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American Journal of Physics

Potential Energies of Identical Motion in Classical and Quantum Mechanics

David B. Berkowitz, Peter D. Skiff

Published Nov 1, 1972• 13 citations
Publisher

Abstract

Potential energies are found for which the stationary-state motion of a particle in a static force field is the same according to the Schrödinger equation as in classical mechanics.
DOI
10.1119/1.1986999

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Stationary states General properties We shall consider first some general characteristics of the ‘stationary states’. These are eigenfunctions of the Hamiltonian operator: The requirement that ψ satisfies the Schrödinger equation fixes its time dependence: Notice that (4.1.2) can only be a solution if the external potential V is independent of the time: V = V ( x ). The initial function ψ 0 is not arbitrary but must satisfy the time-independent Schrödinger equation (4.1.1). Writing ψ = R e iS /ħ one thus has The following deductions may be made from (4.1.3): ( a ) The probability density is independent of the time: |ψ| 2 = R 2 0 ( X ). ( b ) The quantum potential Q = (- ħ2/2 mR 0 )∇ 2 R 0 is time-independent. Therefore so is the total effective potential: ( c ) The velocity field v = (l/ m )∇ S 0 is independent of time and hence the trajectories, found by solving m = v ( x )| x = x ( t ) , are solutions of Moreover, ψ 0 ( x ) is often a real function for many bound state problems of interest (although not for scattering states) which implies that the velocity is zero. The particle is at rest where one would classically expect it to move since the quantum force (- ∇ Q ) cancels the classical force (- ∇ K ). ( d ) The energy of all particles in the ensemble, -∂ S /∂ t , is a constant of the motion and equal to the energy eigenvalue: with (4.1.4), we have an example of the conservation of energy (cf. §3.9.3). ( e ) The total field energy is conserved: 〈Ĥ〉 = E . […]
We consider spatially two-dimensional Madelung fluid whose irrotational motion reduces into the Schr\"odinger equation for a single free particle. In this respect, we regard the former as a direct generalization of the latter, allowing a rotational quantum flow. We then ask for the most likely wave function possessing a given average energy by maximizing the Shannon information entropy over the quantum probability density. We show that there exists a class of solutions in which the wave function is self-trapped, rotationally symmetric, spatially localized with finite support, and spinning around its center, yet stationary. The stationarity comes from the balance between the attractive quantum force field of a trapping quantum potential generated by quantum probability density and the repulsive centrifugal force of a rotating velocity vector field. We further show that there is a limiting case where the wave function is nonspinning and yet still stationary. This special state turns out to be the lowest stationary state of the ordinary Schr\"odinger equation for a particle in a cylindrical tube external classical potential.
Two axioms were excavated. Axiom 1: Only by allowing communication and generating consensus can we ensure equal rights for different individuals. Axiom 2: Objective existence is an original independent existence that is independent of measurement and does not rely on third parties. Axiom 3: All real things have their objective and independent original existence states. Let A and B move relative to each other and observe each other's clocks. In the language of relativity, they all believe that the observed clock slows down. The consensus between A and B is that A clock is both slower and faster than B clock. According to Axiom 2, it can be concluded that quantum mechanics must be a 'local realism'. So, the Copenhagen quantum mechanics explanation violates Axiom 2. One of the methods for qualitatively and quantitatively coordinating relative and absolute is to assume that the clock of absolute motion slows down, thus establishing a relative absolute spatiotemporal view. According to the Dirac equation, the existence of negative mass matter can be predicted. A positive mass particle and a negative mass particle cannot fall at the same acceleration in the induced field. Einstein's explanation of spacetime curvature was fatally challenged. The potential energy of electromagnetic interactions in the Schrödinger equation can be replaced with gravitational potential energy to form the Schrödinger-Tu equation. The potential energy in the equation can be further transformed into force (electromagnetic force and/or gravitational force). the mathematical factors of quantum theory and classical theory can be written into the same equation. The classical laws of mechanics can be expressed using wave mechanics equations.