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

Journal of Physics A Mathematical and General

Wavefunctions and minimum uncertainty states of the harmonic oscillator with an exponentially decaying mass

Chung‐In Um, In-Han Kim, Kyu-Hwang Yeon, Thomas F. George, Lakshmi N. Pandey

Published Apr 7, 1997• 17 citations
Publisher

Abstract

Exact solutions to the harmonic oscillator with an exponentially decaying mass are explicitly represented in terms of Bessel functions. The dynamical invariant quantity of the system has the form of a rosette-shaped orbit in phase space. From this, it is confirmed that this system is bounded. By using the invariant operator represented in terms of lowering and raising operators, we have obtained wavefunctions and the propagator. Finally, minimum uncertainty states and conditions are evaluated by using other operators which are obtained from the above ones.

DOI
10.1088/0305-4470/30/7/031

Related papers

The Wave Function and the Minimum Uncertainty Function of the Time Dependent Harmonic Oscillator(New Developments in Statistical Physics Similarities in Diversities,YITP Workshop)

Kyu Hwang Yeon, Chung‐In Um, Thomas F. George

The time dependent harmonic oscillator is solved explicitly for quantum mechanics by the operator method with an auxiliary condition as the classical solution. Two classical invariant quantities which determine whether or not the system is bound are derived by the classical equation of motion. We obtain the invariant operator from one classical invariant quantity. Its eigenfunction is related to the solution of Schrodinger equation of the system and its eigenvalue is related to another classical quantity. The wave function is evaluated exactly by the eigenfunction of the invariant operator but it is not the eigenfunction of the Hamiltonian of the system. The uncertainty which calculates with the wave function is not a minimum one. We will confirm that the function which holds minimum uncertainty is a eigenfunction of the Hamiltonian.

•Jul 20, 1993•0 citations

Wave function in the invariant representation and squeezed-state function of the time-dependent harmonic oscillator

Kyu Hwang Yeon, Hyon Ju Kim, Chung‐In Um +2 more

The two quantum invariant operators are found from the time-dependent Hamiltonian of the harmonic oscillator with an auxiliary condition. The solution of the Schr\"odinger equation for the system, such as the eigenfunctions, eigenvalues, and minimum uncertainty, is derived by utilizing these invariant operators. The coherent states of this system are not the squeezed states, and the eigenfunction of the invariant operator is not the eigenfunction of the Hamiltonian of the system unless it is in the invariant representation. The squeezing function, which is an eigenfunction of the Hamiltonian of the system in the invariant representation and which also gives the minimum uncertainty, is obtained by a set of unitary transformed operators, i.e., squeezing operators.

Physical Review A•Aug 1, 1994•54 citations

Exact wave functions and coherent states of a damped driven harmonic oscillator

H. G. Oh, H. R. Lee, Thomas F. George +1 more

For a damped harmonic oscillator forced by a time-dependent field, the exact wave function is obtained by three different methods: (i) path integral, (ii) second quantization, and (iii) dynamical invariant. The explicit form of the dynamical invariant involves a solution to a corresponding auxiliary equation. The coherent states, defined as eigenstates of a new destruction operator, form a nonorthogonal, overcomplete set and correspond to the minimum uncertainty states. These coherent states give the exact classical motion of the damped driven harmonic oscillator.

Physical Review A•Jun 1, 1989•40 citations

Path Integral for a Harmonic Oscillator with Time-Dependent Mass and Frequency

Surarit Pepore, P. Winotai, Tanakorn Osotchan +1 more

The exact solutions to the time-dependent Schrodinger equation for a harmonic oscillator with time-dependent mass and frequency were derived in a general form. The quantum mechanical propagator was calculated by the Feynman path integral method, while the wave function was derived from the spectral representation of the obtained propagator. It was shown that the propagator and the wave function depended on the s solution of a classical oscillator, in which the amplitude and phase satisfied the auxiliary equations. To demonstrate the derivation of the solution from our auxiliary equations, exponential and periodic functions of mass with constant frequency were imposed to evaluate the propagator and wave function for the Caldirola-Kanai and pulsating mass oscillators, respectively.

•Jan 1, 2006•6 citations

Quantum theory for coupled harmonic oscillators with exponentially changing masses

Alexander V. Zhukov, P. N. Zhukova

The classical solution is obtained for the system of two coupled harmonic oscillators with exponentially decaying mass. Using the Feynman path-integral method of quantization an exact propagator for the corresponding quantum system is derived.

