Uncertainty Relations in the Madelung Picture

Madelung showed how the complex Schrödinger equation can be rewritten in terms of two real equations, one for the phase and one for the amplitude of the complex wave function, where both equations are not independent of each other, but coupled. Although these equations formally look like classical hydrodynamic equations, they contain all the information about the quantum system. Concerning the quantum mechanical uncertainties of position and momentum, however, this is not so obvious at first sight. We show how these uncertainties are related to the phase and amplitude of the wave function in position and momentum space and, particularly, that the contribution from the phase essentially depends on the position–momentum correlations. This will be illustrated explicitly using generalized coherent states as examples.

Dynamical Invariants for Generalized Coherent States via Complex Quantum Hydrodynamics

For time dependent Hamiltonians like the parametric oscillator with time-dependent frequency, the energy is no longer a constant of motion. Nevertheless, in 1880, Ermakov found a dynamical invariant for this system using the corresponding Newtonian equation of motion and an auxiliary equation. In this paper it is shown that the same invariant can be obtained from Bohmian mechanics using complex Hamiltonian equations of motion in position and momentum space and corresponding complex Riccati equations. It is pointed out that this invariant is equivalent to the conservation of angular momentum for the motion in the complex plane. Furthermore, the effect of a linear potential on the Ermakov invariant is analysed.

Generalized coherent states of exceptional Scarf-I potential: Their spatio-temporal and statistical properties

We construct generalized coherent states for the rationally extended Scarf-I potential. Statistical and geometrical properties of these states are investigated. Special emphasis is given to the study of spatio-temporal properties of the coherent states via the quantum carpet structure and the auto-correlation function. Through this study, we aim to find the signature of the “rationalization” of the conventional potentials and the classical orthogonal polynomials.

Time and classical equations of motion from quantum entanglement via the Page and Wootters mechanism with generalized coherent states

We draw a picture of physical systems that allows us to recognize what “time” is by requiring consistency with the way that time enters the fundamental laws of Physics. Elements of the picture are two non-interacting and yet entangled quantum systems, one of which acting as a clock. The setting is based on the Page and Wootters mechanism, with tools from large-N quantum approaches. Starting from an overall quantum description, we first take the classical limit of the clock only, and then of the clock and the evolving system altogether; we thus derive the Schrödinger equation in the first case, and the Hamilton equations of motion in the second. This work shows that there is not a “quantum time”, possibly opposed to a “classical” one; there is only one time, and it is a manifestation of entanglement.

Two-Qubit Local Fisher Information Correlation beyond Entanglement in a Nonlinear Generalized Cavity with an Intrinsic Decoherence

In this paper, we study a Hamiltonian system constituted by two coupled two-level atoms (qubits) interacting with a nonlinear generalized cavity field. The nonclassical two-qubit correlation dynamics are investigated using Bures distance entanglement and local quantum Fisher information under the influences of intrinsic decoherence and qubit–qubit interaction. The effects of the superposition of two identical generalized coherent states and the initial coherent field intensity on the generated two-qubit correlations are investigated. Entanglement of sudden death and sudden birth of the Bures distance entanglement as well as the sudden changes in local Fisher information are observed. We show that the robustness, against decoherence, of the generated two-qubit correlations can be controlled by qubit–qubit coupling and the initial coherent cavity states.