Quantum Systems Connected by a Time-Dependent Canonical Transformation

We study both classical and quantum relation between two Hamiltonian systems which are mutuallyconnected by time-dependent canonical transformation.One is ordinary conservative system and the other is time-dependentHamiltonian system.The quantum unitary operator relevant to classieai canonical transformation betweenthe two systems are obtained through rigorous evaluation.With the aid of the unitary operator,we have derivedquantum states of the time-dependent Hamiltonian system through transforming the quantum states of the conservativesystem.The invariant operators of the two systems are presented and the relation between them are addressed.Weshowed that there exist numerous Hamiltonians,which gives the same classical equation of motion.Though it isimpossible to distinguish the systems described by these Hamiltonians within the realm of classical mechanics,they canbe distinguishable quantum mechanically.

Testing the validity of the Ehrenfest theorem beyond simple static systems: Caldirola–Kanai oscillator driven by a time-dependent force

The relationship between quantum mechanics and classical mechanics is investigated by taking a Gaussian-type wave packet as a solution of the Schr o¨dinger equation for the Caldirola–Kanai oscillator driven by a sinusoidal force. For this time-dependent system, quantum properties are studied by using the invariant theory of Lewis and Riesenfeld. In particular,we analyze time behaviors of quantum expectation values of position and momentum variables and compare them to those of the counterpart classical ones. Based on this, we check whether the Ehrenfest theorem which was originally developed in static quantum systems can be extended to such time-varying systems without problems.

Nonlinear description of Yang-Mills cosmology： cosmic inflation and the accompanying Hannay's angle

Hannay＇s angle is a classical analogue of the ＂geometrical phase factor＂ found by Berry in his research on the quantum adiabatic theorem. This classical analogue is defined if closed curves of constant action variables return to the same curves in phase space after an adaibatic evolution. Adiabatic evolution of Yang-Mills cosmology, which is described by a time-dependent quartic oscillator, is investigated. Phase properties of the Yang-Mills fields are analyzed and the corresponding Hannay＇s angle is derived from a rigorous evaluation. The obtained Hannay＇s angle shift is represented in terms of several observable parameters associated with such an angle shift. The time evolution of Hannay＇s angle in Yang-Mills cosmology is examined from an illustration plotted on the basis of numerical evaluation,and its physical nature is addressed. Hannay＇s angle, together with its quantum counterpart Berry＇s phase, plays a pivotal role in conceptual understanding of several cosmological problems and indeed can be used as a supplementary probe for cosmic inflation.

An approach to dark energy problem through linear invariants

The time evolution of vacuum energy density is investigated in the coherent states of inflationary universe using a linear invariant approach. The linear invariants we derived are represented in terms of annihilation operators. On account of the fact that the coherent state is an eigenstate of an annihilation operator, the wave function in the coherent state is easily evaluated by solving the eigenvalue equation of the linear invariants. The expectation value of the vacuum energy density is derived using this wave function. Fluctuations of the scalar field and its conjugate momentum are also investigated. Our theory based on the linear invariant shows that the vacuum energy density of the universe in a coherent state is decreased continuously with time due to nonconservative force acting on the coherent oscillations of the scalar field, which is provided by the expansion of the universe. In effect, our analysis reveals that the vacuum energy density decreases in proportion to t-β where β is 3/2 for radiation-dominated era and 2 for matter-dominated era. In the case where the duration term of radiation-dominated era is short enough to be negligible, the estimation of the relic vacuum energy density agrees well with the current observational data.

Quantum-classical correspondence for the inverted oscillator

While quantum-classical correspondence for a system is a very fundamental problem in modern physics,the understanding of its mechanism is often elusive,so the methods used and the results of detailed theoretical analysis have been accompanied by active debate.In this study,the differences and similarities between quantum and classical behavior for an inverted oscillator have been analyzed based on the description of a complete generalized Airy function-type quantum wave solution.The inverted oscillator model plays an important role in several branches of cosmology and particle physics.The quantum wave packet of the system is composed of many sub-packets that are localized at different positions with regular intervals between them.It is shown from illustrations of the probability density that,although the quantum trajectory of the wave propagation is somewhat different from the corresponding classical one,the difference becomes relatively small when the classical excitation is sufficiently high.We have confirmed that a quantum wave packet moving along a positive or negative direction accelerates over time like a classical wave.From these main interpretations and others in the text,we conclude that our theory exquisitely illustrates quantum and classical correspondence for the system,which is a crucial concept in quantum mechanics.