non-unitary transformations. The importance of non-unitary transformations for constructing solutions of the Schrödinger equation is discussed. Three elementary canonical transformations are shown both to have quantum implementations as finite transformations and to generate, classically and infinitesimally, the full canonical algebra. A general canonical transformation can be realized quantum mechanically as a product of these transformations. Each transformation corresponds to a familiar tool used in solving differential equations, and

Maciej Błaszak, Ziemowit Domański

Chung‐In Um

In a general time-dependent quadratic Hamiltonian, we demonstrate that canonical transformations in classical mechanics correspond to unitary transformations in quantum mechanics. The values of some functions of canonical variables q and p are generally not equal to the values of same functions of the other canonical variables Q and P. The wave functions, propagators and uncertainty relations are evaluated explicitly using the dynamical invariant method.

A. Anderson

Wenwen LYU, Feng Wang

This paper introduces the development of the principle of relativity from Galileo's principle of relativity to special relativity and then to general relativity. Inspired by this, we explored the transformation form of quantum mechanics to meet Galileo's principle of relativity. Firstly, from the point that unitary transformation can keep the physical laws of quantum mechanics unchanged, we correspond the momentum translation unitary transformation in quantum mechanics to the Galileo transformation in classical mechanics, and reveal that the transformation form of quantum mechanics that meets the Galileo principle of relativity can be realized through the momentum translation unitary transformation. This inquiry is conducive to students' in-depth thinking about the corresponding relationship between quantum mechanics and classical physics, can help beginners reduce the conflict between quantum physics and classical physics, and help students build a bridge from classical to quantum thinking.