In this work we compute, via the quantum potential approach, the Hamiltonian system determined by Hermite–Gaussian beams. Then we show that the integral curves of the Poynting vector, exact optics energy trajectories, conform to a subset of solutions to the corresponding Hamilton equations lying on hyperboloidal surfaces. The geometrical light rays associated with these beams are given by the tangent lines to the integral curves of the Poynting vector at the zeroes of the quantum potential, and the caustic region coincides with the zeroes of quantum potential and quantum force. One of the main contributions of this work is to present the relationship between the physical phase k Φ, the geometrical-optics phase k Φ G , and the quantum potential Q HG in the Hermite–Gaussian beams. Furthermore, note that for any solution to the paraxial wave equation in free space, the tangent lines to the integral curves of the Poynting vector that correspond to the geometric light rays are those that pass through the points where the region determined by zeroes of the quantum potential is tangent to the geometrical caustic determined by the geometric light rays .
In this work we compute, via the quantum potential approach, the Hamiltonian system determined by Hermite–Gaussian beams. Then we show that the integral curves of the Poynting vector, exact optics energy trajectories, conform to a subset of solutions to the corresponding Hamilton equations lying on hyperboloidal surfaces. The geometrical light rays associated with these beams are given by the tangent lines to the integral curves of the Poynting vector at the zeroes of the quantum potential, and the caustic region coincides with the zeroes of quantum potential and quantum force. One of the main contributions of this work is to present the relationship between the physical phase k Φ, the geometrical-optics phase k Φ G , and the quantum potential Q HG in the Hermite–Gaussian beams. Furthermore, note that for any solution to the paraxial wave equation in free space, the tangent lines to the integral curves of the Poynting vector that correspond to the geometric light rays are those that pass through the points where the region determined by zeroes of the quantum potential is tangent to the geometrical caustic determined by the geometric light rays .
The aim of the present work is to show that any monochromatic solution to the scalar wave equation in free space defines a conservative Hamiltonian system, describing a particle of mass m = 1 / 2 and energy E = 1 , under the influence of the so-called quantum potential. We remark that the integral curves of its Poynting vector, exact optics energy trajectories, define a particular subset of solutions to the corresponding Hamilton equations. Furthermore, we introduce the zero quantum potential straight lines concept, as the family of tangent lines to the integral curves of the Poynting vector at the zeros of the quantum potential. These general results are applied to a family of plane waves and to Bessel beams. In the case of Bessel beams, we present a detailed study of the trajectories determined by the corresponding Hamiltonian system, and we show that the zero quantum potential straight lines coincide with the geometrical light rays, geometrical optics energy trajectories. Furthermore, we show that the areal velocity , determined by the exact optics energy trajectories, for non-zero order Bessel beams is not a constant of motion. However, its projection along the z ^ direction is a constant of motion because L z is a constant.
The aim of the present work is to show that the Laguerre–Gauss beams determine a Hamiltonian system with two degrees of freedom for a particle of mass m = 1 under the action of the quantum potential determined by these beams. We show that the integral curves of the Poynting vector constitute a particular subset of solutions to the corresponding Hamilton equations and that the geometrical light rays associated with these twisted beams turn out to be the tangent straight lines to the exact optics energy trajectories at the zeroes of the quantum potential . By using the Hamiltonian formulation, we determine the velocity, the linear momentum, the angular momentum, the torque, and the areal velocity characterizing the particle associated with the Laguerre–Gauss beams.
In this work, we show that the Madelung–Bohm trajectories determined by any solution, Ψ( ξ , ζ ), of the paraxial wave equation in an arbitrary two-dimensional optical medium, where ζ is the propagation coordinate and ξ the transversal one, are given by the contours of the function G ( ξ , ζ )≡∫|Ψ( ξ , ζ )| 2 d ξ . Furthermore, we remark that an analogous result follows directly for the one-dimensional Schrödinger equation for a particle evolving in an arbitrary classical potential, where ξ gives the position of the particle at time ζ . We apply this result to several examples in free and quadratic optical media. They include the Airy beam, Hermite–Gaussian beams, and some of their superpositions. Additionally, this work provides an alternative derivation of Hermite–Gaussian (HG) beams in free space by employing the quantum potential equations and the framework of quantum trajectories. By analyzing known trajectories, we demonstrate that the fundamental parameters characterizing these beams—the Gouy phase, beam radius, and wavefront curvature—emerge naturally within the quantum potential formalism.
From a geometric perspective, the caustic is the most classical description of a wave function since its evolution is governed by the Hamilton-Jacobi equation. On the other hand, according to the Madelung-de Broglie-Bohm equations, the most classical description of a solution to the Schrödinger equation is given by the zeros of the Madelung-Bohm potential. In this work, we compare these descriptions, and, by analyzing how the rays are organized over the caustic, we find that the wave functions with fold caustic are the most classical beams because the zeros of the Madelung-Bohm potential coincide with the caustic. For another type of beam, the Madelung-Bohm potential is in general distinct to zero over the caustic. We have verified these results for the one-dimensional Airy and Pearcey beams, which, according to the catastrophe theory, have stable caustics. Similarly, we introduce the optical Madelung-Bohm potential, and we show that if the optical beam has a caustic of the fold type, then its zeros coincide with the caustic. We have verified this fact for the Bessel beams of nonzero order. Finally, we remark that for certain cases, the zeros of the Madelung-Bohm potential are linked with the superoscillation phenomenon.
Abstract In the first part of this work, using the quantum potential approach, we show that a solution to the time-independent Schrödinger equation determines a subset of classical solutions, only if the region corresponding to the zeroes of the quantum potential is tangent to the caustic region determined by the classical trajectories. Thus, the solutions of the time-independent Schrödinger equation, according to their caustic and the zeros of the quantum potential, can be classified in three different cases given by the following conditions: the two regions coincide, they are tangent at certain subset of points, and the two regions are not tangent at any point. In the second part, as examples of the first type of wave functions, we present the solutions of the Schrödinger equation for the 2D isotropic harmonic oscillator, which are eigenfunctions of both the Hamiltonian and the angular momentum operators. That is, we show that for this family of solutions, the zeroes of the quantum potential coincide with the caustic region. Furthermore, we find that the classical trajectories, determined from the quantum ones and the zeroes of the quantum potential, conform to a family of elliptical curves for a particle with energy, (2n + l + 1)ℏ ω, and orbital angular momentum l ℏ.
In this work, we use the quantum potential approach to quantum mechanics to show that the Madelung–Bohm trajectories for a particle in a constant gravitational field can be related to those of a free particle by means of a quantum point transformation defined in the extended configuration space. We find that the point transformation also gives a connection between the corresponding quantum Hamiltonians determined by the solutions of the corresponding Schrödinger equations for these two problems. We show that the Madelung–Bohm trajectories determined by the stationary solutions to the Schrödinger equation for the particle in a constant gravitational field are straight ones, while the corresponding ones for the free particle are parabolic trajectories. The Airy beam is one example of this type of solution. We study the properties of a solution to the Schrödinger equation for a free particle with phase singularities (zeroes), and we find that the corresponding solution to the Schrödinger equation for the particle in a constant gravitational field also has zeroes at the same spacetime points. However, the Madelung–Bohm trajectories determined by the two solutions are totally different. Furthermore, we remark that similar results can be directly obtained for the paraxial wave equation.