Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physics Research International

Quantum Analysis on Time Behavior of a Lengthening Pendulum

Jeong Ryeol Choi, Ji Nny Song

Published Feb 3, 2016• 0 citations
Publisher

Abstract

Quantum properties of a lengthening pendulum are studied under the assumption that the length of the string increases at a steady rate. Advanced analysis for various physical problems in several types of quantum states, such as propagators, Wigner distribution functions, energy eigenvalues, probability densities, and dispersions of physical quantities, is carried out using quantum wave functions of the system. In particular, the time behavior of Gaussian-type wave packets is investigated in detail. The probability density for a Gaussian wave packet displaced in the positive θ at t=0 oscillates back and forth from the center ( θ=0 ). This phenomenon is very similar to the classical motion of the pendulum. As a consequence, we can confirm that there is a correspondence between its quantum and classical behaviors. When we analyze a dynamical system in view of quantum mechanics, the quantum and classical correspondence is very important in order for the associated quantum theory to be valid and viable.

DOI
10.1155/2016/1696105

Related papers

Quantum-classical correspondence for the inverted oscillator

Mustapha Maamache, Jeong Ryeol Choi

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.

Chinese Physics C•Oct 24, 2017•10 citations

Spectrum of Quantized Energy for a Lengthening Pendulum

Jeong Ryeol Choi, Ji Nny Song, Seong Ju Hong +3 more

We considered a quantum system of simple pendulum whose length of string is increasing at a steady rate. Since the string length is represented as a time function, this system is described by a time‐dependent Hamiltonian. The invariant operator method is very useful in solving the quantum solutions of time‐dependent Hamiltonian systems like this. The invariant operator of the system is represented in terms of the lowering operator ǎ(t) and the raising operator ǎ†(t). The Schrödinger solutions ψn(θ, t) whose spectrum is discrete are obtained by means of the invariant operator. The expectation value of the Hamiltonian in the ψn(θ, t) state is the same as the quantum energy. At first, we considered only θ2 term in the Hamiltonian in order to evaluate the quantized energy. The numerical study for quantum energy correction is also made by considering the angle variable not only up to θ4 term but also up to θ6 term in the Hamiltonian, using the perturbation theory.

AIP conference proceedings•Jan 1, 2010•0 citations

Time behavior of a Gaussian wave packet accompanying the generalized coherent state for the inverted oscillator

Mustapha Maamache, Yacine Bouguerra, Jeong Ryeol Choi

A Gaussian wave packet of the inverted oscillator is investigated using the invariant operator method together with the unitary transformation method. A simple wave packet directly derived from the eigenstates of the invariant operator of the system corresponds to a plane wave that is fully delocalized. However, we can construct a weighted wave packet in terms of such plane waves, which corresponds to a Gaussian wave. This wave packet is associated with the generalized coherent state, which can be crucially utilized for investigating the classical limit of quantum wave mechanics. Various quantum properties of the system, such as fluctuations of the canonical variables, the uncertainty product, and the motion of the wave packet or quantum particle, are analyzed by means of this wave packet. We have confirmed that the time behavior of such a wave packet is very similar to the counterpart classical state. The wave packet runs away from the origin in the positive or negative direction in the 1D coordinate depending on the condition of the initial state. We have confirmed that this wave packet not only moves acceleratively but also spreads out during its propagation.

Progress of Theoretical and Experimental Physics•Jun 1, 2016•8 citations

Quantum-classical correspondence and quantum chaos in the periodically kicked pendulum

Boon Leong Lan, Ronald F. Fox

The behavior of the periodically kicked pendulum is determined classically and quantum mechanically. Our earlier analytic study [Phys. Rev. A 41, 2952 (1990)] is confirmed and extended by the numerical results presented here. We are able to construct minimum uncertainty, Gaussian wave packets that maintain their form, and shadow the corresponding classical motion for long times when the classical motion is nonchaotic or only weakly chaotic. However, when the classical motion is robustly chaotic, the quantum-mechanical probability distribution grows macroscopically broad exponentially fast and the quantum-classical correspondence breaks down.

Physical Review A•Jan 1, 1991•23 citations

Wave packet revivals and the energy eigenvalue spectrum of the quantum pendulum

Doncheski, M A, Robinett, R W

The rigid pendulum, both as a classical and as a quantum problem, is an interesting system as it has the exactly soluble harmonic oscillator and the rigid rotor systems as limiting cases in the low- and high-energy limits respectively. The energy variation of the classical periodicity ($\\tau$) is also dramatic, having the special limiting case of $\\tau \\to \\infty$ at the 'top' of the classical motion (i.e. the separatrix.) We study the time-dependence of the quantum pendulum problem, focusing on the behavior of both the (approximate) classical periodicity and especially the quantum revival and superrevival times, as encoded in the energy eigenvalue spectrum of the system. We provide approximate expressions for the energy eigenvalues in both the small and large quantum number limits, up to 4th order in perturbation theory, comparing these to existing handbook expansions for the characteristic values of the related Mathieu equation, obtained by other methods. We then use these approximations to probe the classical periodicity, as well as to extract information on the quantum revival and superrevival times. We find that while both the classical and quantum periodicities increase monotonically as one approaches the 'top' in energy, from either above or below, the revival times decrease from their low- and high-energy values until very near the separatrix where they increase to a large, but finite value.

