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Revista Mexicana de Física E

The propagator of the inverted Caldirola-Kanai Oscillator

Umpon Pon Jairuk, Surarit Pepore

Published Jul 1, 2024• 0 citations
Publisher

Abstract

In this paper, we will derive the propagators for an inverted Caldirola-Kanai oscillator by the Feynman path integral, the Schwinger method, and the Dodonov method. In Feynman path inte-gral, the propagator can be calculated from the functional integrals while in the Schwinger method the propagator can be obtained from basic operator algebra and elementary integrations. In Dodonov method, the propagator can be derived from the application of the integrals of the motion of quantum systems. Furthermore, the connection between these methods is also discussed.

DOI
10.31349/revmexfise.21.020214

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In this paper, we will derive the propagators for an inverted Caldirola-Kanai oscillator by the Feynman path integral, the Schwinger method, and the Dodonov method. In Feynman path inte-gral, the propagator can be calculated from the functional integrals while in the Schwinger method the propagator can be obtained from basic operator algebra and elementary integrations. In Dodonov method, the propagator can be derived from the application of the integrals of the motion of quantum systems. Furthermore, the connection between these methods is also discussed.
A calculation for the Kanai–Caldirola propagator in the de Broglie–Bohm theory is proposed. The technique of space-time transformations used in the Feynman path integral is adapted and then the problem is converted to that of the harmonic oscillator. The quantum propagator is viewed as an expansion of the guiding wave function over the velocity space. The construction is re-examined by replacing the integration over the initial velocity of the classic path by its final extremity. This gives a certain affinity with the Feynman path integral approach.
In this paper, the propagators for a particle moving in a time-dependent linear potential and a free particle with linear damping are calculated by the application of the integrals of the motion of a quantum system. The propagator for a charged harmonic oscillator is derived from the Feynman path integrals and the propagator for a damped harmonic oscillator is evaluated by the Schwinger method. The relation between the integrals of the motion, Feynman path integrals, and Schwinger method are also described.
Using the Caldirola-Kanai Hamiltonian for the quantum dissipative system, the author expresses the propagator as the modified Feynman path integral in the configuration space, which can then be evaluated for the damped harmonic oscillator by Montroll's method. The propagator of the damped harmonic oscillator can also be calculated beyond and at caustics with the help of Horvathy-Feynman formula. The new results are confirmed by investigating the classical paths joining two fixed end-point positions, Finally the author obtains the time-dependent wavefunctions from the propagator of the dynamical system.
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.