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.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.

Journal of Physics A Mathematical and GeneralApr 7, 199717 citations

Minimum-uncertainty states and pseudoclassical dynamics

V. Canivell, Pere Seglar

All the minimum-uncertainty states $|z(\ensuremath{\alpha})〉$ are built as the eigenstates of generalized annihilation operators $a(\ensuremath{\alpha})$. We obtain the most general Hamiltonian which verfies the following property: Given an initial minimum-uncertainty state $|z(\ensuremath{\alpha})〉$ at $t = 0$ , the ${z}^{\ensuremath{'}}$ which corresponds to the minimum-uncertainty state that maximizes ${|〈{z}^{\ensuremath{'}}(\ensuremath{\alpha})|z(\ensuremath{\alpha}),t〉|}^{2}$ follows the classical trajectory in phase space. Finally, we comment on its relationship with the most general Hamiltonian which preserves minimality.All the minimum-uncertainty states $|z(\ensuremath{\alpha})〉$ are built as the eigenstates of generalized annihilation operators $a(\ensuremath{\alpha})$. We obtain the most general Hamiltonian which verfies the following property: Given an initial minimum-uncertainty state $|z(\ensuremath{\alpha})〉$ at $t = 0$, the ${z}^{\ensuremath{'}}$ which corresponds to the minimum-uncertainty state that maximizes ${|〈{z}^{\ensuremath{'}}(\ensuremath{\alpha})|z(\ensuremath{\alpha}),t〉|}^{2}$ follows the classical trajectory in phase space. Finally, we comment on its relationship with the most general Hamiltonian which preserves minimality.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fieldsFeb 15, 19778 citations

Classical properties of quantum systems in minimum uncertainty states

Alexander Bach, Helmut Francke

Physics Letters AOct 1, 19861 citation

Quantum Dynamics for Classical Systems

Fabio Bagarello

Wiley eBooksOct 10, 201290 citations

Classical-quantum correspondence in the driven surface-state-electron model

Miroslaw Latka, Paolo Grigolini A, Bruce J. West

The quantum dynamics of minimum uncertainty wave packets in a system described by the surface-state-electron Hamiltonian are studied herein. The quantum evolution is found to be strongly dependent upon the orbit structure in the classical phase space. A Gaussian wave packet, initially centered on the part of the phase space incased by Kolmogorov-Arnold-Moser tori remains localized, while the packets located in the chaotic region spread rapidly, filling the available phase space. To quantify the observed spreading quantum uncertainty is introduced as a dynamical variable, and its time development is determined by the ballistic growth of the position uncertainty and the erratic oscillations of the momentum uncertainty. The quantum evolution is analyzed within the framework of the Floquet formalism. The Floquet spectrum of the stable packet is shown to be dominated by a few quasienergy states with the corresponding Husimi distribution embedded in the classical stability island. Further, the classical concept of diffusion previously used in this context is shown to be inappropriate.The quantum dynamics of minimum uncertainty wave packets in a system described by the surface-state-electron Hamiltonian are studied herein. The quantum evolution is found to be strongly dependent upon the orbit structure in the classical phase space. A Gaussian wave packet, initially centered on the part of the phase space incased by Kolmogorov-Arnold-Moser tori remains localized, while the packets located in the chaotic region spread rapidly, filling the available phase space. To quantify the observed spreading quantum uncertainty is introduced as a dynamical variable, and its time development is determined by the ballistic growth of the position uncertainty and the erratic oscillations of the momentum uncertainty. The quantum evolution is analyzed within the framework of the Floquet formalism. The Floquet spectrum of the stable packet is shown to be dominated by a few quasienergy states with the corresponding Husimi distribution embedded in the classical stability island. Further, the classical concept of diffusion previously used in this context is shown to be inappropriate.

Physical Review AJun 1, 199318 citations

t = 0

, the ${z}^{\ensuremath{'}}$ which corresponds to the minimum-uncertainty state that maximizes ${|〈{z}^{\ensuremath{'}}(\ensuremath{\alpha})|z(\ensuremath{\alpha}),t〉|}^{2}$ follows the classical trajectory in phase space. Finally, we comment on its relationship with the most general Hamiltonian which preserves minimality.