Bulletin of Science and Technology

The Study of the System with Varying Mass by Making Use of the Time-dependent Unitary Transformations

Yu You

Published Jan 1, 2000• 0 citations

Abstract

Method of the time dependent unitary transformation is one of powerful tools for the study of the time dependent system. The time evolution of a harmonic oscillator with varying mass was studied by making use of a sequence of the time dependent unitary transformations and the exact solution of the system obtained. An explicit expression for the exact evolving state of a harmonic oscillator, whose mass varies as non negtive powers of time was also given.

Related papers

Time Evolution of the Time-Dependent Harmonic Oscillator

Jing‐Bo Xu, YU You-hong

The time evolution of the time-dependent harmonic oscillator is studied by a sequence of unitary transformations and the exact evolving state for the system is obtained. A specific model of frequency variation for the time-dependent harmonic oscillator is discussed as an illustrative example.

Communications in Theoretical PhysicsApr 30, 19984 citations

Unitary relations in time-dependent harmonic oscillators

Dae-Yup Song

For a harmonic oscillator with time-dependent (positive) mass and frequency, a unitary operator is shown to transform the quantum states of the system to those of a harmonic oscillator system of unit mass and time-dependent frequency, as well as operators. For a driven harmonic oscillator, a unitary transformation which relates the driven system and a system of the same mass and frequency without a driving force is given, as a generalization of previous results, in terms of the solution of classical equation of motion of the driven system. Thus transformations give a simple way of finding exact wavefunctions of a driven harmonic oscillator system, provided the quantum states of the corresponding system of unit mass are given.

arXiv:quant-ph/9812038Journal of Physics A Mathematical and GeneralJan 1, 199924 citations

Unitary relation between a harmonic oscillator of time-dependent frequency and a simple harmonic oscillator with and without an inverse-square potential

Dae-Yup Song

The unitary operator that transforms a harmonic oscillator system of time-dependent frequency into that of a simple harmonic oscillator of different timescale is found, with and without an inverse-square potential. It is shown that for both cases, this operator can be used in finding complete sets of wave functions of a generalized harmonic oscillator system from the well-known sets of the simple harmonic oscillator. Exact invariants of the time-dependent systems can also be obtained from the constant Hamiltonians of unit mass and frequency by making use of this unitary transformation. The geometric phases for the wave functions of a generalized harmonic oscillator with an inverse-square potential are given.

arXiv:quant-ph/0002065Physical Review AJun 14, 200041 citations

Path integral for new models of time-dependent harmonic oscillators

Badri Berrabah, Baya Bentag

We have presented a particular direct method to solve time-dependent quantum problems within the framework of path integrals by using explicitly time-dependent transformations. We have applied it to the case of harmonic oscillator with time-dependent mass and frequency. Some examples have illustrated the method used. From their exact propagators, normalized solutions have been obtained.

International Journal of Geometric Methods in Modern PhysicsDec 12, 20191 citation

Time-dependent coupled harmonic oscillators: Classical and quantum solutions

D. X. Macedo, I. Guedes

In this work we present the classical and quantum solutions for an arbitrary system of time-dependent coupled harmonic oscillators, where the masses (m), frequencies (ω) and coupling parameter (k) are functions of time. To obtain the classical solutions, we use a coordinate and momentum transformations along with a canonical transformation to write the original Hamiltonian as the sum of two Hamiltonians of uncoupled harmonic oscillators with modified time-dependent frequencies and unitary masses. To obtain the exact quantum solutions we use a unitary transformation and the Lewis and Riesenfeld (LR) invariant method. The exact wave functions are obtained by solving the respective Milne–Pinney (MP) equation for each system. We obtain the solutions for the system with m1 = m2 = m0eγt, ω1 = ω01e-γt/2, ω2 = ω02e-γt/2 and k = k0.

International Journal of Modern Physics EAug 25, 20144 citations

Harmonic oscillators coupled time-dependent

Diego Ximenes Macedo

In this work we present the classical and quantum solutions of time-dependent coupled harmonic oscillators. In these systems the masses, frequencies and coupling parameter (k) are functions of time. Four systems are investigated. To obtain the classical solutions we use a coordinate and momentum transformations along with a canonical transformation to write the original Hamiltonian as the sum of two Hamiltonians of uncoupled harmonic oscillators with modified time-dependent frequencies and unitary masses. We find the analytical expression for position and velocity of each oscillator of the systems. To obtain the exact quantum solutions we use a unitary transformation and the Lewis and Riesenfeld invariant method. The wave functions obtained are written in terms of a c-number quantity () which is solution of the Milne-Pinney equation. For each system we solve the respective Milne-Pinney equation and discuss how the quantum fluctuations and the uncertainty product evolve with time.

Jan 1, 20120 citations