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Communications in Theoretical Physics

A Method for Avoiding the Divergence Difficulties of Quant urn Mechanics

Wenhua Hai

Published Mar 15, 1999• 4 citations
Publisher

Abstract

High-ordered correction of wavefunction for Schrödinger equation with one dimensional potential V(x) and interaction Hamiltonian has been found by introducing a new particular solution for . Convergence conditions of the wavefunction lead to the formulas of energy corrections and scattering amplitudes. It is shown that the result can avoid some divergence difficulties of quantum mechanics.

DOI
10.1088/0253-6102/31/2/297

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High-ordered correction of wavefunction for Schrödinger equation with one dimensional potential V(x) and interaction Hamiltonian has been found by introducing a new particular solution for . Convergence conditions of the wavefunction lead to the formulas of energy corrections and scattering amplitudes. It is shown that the result can avoid some divergence difficulties of quantum mechanics.
A different integral form of the Schr\"odinger equation and some boundedness conditions of the wave functions are proved, which exhibit the origin of some divergences in quantum mechanics. The iterations from the integral equation and the boundedness conditions lead to a method for avoiding the divergence difficulties. The result is extended to the time-dependent and the multidimensional cases. Examples of an electron in the Wigner spherical ``box'' with a 1/r1/r1/r perturbed potential, some heliumlike ions in the ground state, and the Stark effect of the hydrogen atom show that this different perturbation method can give rational multiorder corrections of the wave function and energy.
A new approach to find exact solutions to one--dimensional quantum mechanical systems is devised. The scheme is based on the introduction of a potential function for the wavefunction, and the equation it satisfies. We recover known solutions as well as to get new ones for both free and interacting particles with wavefunctions having vanishing and non--vanishing Bohm potentials. For most of the potentials, no solutions to the Schrödinger equation produce a vanishing Bohm potential. A (large but) restricted family of potentials allows the existence of particular solutions for which the Bohm potential vanishes. This family of potentials is determined, and several examples are presented. It is shown that some quantum, such as accelerated Airy wavefunctions, are due to the presence of non--vanishing Bohm potentials. New examples of this kind are found and discussed.
THE classical theory of a point charge and the quantum theory of wave packets contain in their usual form well-known divergences. Dirae1 has elaborated a classical particle theory which avoids these difficulties, but his results cannot as yet be applied to quantum theory. On the other hand, Heisenberg1 has recently proposed a new formalism which permits the calculation of collision cross-sections in quantum theory without being disturbed by diverging terms. However, no connecting link such as the correspondence principle has been given in order to apply this quantum formalism to a given problem such as Rutherford scattering, Compton effect or radiation damping.
Abstract This work extended the simple diffusion QMC method of solving the Schrödinger equation for a wavefunction ψ; to that of solving for a correction to a trial wavefunction ψo such that ψ= ψo+ . The method may be applied in the manner of successive corrections to obtain very high accuracies. As described, the method is limited to solutions for systems with fixed nodes. The method was illustrated for the cases of the particle-in-a-box and the hydrogen atom. For the latter, successive corrections improved the accuracy in energy by about a factor of ten with each iteration. For higher dimensional systems, difficulties in the fitting procedure may be expected to limit the applicability of the method.
This solves quantum physics problems using the integral Rohit transform, including the scattering of low energy particles by a completely rigid sphere and particle behavior in a one-dimensional infinitely high potential box. The standard calculus approach is typically used to solve these quantum mechanics issues. To solve these quantum physics challenges, this research presents a new method: the integral Rohit transform. When compared to other approaches that are currently in the literature, the obtained solutions show how accurate the suggested method is. This demonstrates the potential and efficacy of the suggested approach to overcoming quantum mechanical issues, such as low-energy particle scattering by a completely rigid sphere and particle behavior in a one-dimensional infinitely high potential box. In this paper, the successful application of the integral Rohit Transform has been demonstrated in solving the one-dimensional time-independent Schrodinger's equation. This application has yielded results that include the determination of eigen energy values and eigen functions for a particle confined within an infinitely high potential well, as well as the calculation of the total scattering cross-section for low energy particles interacting with a perfectly rigid sphere. In the case of low energy limit, the total scattering cross-section for low energy particles due to a perfectly rigid sphere, as determined through quantum mechanics, is equivalent to the geometrical cross-section of said sphere. Additionally, the energy values that the particle can possess within a 1d infinitely high potential well demonstrate that the energy of said particle, when confined within this potential well, is quantized.