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arXiv:2411.03261

On the Equivalence Between the Schrodinger Equation in Quantum Mechanics and the Euler-Bernoulli Equation in Elasticity Theory

Igor Vasil'evich Volovich

Published Nov 5, 2024• 1 citation
arXivHTMLPDF

Abstract

In this note, we show that the Schrodinger equation in quantum mechanics is mathematically equivalent to the Euler-Bernoulli equation for vibrating beams and plates in elasticity theory, with dependent initial data. Remarks are made on potential applications of this equivalence for symplectic and quantum computing, the two-slit experiment using vibrating beams and plates, and the p-adic Euler-Bernoulli equation.

DOI
10.48550/arxiv.2411.03261

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In this note, we show that the Schrodinger equation in quantum mechanics is mathematically equivalent to the Euler-Bernoulli equation for vibrating beams and plates in elasticity theory, with dependent initial data. Remarks are made on potential applications of this equivalence for symplectic and quantum computing, the two-slit experiment using vibrating beams and plates, and the p-adic Euler-Bernoulli equation.
Here we present a transformation that maps the Schrodinger equation of quantum mechanics to the incompressible Euler equations of fluid mechanics. The transformation provides a wave solution and a potential function based on fluid properties that satisfy the Schrodinger equation given that the fluid velocity potential and pressure satisfy the Euler equations. Interestingly, in our transformation, the equivalent of quantum potential becomes the physical surface tension. This is contrary to the Madelung transformation that maps the Schrodinger equation to the compressible Euler equations where there is no physical counterpart for the quantum potential. Lastly, we show that using this transformation, the Bohm equation can be mapped to a particle's equation of motion moving on the free surface of the fluid.
This chapter concerns the basic understanding of quantum mechanics from an engineering perspective. Understanding the topological behavior of elastic and acoustic waves demands physical knowledge and the mathematical treatments developed for quantum mechanics. This chapter thus presents quantum mechanics and its relation to the understanding of acoustic and elastic wave propagation for quantum analog applications. It is assumed in this chapter that the readers are familiar with the fundamentals of modern physics and the construct of an atom with nucleus and cloud of electrons. Assuming this familiarities, first the Schrödinger equation is derived. The derivation employed the concepts of relativistic particles and their properties of wave-particle-duality. Schrödinger wave equation and an elastic wave equation are solved in a one-dimensional system, simultaneously, to show the respective equivalency. Specific features such as the uncertainty principle of angular momentum and the relation to the Pauli matrices are derived for the eigenspinors. Next the Schrödinger equation is solved in a periodic crystal with the Bloch solution, and sample energy bands are presented., It is explained how, in these energy bands at a specific energy level, the Dirac equation for relativistic particle can help find the Hamiltonian operator. It is explained how using the Hamiltonian for spinors could help understand the band structure with their respective spin behavior at Dirac cones. These features and understanding will be useful for quantum analog applications of acoustic and elastic waves.
A generalized Euler equation of fluid dynamics is derived for describing many-body states of quantum mechanics. The Eulerian Eq. can be viewed as representing the interaction of two substates, where each substate has its own velocity and pressure fields. These field quantities are given by maps of the wavefunction. For one-body systems, the Eulerian Eq. can model either a fluid or particle description of quantum states. The generalized Euler Eq. is shown to be the gradient of an equation representing the total-energy of the two substates, having two energy fields. This total-energy Eq. is a generalization of the Bernoulli Eq. of fluid dynamics. The total-energy Eq., along with a continuity-equation, is equivalent to the time-dependent Schroedinger Eq. An equation is also derived that is equivalent to the main equation of Bohmian mechanics with additional identifications: The quantum potential of Bohmian mechanics is given as a sum of a kinetic energy and pressure fields. Also, the time derivative of the wavefunction phase is replaced by an energy field. In the formalism, field quantities are identified from their placement in equations of classical mechanics and separately, by definitions that involve the wavefunction and operators of quantum mechanics. This approach yields, unintended, and unknown energy and pressure fields. These fields, however, are shown to satisfy a continuity Eq., an equation that is equivalent to the other equation of Bohmian mechanics. It is also demonstrated that energy conservation holds for both of these energy fields, if the wavefunction is a linear-combination of eigenvectors, where the eigenvectors can be nondegenerate. A detailed investigation is given on the possible behavior, or source, of an electron that has one of the velocity fields. Alternate formulae for this velocity fields are also considered.
The basic equation of quantum mechanics is the Schrödinger equation which expresses the wave function Ψ of a quantum system as an eigenfunction of a quantized Hamiltonian operator H : HΨ = λΨ where the (real) eigenvalue λ is the quantum energy of the system in the state Ψ see equations (1.3.2), (1.3.4). Embodied already in this equation is the basic quantum mechanical principle that quantum energies cannot take on arbitrary values but are quantized : they are given by a discrete set of eigenvalues of a suitable second-order differential operator. This mathematical phenomenon of the discreteness of eigenvalues explains, for example, the observed discreteness of absorption and emission atomic spectral lines; compare remarks in Sections 1.2 and 1.3 of Chapter 1. The Schrödinger theory, and the equivalent theory of Heisenberg, Born and Jordan, represents a distinct advancement of the Bohr theory. Some early basic papers on quantum mechanics are compiled in the book [82], which includes a historic introduction by B. van der Waerden. Also see [6, 8, 9, 24, 75, 76]. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is the fourth paper of a series devoted to mathematically derive the Schrödinger equation using stochastic constructs. We then show that Quantum Mechanics has a stochastic support by following two paths: we first present a stochastic derivation already known in the literature and show that it is equivalent to the characteristic function derivation, made by us in the first paper of this series. However, this approach does not furnish the true stochastic dynamics of the quantum mechanical system, but only an averaged one. We then present a new derivation of the Schrödinger equation based on Langevin equations that present all features of a true stochastic system. These two approaches then improve our understanding of Quantum Mechanics to encompass stochastic behavior.