Classical distributions, Wigner, and Husimi functions for the Generalized Caldirola-Kanai Hamiltonian~

H. N. Ioannidou

D. Skaltsas, H. N. Ioannidou

Pinelopi Angelopoulou, Sotirios Baskoutas, A. D. Jannussis +1 more

Ty Beus, Jean-François S. Van Huele, Manuel Berrondo

Abstract In this paper, we focus on the effects of temporally compact fluctuations on the parameters m ( t ) (mass), ω ( t ) (frequency), and γ ( t ) (spatial dilation coefficient), for the generalized Caldirola-Kanai Hamiltonian. We propose an algorithm such that given control over the timing of repeated compact fluctuations, one can produce any time-evolution associated with complete parameter control of the generalized Caldirola-Kanai Hamiltonian. Computational simplicity is achieved in this endeavor using a faithful representation of the Hamiltonian’s associated Lie algebra. Further simplification occurs as a result of using unitarity conditions for the time-evolution operator. We apply our analysis to the effects of Dirac delta fluctuations in mass and frequency, both separately and simultaneously. We also numerically demonstrate control of the generalized Caldirola-Kanai system for the case of timed Gaussian fluctuations in the mass term. Furthermore, we use the faithful Lie algebra representation to show that any unitary evolution of this generalized Caldirola-Kanai system is given by the squeeze operator multiplied with a phase factor. We also show that the classical evolution of a probabilistic Gaussian phase space distribution and the evolution of the Wigner function of a coherent state in the quantum Caldirola-Kanai system are identical.

Lorenzo Galleani, Leon W. Cohen

Shuai Wang, Hong‐Chun Yuan, Hong-Yi Fan

Based on the technique of integration within an ordered product (IWOP) of operators, we introduce the Fresnel operator for converting a kind of time-dependent Hamiltonian into the standard harmonic oscillator Hamiltonian. The Fresnel operator with the parameters A, B, C, D corresponds to classical optical Fresnel transformation, these parameters are the solution to a set of coupled partial differential equations set up in the above-mentioned converting process. In this way, the Caldirola–Kanai Hamiltonian has been easily converted into the standard harmonic oscillator Hamiltonian. And then the exact wave function solution of the Schrödinger equation governed by the Caldirola–Kanai Hamiltonian is obtained, which represents a squeezed number state. The corresponding Wigner function is derived by virtue of the Weyl ordered form of the Wigner operator and the order-invariance of Weyl ordered operators under similar transformations.