•Jul 10, 2003•17 citations

Quantum ergodicity for time-dependent wave-packet dynamics

John S. Hutchinson, Róbert E. Wyatt

The dynamics of nonstationary wave packets in model quantum-mechanical systems are studied for the properties of ergodicity and instability. The long-time-average Wigner phase-space distribution is calculated for each wave packet and analyzed for uniformity on a surface of section. A mild transition from nonergodic to ergodic behavior is observed with increasing energy. These results are compared to long-time averages of the wave-function probability density. The "stability" of each packet is also analyzed by several methods, including the "survival" of the initial state and a new measure of "separation" of similar wave packets. The relationships between the wave-packet ergodicity and instability and the corresponding classical dynamics are developed. Comparisons to other quantum-ergodic studies, and implications for intramolecular energy transfer and classical-ergodic studies are discussed.

Physical Review A•Apr 1, 1981•76 citations
Quantum properties of a lengthening pendulum are studied under the assumption that the length of the string increases at a steady rate. Advanced analysis for various physical problems in several types of quantum states, such as propagators, Wigner distribution functions, energy eigenvalues, probability densities, and dispersions of physical quantities, is carried out using quantum wave functions of the system. In particular, the time behavior of Gaussian-type wave packets is investigated in detail. The probability density for a Gaussian wave packet displaced in the positive θ at t=0 oscillates back and forth from the center ( θ=0 ). This phenomenon is very similar to the classical motion of the pendulum. As a consequence, we can confirm that there is a correspondence between its quantum and classical behaviors. When we analyze a dynamical system in view of quantum mechanics, the quantum and classical correspondence is very important in order for the associated quantum theory to be valid and viable.
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.
We considered a quantum system of simple pendulum whose length of string is increasing at a steady rate. Since the string length is represented as a time function, this system is described by a time‐dependent Hamiltonian. The invariant operator method is very useful in solving the quantum solutions of time‐dependent Hamiltonian systems like this. The invariant operator of the system is represented in terms of the lowering operator ǎ(t) and the raising operator ǎ†(t). The Schrödinger solutions ψn(θ, t) whose spectrum is discrete are obtained by means of the invariant operator. The expectation value of the Hamiltonian in the ψn(θ, t) state is the same as the quantum energy. At first, we considered only θ2 term in the Hamiltonian in order to evaluate the quantized energy. The numerical study for quantum energy correction is also made by considering the angle variable not only up to θ4 term but also up to θ6 term in the Hamiltonian, using the perturbation theory.
A Gaussian wave packet of the inverted oscillator is investigated using the invariant operator method together with the unitary transformation method. A simple wave packet directly derived from the eigenstates of the invariant operator of the system corresponds to a plane wave that is fully delocalized. However, we can construct a weighted wave packet in terms of such plane waves, which corresponds to a Gaussian wave. This wave packet is associated with the generalized coherent state, which can be crucially utilized for investigating the classical limit of quantum wave mechanics. Various quantum properties of the system, such as fluctuations of the canonical variables, the uncertainty product, and the motion of the wave packet or quantum particle, are analyzed by means of this wave packet. We have confirmed that the time behavior of such a wave packet is very similar to the counterpart classical state. The wave packet runs away from the origin in the positive or negative direction in the 1D coordinate depending on the condition of the initial state. We have confirmed that this wave packet not only moves acceleratively but also spreads out during its propagation.
The behavior of the periodically kicked pendulum is determined classically and quantum mechanically. Our earlier analytic study [Phys. Rev. A 41, 2952 (1990)] is confirmed and extended by the numerical results presented here. We are able to construct minimum uncertainty, Gaussian wave packets that maintain their form, and shadow the corresponding classical motion for long times when the classical motion is nonchaotic or only weakly chaotic. However, when the classical motion is robustly chaotic, the quantum-mechanical probability distribution grows macroscopically broad exponentially fast and the quantum-classical correspondence breaks down.
The rigid pendulum, both as a classical and as a quantum problem, is an interesting system as it has the exactly soluble harmonic oscillator and the rigid rotor systems as limiting cases in the low- and high-energy limits respectively. The energy variation of the classical periodicity (tau\\tautau) is also dramatic, having the special limiting case of tautoinfty\\tau \\to \\inftytautoinfty at the 'top' of the classical motion (i.e. the separatrix.) We study the time-dependence of the quantum pendulum problem, focusing on the behavior of both the (approximate) classical periodicity and especially the quantum revival and superrevival times, as encoded in the energy eigenvalue spectrum of the system. We provide approximate expressions for the energy eigenvalues in both the small and large quantum number limits, up to 4th order in perturbation theory, comparing these to existing handbook expansions for the characteristic values of the related Mathieu equation, obtained by other methods. We then use these approximations to probe the classical periodicity, as well as to extract information on the quantum revival and superrevival times. We find that while both the classical and quantum periodicities increase monotonically as one approaches the 'top' in energy, from either above or below, the revival times decrease from their low- and high-energy values until very near the separatrix where they increase to a large, but finite value.
The dynamics of nonstationary wave packets in model quantum-mechanical systems are studied for the properties of ergodicity and instability. The long-time-average Wigner phase-space distribution is calculated for each wave packet and analyzed for uniformity on a surface of section. A mild transition from nonergodic to ergodic behavior is observed with increasing energy. These results are compared to long-time averages of the wave-function probability density. The "stability" of each packet is also analyzed by several methods, including the "survival" of the initial state and a new measure of "separation" of similar wave packets. The relationships between the wave-packet ergodicity and instability and the corresponding classical dynamics are developed. Comparisons to other quantum-ergodic studies, and implications for intramolecular energy transfer and classical-ergodic studies are discussed.