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Journal of Nantong University

Continuity of Wave-function and Its First Rank Differential

Fang Jing-huai

Published Jan 1, 2007• 0 citations
OpenAlex

Abstract

According to mathematical principles the article rigidly deduces from Schrodinger equation the continuity conditions of wave-function and its first rank differential.They will continue if potential function continues,or if it doesn't continue but has a bound in the contiguity of the discontinuous point.However,the continuity of wave-function and its first rank differential will be uncertain if potential function is discontinuous,and boundless in the contiguity of the discontinuous point.

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According to mathematical principles the article rigidly deduces from Schrodinger equation the continuity conditions of wave-function and its first rank differential.They will continue if potential function continues,or if it doesn't continue but has a bound in the contiguity of the discontinuous point.However,the continuity of wave-function and its first rank differential will be uncertain if potential function is discontinuous,and boundless in the contiguity of the discontinuous point.
The method of convolutions is used to form smoothed functions from approximate wave functions which are discontinuous or have discontinuous first derivatives. As a parameter ε in the smoothed function decreases, the smoothed function approaches the approximate wave function. The expectation values for physical properties corresponding to the approximate wave function are defined to be the limit as ε approaches zero of the expectation values corresponding to the smoothed function. It is found that if the approximate wave function is discontinuous, the corresponding expectation value for the kinetic energy is infinite. Therefore, it seems unlikely that discontinuous approximate wave functions can ever be useful. However, if the approximate wave function is continuous but has a discontinuity in its first derivative, then, as a result of the discontinuity, there is a contribution δT̄12 to the expectation value of the kinetic energy. For a one-dimensional problem δT̄12=−(ℏ2/2m)ψ*(0) [ψ2′(0)−ψ1′(0)].Here ψ(0) is the value of the approximate wave function at the point zero where the discontinuity in its first derivative occurs, and ψ1′(0) and ψ2′(0) are the first derivatives of ψ taken from the left and from the right, respectively, at this point. Similarly, for an N-dimensional problem having a surface S12=0 over which the first derivatives of the approximate wave function are discontinuous, δT̄12= ∫ surfaceψ*[(∂/∂n) (ψ2−ψ1)]dS12.Here the ∂/∂n is the normal derivative with the normal pointed from region 1 towards region 2.
A general formulation of the eigenvalue condition is pointed out for a one-dimensional stationary part of a separable Schrodinger equation with finite bounds, i.e. where the wavefunction vanishes at two arbitrary values of the independent variable. The procedure is general and applies equally to other eigenvalue problems of second-order, homogeneous differential equations. Where appropriate a continuous density of eigenvalues can be defined and evaluated without first solving the eigenvalue problem. Illustrative examples are provided, including gravitational and harmonic oscillator potentials.
Abstract: Quantum theory does not require the existence of discontinuities: neither in time (quantum jumps), nor in space (particles), nor in spacetime (quantum events). These apparent discontinuities are readily described objectively by the continuous process of decoherence occurring locally on a very short time scale according to the Schrdinger equation for interacting systems, while the observerÕs Ôincrease of informationÕ is appropriately represented by the resulting dynamical decoupling of the corresponding components of the global wave function. Physics Letters A172, 189 (1993) 2As far as is known, all properties of closed quantum systems are perfectly described by means of wave functions in configuration space (in general, wave functionals of certain fields) dynamically evolving smoothly according to the time-dependent Schrdinger equation. However, the condition of being closed (or shielded against interactions with the environment) can easily be estimated to be quite exceptional. It characterizes very special (usually atomic) systems from which the laws of quantum mechanics were derived. When the shielding ceases, most notably during measurements, discontinuous