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AIP conference proceedings

Canonical and unitary transformation for a general time-dependent quadratic Hamiltonian system

Chung‐In Um

Published Jan 1, 2000• 0 citations
Publisher

Abstract

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.

DOI
10.1063/1.59946

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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.
We consider general time-dependent quadratic Hamiltonian systems which are connected by canonical transformations and give the same classical equations of motion. In those systems, we demonstrate that canonical transformations in classical mechanics correspond to unitary transformations in quantum mechanics. The wave functions and the propagators are evaluated using the invariant operator method. However, the values of some functions of the canonical variables q and p are not equal to the values of the same functions of the other canonical variables Q and P, but the values of the functions of q and the kinetic momentum pk{\mathrm{p}}_{\mathrm{k}}pk​ are equal to those of the other Q and Pk{\mathrm{P}}_{\mathrm{k}}Pk​ in classical mechanics. We prove that these also hold in the quantum treatment. The uncertainty relations of momentum and position are evaluated for the two Hamiltonians.
We study both classical and quantum relation between two Hamiltonian systems which are mutually connected by time-dependent canonical transformation. One is ordinary conservative system and the other is time-dependent Hamiltonian system. The quantum unitary operator relevant to classical canonical transformation between the two systems are obtained through rigorous evaluation. With the aid of the unitary operator, we have derived quantum states of the time-dependent Hamiltonian system through transforming the quantum states of the conservative system. The invariant operators of the two systems are presented and the relation between them are addressed. We showed that there exist numerous Hamiltonians, which gives the same classical equation of motion. Though it is impossible to distinguish the systems described by these Hamiltonians within the realm of classical mechanics, they can be distinguishable quantum mechanically.
General Hamiltonian systems related by linear canonical transformations, which are combinations of the scale and gauge transformations, are considered. Using path integrals, it is proven that each quantum Hamiltonian can be expressed by the classical Hamiltonian whose canonical variables are replaced by their corresponding quantum operators. The relation between the Schr\"odinger solution and the propagator for the transformed (new) and for the original (old) systems are evaluated by applying the unitary operator which describes the linear relationship between their quantum operators as the corresponding relations between their classical forms. It is shown that the uncertainty relations between the canonical position and momentum operator depend on the gauge function chosen, and satisfy Heisenberg's uncertainty principle. The uncertainty relations between the canonical position and kinetic momentum operator (distinguished from the momentum operator) do not depend on the gauge function chosen, and may not satisfy Heisenberg's uncertainty principle. By the gauge transformation, a single system has innumerable Schr\"odinger equations, but the quantum averages of the function of the position and kinetic momentum operators are invariant for all solutions, as for classical cases.
The time evolution of a quantum system with at most quadratic Hamiltonian is described with the help of different methods, namely the time-dependent Schr\"odinger equation, the time propagator or Feynman kernel, and the Wigner function. It is shown that all three methods are connected via a dynamical invariant, the so-called Ermakov invariant. This invariant introduces explicitly the quantum aspect via the position uncertainty and its possible time dependence. The importance of this aspect, also for the difference between classical and quantum dynamics, and in particular the role of the initial position uncertainty is investigated.