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Physica Scripta

The Kanai–Caldirola propagator in the de Broglie–Bohm theory

A. Tilbi, Tahar Boudjedaa, M. Merad, L. Chétouani

Published Feb 21, 2007• 7 citations
Publisher

Abstract

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.

DOI
10.1088/0031-8949/75/4/018

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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.
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.
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.
The phase corresponding to a certain path in the Feynman path integral is derived with the help of the de Broglie hypothesis. Zusammenfassung. Die einem bestimmten Pfad entsprechende Phase im Feynman-Pfadintegral wird mit Hilfe der de Broglie-Hypothese abgeleitet. In the beginning quantum mechanics was built up by analogy with the formulation of classical mechanics in terms of the Hamilton function. In 1933, Dirac [2] announced the idea of formulating quantum mechanics on the basis of the Lagrange function. He started from the analogy between the treatment of classical motion as a canonical transformation and the transformational properties of operators with respect to a time translation. This idea of Dirac was developed further by Feynman [3], who regarded a quantum-mechanical motion from one space-time point to another as a superposition of motions along all possible paths connecting the two points. Each path contributes to the total probability amplitude a term proportional to where is the classical action corresponding to this path and is Planck's constant. Feynman [3] adopted the form of the phase in the expression (1) as a postulate. Another way, however, of deriving the phase is to use the de Broglie hypothesis [1], as shown below. According to this hypothesis, it is possible to associate any particle with a certain frequency proportional to its energy (Hamiltonian function), i.e., and a wave vector proportional to the particle's momentum as Now consider, in the sense of Feynman's formulation, a particle moving along a certain path from point 1 to point 2. During the motion, let and be the radius vectors of the particle at the times t, t+ dt respectively. According to the de Broglie hypothesis, we can associate a plane wave with the particle in the neighbourhood of these points. The phase change of this wave along the particle's path during the motion between the points and consists of space and time phase changes: The term in the parentheses is precisely the value of the Lagrange function at the point . The phase change of the wavefunction corresponding to the whole path can be obtained by integration of the term expressing the infinitesimal phase change: In this way, on the basis of the de Broglie hypothesis, we have reached full agreement with Feynman's postulate: the phase of the probability amplitude corresponding to a certain path is equal to , where is the classical action corresponding to this path. References [1] de Broglie L 1923 Comptes rendus 177 548 - 550 [2] Dirac P A M 1933 Phys. Z. der Sowjetunion 3 64 [3] Feynman R 1948 Rev. Mod. Phys. 2 367
This study addresses a significant gap in the literature by extending the path integral formalism to time-dependent systems within the Wigner–Dunkl framework, deriving exact propagator solutions for analytically solvable cases. By employing generalized canonical transformations, we reformulated the path integral to develop an explicit expression for the propagator. This formalism is applied to specific cases, including a Dunkl-harmonic oscillator with time-dependent mass and frequency. Solutions for the Dunkl–Caldirola-Kanai oscillator and a model with a strongly pulsating mass are derived, providing exact propagator expressions and corresponding wave functions. These findings extend the utility of Dunkl operators in quantum mechanics, offering new insights into the dynamics of time-dependent quantum systems and possibly find application in quantum optics, plasma physics, and other fields.
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.