Geometrical optics can be regarded both as the short-wavelength approximation of the propagation of electromagnetic waves, and as the exact way in which propagate the surfaces of discontinuity of the classical electromagnetic field. In this work we translate this last idea to quantum mechanics (both relativistic and nonrelativistic). We find that the surfaces of discontinuity of the wave function propagate exactly following the classical trajectories determined by the Hamilton-Jacobi equation. As an example, we consider the lack of diffraction of abrupt wave fronts.
Geometrical optics can be regarded both as the short-wavelength approximation of the propagation of electromagnetic waves, and as the exact way in which propagate the surfaces of discontinuity of the classical electromagnetic field. In this work we translate this last idea to quantum mechanics (both relativistic and nonrelativistic). We find that the surfaces of discontinuity of the wave function propagate exactly following the classical trajectories determined by the Hamilton-Jacobi equation. As an example, we consider the lack of diffraction of abrupt wave fronts.
In this article, we will answer a question posed in the book Classical Mechanics by H. Goldstein: ``"Is the Hamilton-Jacobi equation the short wavelength limit of the Schrödinger equation?" But, before that, we will identify an essential element that will take us from the Hamilton-Jacobi equation to the dynamic equation of non-relativistic quantum mechanics for a function ψ through an exact procedure. This element is the linear independence of the functions ψ and ψ* (their complex conjugate). Their independence is demonstrated for physical systems where the acting physical potential does not explicitly depend on time. Proceeding in reverse, from the Schrödinger equation, we obtain the Hamilton-Jacobi equation, exactly, without additional terms.
A well-known formulation of classical mechanics of system of particles is based on solving the Hamilton-Jacobi equation ∂S ∂t + H ( qi , ∂S ∂qi , t ) = 0, (19.1) where S(qi , αi , t) is Hamilton’s principal function, H is the Hamiltonian, qi are the configuration space variables, αi are constants of integration, and t denotes time. If the Hamiltonian is independent of t and is a constant of motion equal to the total energy E, then S and Hamilton’s characteristic function W are related by S = W − Et . (19.2) Since W is independent of time, the surfaces with W = constant have fixed locations in configuration space. However, the S = constant surfaces move in time and may be considered as wavefronts propagating in this space [1]. For a single particle one can show that the wave velocity at any point is given by u = E|∇W | = E p = E mv . (19.3) T&F Cat # K11224, Chapter 19, Page 302, 15-9-2010 This shows that the velocity of a point on this surface is inversely proportional to the spatial velocity of the particle. Also one can show that the trajectories of the particle must always be normal to the surfaces of constant S. The momentum of the particle is obtained as [1] p = ∇W . (19.4) It is not clear whether this wave picture was known to classical physicists. However, in hindsight, we may now observe that there exists a correspondence between classical mechanics and geometrical optics. That is, the particle trajectories orthogonal to surfaces of constant S in classical mechanics, as we have discussed above, are similar to light rays traveling orthogonal to Huygens’ wavefronts.
Most authors of textbooks on quantum mechanics either postulate or sketch a short `ad hoc` derivation of Schrodinger's equation. In this work we give a detailed derivation of Schrodinger's equation from the Hamilton-Jacobi equation and the Eikonal equation in geometrical optics. We start from the historical debates on the nature of light -- whether it is a beam of particles, or waves in the aether. We derive the Eikonal equation and show the conditions when a wave can behave as a beam of particles. Then we discuss several experiments with an electron gun, that show clearly diffraction and interference of a single electron. Next, in order to explain these experiments, we derive Schrodinger's equation by comparing Hamilton-Jacobi equation in classical mechanics with the Eikonal equation in geometrical optics. To do that, we first show how to derive the wave equation from the Eikonal equation (not the other way around!). Second, we use this method to derive Schrodinger's equation from the Hamilton-Jacobi equation. Next, we derive Born's statistical rule using the early understanding of de Broglie that both particles and waves exist. Afterwards, we show that historically people preferred to remove the particles (as well as their trajectories) altogether from de Broglie's ideas but retained Born's rule (the so called Copenhagen interpretation). These derivations of the foundations of quantum mechanics do not follow precisely the history of the subject. Rather we select some early ideas and experiments in a judicious manner to present Schrodinger's equation in a logical and ordered way. We use the electron gun experiments instead of black body radiation and photoelectric effect. Our derivation may bring more light and satisfaction for the undergraduate students about the confusing and rather mysterious subject of quantum mechanics.
The Hamilton–Jacobi equation in classical mechanics and the related eikonal equation in geometrical optics are often described as the “point of closest approach” between classical and quantum mechanics. Most textbook treatments of Hamilton–Jacobi theory are aimed at graduate students and derive the equation only after a long introduction to canonical transformations. Most treatments of the eikonal equation only emphasize its use in geometrical optics. We show that both the Hamilton–Jacobi equation and the eikonal equation can be derived by a common procedure using only elementary aspects of the Lagrangian and Hamiltonian formalisms introduced in undergraduate classical mechanics courses. Through this common approach, we hope to highlight to undergraduates the deep connections between classical mechanics, classical wave theory, and Schrödinger’s wave mechanics.
Some recent experiments claim to show that any model in which a quantum state represents mere information about an underlying physical reality of the system must make predictions which contradict those of quantum theory. The present work undertakes to investigate the issue of reality, treading a more fundamental route from the Hamilton-Jacobi equation of classical mechanics to the Schrodinger equation of quantum mechanics. Motivation for this is a similar approach from the eikonal equation in geometrical optics to the wave equation in electromagnetic theory. We rewrite the classical Hamilton-Jacobi equation as a wave equation and seek to generalise de Broglie's wave particle duality by demanding that both particle and light waves have the freedom of being described by any square-integrable function. This generalisation, which allows superposition also for matter wave functions, helps us to obtain the Schrodinger equation, whose solution can be seen to be as much objective as the classical mechanics wave function. Several other equations which one writes in quantum mechanics, including the eigenvalue equations for observables, series expansion of energy states in terms of eigenstates of observables other than energy, etc., can be written in the classical case too. Absence of any collapse of the wave function, entanglement, etc. in the classical realm have their origin in the nonlinearity of the classical wave equation. These considerations indicate that many of the puzzles in quantum mechanics are present also in classical mechanics in a dormant form, which fact shall help to demystify quantum mechanics to a great extent.
This paper explores the profound connection between the Eikonal principle, primarily known in optics for describing wavefront propagation, and the Hamilton-Jacobi formulation of classical mechanics. We explicitly demonstrate how the Hamilton-Jacobi equation, a powerful partial differential equation that unifies various aspects of classical dynamics, naturally gives rise to an Eikonal-like behavior when considering the surfaces of constant action. By drawing an explicit analogy between the optical path length and Hamilton's principal function, we show that the trajectories of particles in classical mechanics can be interpreted as "rays" that are orthogonal to these propagating "wavefronts" of constant action in configuration space. This framework provides an elegant geometric interpretation of classical mechanics, where the evolution of a system is described by the propagation of a scalar field. The paper elucidates how the principle of least action, a cornerstone of classical mechanics, finds its optical counterpart in Fermat's principle of least time, thereby achieving a deeper, unified understanding of these seemingly disparate physical phenomena. This unification not only offers a powerful method for solving complex mechanical problems but also provides conceptual insights bridging classical wave theory and particle dynamics, laying foundational groundwork for quantum mechanics.