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The Journal of Chemical Physics

Position moments linearly averaged over the wave function. II

Toshikatsu Koga

Published May 15, 1989• 3 citations
Publisher

Abstract

Based on the Fourier transform of the position–space Schrödinger equation in a finite domain, a set of relations is derived among position moments linearly averaged over the wave function in that domain. The present results add nontrivial information for the structure of the Schrödinger equation, including the known relations for an infinite domain. A special case provides a regional form of the zero momentum energy expression, which measures the accuracy of wave functions in some specific region.

DOI
10.1063/1.456401

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Based on the Fourier transform of the position–space Schrödinger equation in a finite domain, a set of relations is derived among position moments linearly averaged over the wave function in that domain. The present results add nontrivial information for the structure of the Schrödinger equation, including the known relations for an infinite domain. A special case provides a regional form of the zero momentum energy expression, which measures the accuracy of wave functions in some specific region.
Information‐theoretical concepts are employed for the analysis of the interplay between a transverse electric field applied to a one‐dimensional surface and Robin boundary condition (BC), which with the help of the extrapolation length Λ zeroes at the interface a linear combination of the quantum mechanical wave function and its spatial derivative, and its influence on the properties of the structure. For doing this, exact analytical solutions of the corresponding Schrödinger equation are derived and used for calculating energies, dipole moments, position and momentum quantum information entropies and their Fisher information and and Onicescu information energies and counterparts. It is shown that the weak (strong) electric field changes the Robin wall into the Dirichlet, (Neumann, ), surface. This transformation of the energy spectrum and associated waveforms in the growing field defines an evolution of the quantum‐information measures; for example, it is proved that for the Dirichlet and Neumann BCs the position (momentum) quantum information entropy varies as a positive (negative) natural logarithm of the electric intensity what results in their field‐independent sum . Analogously, at and the position and momentum Fisher informations (Onicescu energies) depend on the applied voltage as ( ) and its inverse, respectively, leading to the field‐independent product ( ). Peculiarities of their transformations at the finite nonzero Λ are discussed and similarities and differences between the three quantum‐information measures in the electric field are highlighted with the special attention being paid to the configuration with the negative extrapolation length. image
Abstract In stark contrast to classical Newtonian mechanics, the standard interpretation of quantum mechanics associates the Schrödinger wavefunction, not in a determ- inistic way with definite predictions about a specific, individual experimental result, but only in a probabilistic manner, with |ψ(x, t )|2 acting as a probability density, in this case, for position measurements. Before analyzing this approach in more detail, we will find it profitable to briefly review some of the basic concepts of discrete and
Abstract In this paper, the finite square well and its application are investigated, namely the quantum state discrimination. The finite square well is treated in all standard textbooks on introductory quantum mechanics. It is used as a simple ‘model of departure’ in many areas of physics. In atomic and molecular physics, it may be used as a model of an electron moving in the mean field of a linear molecule It also arises as the partial wave radial equation for a spherically symmetric, finite square-well potential. The Schrodinger equation for finite square well is solved. The domain is divided into three regions by the existing potential V 0, so for convenience, those three regions are named Region I, Region II, and Region III, respectively. Specifically, the potential of Region I and III are V 0, and that of Region II is 0. Also, a constant is needed to make this wave function normalized. Then since it is a wave function, it is necessary to make sure the function is continuous and differentiable everywhere within the domain. Besides, the wave function needs to be either odd or even just like infinite square well. After that, the wave functions for the three intervals can be obtained, and the exact quantum state is distinguished from a group of different quantum states. In the end, two wave functions are obtained; one for even form and one for odd form. Then all the energy level of the corresponding function with different wavelength needs to be found and listed. Local discrimination of orthogonal quantum states has attracted much attention during the last twenty years. The results are applied to the quantum-information task of state discrimination, by using the obtained six states in finite square well. It is assumed that all the quantum states are locally distinguishable, and the six states are distinguished using the hypothesis of quantum measurements.