Exact Bateman-Hillion solutions of the wave equation are applied to Hermite-Gaussian beams using a space-time constraint condition that requires the field density to fall as the inverse square of distance from the focal point of the beam at large distances from it. Following a familiar practice, the constraint is implemented in integrals through the use of a Dirac delta function. It is shown the Hermite-Gaussian functions evolve to become pure functions of angular position on the fully developed spherical phase fronts. Under the paraxial approximation it is further shown the wave equation and Schrodinger equation are interchangeable within the constraint space in correspondence to a recent paper claiming indirect evidence of Gouy phase in matter waves.
Exact Bateman-Hillion solutions of the wave equation are applied to Hermite-Gaussian beams using a space-time constraint condition that requires the field density to fall as the inverse square of distance from the focal point of the beam at large distances from it. Following a familiar practice, the constraint is implemented in integrals through the use of a Dirac delta function. It is shown the Hermite-Gaussian functions evolve to become pure functions of angular position on the fully developed spherical phase fronts. Under the paraxial approximation it is further shown the wave equation and Schrodinger equation are interchangeable within the constraint space in correspondence to a recent paper claiming indirect evidence of Gouy phase in matter waves.
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.
We present exact solutions of the wave equation involving an arbitrary wave form with a phase closely similar to the general astigmatic phase of paraxial wave optics. Special choices of the wave form allow general astigmatic beamlike and pulselike waves with a Gaussian-type unrestricted localization in space and time. These solutions are generalizations of the known Bateman-type waves obtained from the connection existing between beamlike solutions of the paraxial parabolic equation and relatively undistorted wave solutions of the wave equation. As a technical tool, we present a full description of parametrizations of $2\ifmmode\times\else\texttimes\fi{}2$ symmetric matrices with positive imaginary part, which arise in the theory of Gaussian beams.
In this work, the paraxial approximation of the free Dirac equation is examined. The results are first obtained by constructing superpositions of exact solutions with suitable profiles, which are borrowed from paraxial optics. In this manner, the paraxial Dirac beams are obtained in four cases: as Gaussian, Bessel-Gaussian, modified Bessel-Gaussian and elegant Laguerre-Gaussian beams. In the second part of the work, the paraxial Dirac equation is derived, and then its solutions in the aforementioned cases are directly obtained. All the resulting wave functions conform to those derived formerly by virtue of superpositions, except for terms that are negligible upon the assumption that the paraxial functions along the propagation axis vary only slightly over a distance equal to the de Broglie wavelength, which is the standard paraxial requirement.
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 two recent papers exact Hermite-Gaussian solutions to relativistic wave equations were obtained for both electromagnetic and particle beams. The solutions for particle beams correspond to those of the Schr\"odinger equation in the nonrelativistic limit. Here, it will be shown that each beam particle has additional 4-momentum resulting from transverse localization compared to a free particle traveling in the same direction as the beam with the same speed. This will be referred to as the quantum 4-potential term since it will be shown to play an analogous role in relativistic Hamiltonian quantum mechanics as the Bohm potential in the nonrelativistic quantum Hamilton-Jacobi equation. Low-order localization effects include orbital angular momentum, Gouy phase, and beam spreading. Toward a more systematic approach for calculating localization effects at all orders, it will be shown that both the electromagnetic and quantum 4-potentials couple into the canonical 4-momentum of a particle in a similar way. This offers the prospect that traditional methods used to calculate the affect of an electromagnetic field on a particle can now be adapted to take localization effects into account. The prospects for measuring higher order quantum 4-potential related effects experimentally are also discussed alongside some questions to challenge the quantum information and quantum field theorists.
Abstract The two‐dimensional paraxial equation of optics and the two‐dimensional time‐dependent Schrödinger equation, derived as approximations of the three‐dimensional Helmholtz equation and the three‐dimensional time‐independent Schrödinger equation, respectively, are identical. Here the free propagation in space and time of Hermite–Gauss wave packets (optics) or harmonic oscillator eigenfunctions (quantum mechanics) is examined in detail. The Gouy phase is shown to be a dynamic phase, appearing as the integral of the adiabatic eigenfrequency or eigenenergy. The wave packets propagate adiabatically in that at each space or time point they are solutions of the instantaneous harmonic problem. In both cases, it is shown that the form of the wave function is unchanged along the loci of the normals to wave fronts. This invariance along such trajectories is connected to the propagation of the invariant amplitude of the corresponding free wave number (optics) or momentum (quantum mechanics) wave packets. It is shown that the van Vleck classical density of trajectories function appears in the wave function amplitude over the complete trajectory. A transformation to the co‐moving frame along a trajectory gives a constant wave function multiplied by a simple energy or frequency phase factor. The Gouy phase becomes the proper time in this frame. Key Points This paper builds a bridge between quantum mechanics (QM) and classical optics in that the identity of the paraxial equation of optics and the Schrödinger equation of QM is shown, the Bohmian trajectories of QM are defined in optics, the Gouy phase of optics is defined in QM and given a new interpretation and The space and momentum wave functions are equivalent along a trajectory.