General Unitary Transformations in Quantum Mechanics

Abstract

Unitary transformations are a cornerstone of quantum mechanics. The special unitary transformations which depend on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{x}$$\end{document} and t have been established from the perspective of the equivalent Lagrangian transformations. The general unitary transformations which depend on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{x}$$\end{document}, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{p}_{x}$$\end{document} and t, however, are not associated with a change of Lagrangian. In this paper, we elucidate how to construct the unitary transformations from the perspective of the canonical transformations. In particular, we demonstrate that the general unitary transformations are induced by an infinite succession of infinitesimal canonical transformations. The generators of the general unitary transformations coincide with those of the infinitesimal canonical transformations. Thus we verify, from the perspective of the canonical transformations, the form invariance of the Schr & ouml;dinger equation under the general unitary transformations. We conclude that the form invariance of the Hamilton's equations under an infinite succession of infinitesimal canonical transformations ensures, after the canonical quantization, the form invariance of the Schr & ouml;dinger equation under the general unitary transformations.