Journal of Physics A Mathematical and General•Jan 1, 1999•3 citations

Exact solution of a time-dependent quantal harmonic oscillator with damping and a perturbative force

Dinkar C. Khandekar, S. V. Lawande

The problem of a quantal harmonic oscillator with damping and a time-dependent frequency acted on by a time-dependent perturbative force is exactly solved. The wavefunctions are found in Schrödinger representation using the theory of explicitly time-dependent invariants and also by an expansion of the Feynman propagator. The propagator is obtained in exactly closed form by an explicit path integration of the classical Lagrangian. It is found that the wavefunctions and the propagator depend only on the solution of classical damped oscillator through a single function ρ (t). The function ρ (t) itself may be obtained as a solution of a second order nonlinear differential equation under the appropriate set of initial conditions.

Journal of Mathematical Physics•Sep 1, 1979•85 citations
Exact solutions to the harmonic oscillator with an exponentially decaying mass are explicitly represented in terms of Bessel functions. The dynamical invariant quantity of the system has the form of a rosette-shaped orbit in phase space. From this, it is confirmed that this system is bounded. By using the invariant operator represented in terms of lowering and raising operators, we have obtained wavefunctions and the propagator. Finally, minimum uncertainty states and conditions are evaluated by using other operators which are obtained from the above ones.
The time dependent harmonic oscillator is solved explicitly for quantum mechanics by the operator method with an auxiliary condition as the classical solution. Two classical invariant quantities which determine whether or not the system is bound are derived by the classical equation of motion. We obtain the invariant operator from one classical invariant quantity. Its eigenfunction is related to the solution of Schrodinger equation of the system and its eigenvalue is related to another classical quantity. The wave function is evaluated exactly by the eigenfunction of the invariant operator but it is not the eigenfunction of the Hamiltonian of the system. The uncertainty which calculates with the wave function is not a minimum one. We will confirm that the function which holds minimum uncertainty is a eigenfunction of the Hamiltonian.
The two quantum invariant operators are found from the time-dependent Hamiltonian of the harmonic oscillator with an auxiliary condition. The solution of the Schr\"odinger equation for the system, such as the eigenfunctions, eigenvalues, and minimum uncertainty, is derived by utilizing these invariant operators. The coherent states of this system are not the squeezed states, and the eigenfunction of the invariant operator is not the eigenfunction of the Hamiltonian of the system unless it is in the invariant representation. The squeezing function, which is an eigenfunction of the Hamiltonian of the system in the invariant representation and which also gives the minimum uncertainty, is obtained by a set of unitary transformed operators, i.e., squeezing operators.
For a damped harmonic oscillator forced by a time-dependent field, the exact wave function is obtained by three different methods: (i) path integral, (ii) second quantization, and (iii) dynamical invariant. The explicit form of the dynamical invariant involves a solution to a corresponding auxiliary equation. The coherent states, defined as eigenstates of a new destruction operator, form a nonorthogonal, overcomplete set and correspond to the minimum uncertainty states. These coherent states give the exact classical motion of the damped driven harmonic oscillator.
The exact solutions to the time-dependent Schrodinger equation for a harmonic oscillator with time-dependent mass and frequency were derived in a general form. The quantum mechanical propagator was calculated by the Feynman path integral method, while the wave function was derived from the spectral representation of the obtained propagator. It was shown that the propagator and the wave function depended on the s solution of a classical oscillator, in which the amplitude and phase satisfied the auxiliary equations. To demonstrate the derivation of the solution from our auxiliary equations, exponential and periodic functions of mass with constant frequency were imposed to evaluate the propagator and wave function for the Caldirola-Kanai and pulsating mass oscillators, respectively.
The problem of a quantal harmonic oscillator with damping and a time-dependent frequency acted on by a time-dependent perturbative force is exactly solved. The wavefunctions are found in Schrödinger representation using the theory of explicitly time-dependent invariants and also by an expansion of the Feynman propagator. The propagator is obtained in exactly closed form by an explicit path integration of the classical Lagrangian. It is found that the wavefunctions and the propagator depend only on the solution of classical damped oscillator through a single function ρ (t). The function ρ (t) itself may be obtained as a solution of a second order nonlinear differential equation under the appropriate set of initial conditions.