Quantum Theory from a Nonlinear Perspective

Abstract

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Quantum nonstationary oscillators: Invariants, dynamical algebras and coherent states via point transformations

“…is well known in the literature and finds many application in physics [22-25, 30, 57, 64, 65, 82-85, 90, 91]. It arises quite naturally in the studies of parametric oscillators [22][23][24][25]30], in the description of structured light in varying media [82][83][84][85], and in the study of non-Hermitian Hamiltonians with real spectrum [57,64,65]. The key to solve (C-1) is to consider the homogeneous linear equation q + Ω 2 (t) q = 0, (C-2) which coincides with the equation of motion for a classical parametric oscillator.…”

Dynamical Invariants for Generalized Coherent States via Complex Quantum Hydrodynamics

“…To our opinion, the closest connection was achieved by Lidsey [30] who could show that the Friedmann-Lemaître equations could be brought into the form of the Ermakov Equation ( 10), only replacing the position uncertainty α(t) by the scale factor a(t). Corresponding "coherent states of the universe" could be constructed using adequate creation and anihilation operators (for details, see Section 7.7 of [15]).…”

“…The invariant is present in several different technological applications [8]: "The extensions from single harmonic oscillators to coupled time dependent harmonic oscillators may be found in ion-laser interactions [9][10][11], quantized fields propagating through dielectric media [12], shortcuts to adiabaticity [13], the Casimir effect [14] to name some". For further details on the historical development of the Ermakov invariant, see [15] and the literature quoted therein.…”

Cosmological analogies, Lagrangians, and symmetries for convective–radiative heat transfer

“…If the coefficients are allowed to change signs, the equation describing the truncated Newton–Stefan model appears in other field of physics and mathematics. For example, the logistic equation can be reproduced or, in a simplified laser emission model, the photon emission rate dn / dt is related to the number of photons n ( t ) in an excited state by [ 36 , 37 ] …”