Certain Functions used in Quantum Mechanics

D. A. Slavnov

It is proposed the scheme of quantum mechanics, in which a Hilbert space and the linear operators are not primary elements of the theory. Instead of it certain variant of the algebraic approach is considered. The elements of noncommutative algebra (observables) and the nonlinear functionals on this algebra (physical states) are used as the primary constituents. The functionals associate with results of a particular measurement. It is suggested to consider certain ensembles of the physical states as quantum states of the standart quantum mechanics. It is shown that in such scheme the mathematical formalism of the standart quantum mechanics can be reproduced completely.

Claude N. Cohen-Tannoudji, Bernard Diu, Franck Laloë

D. Terasawa

Abstract The number of electrons in a homogeneous electric field is derived quantum mechanically for a certain range. Using the delta function normalization method, which is described in detail in Landau–Lifshitz’s Quantum Mechanics, the normalization coefficient of the wavefunctions with continuous energy eigenvalue spectra is obtained. Subsequently, the number of electrons in a one-dimensional homogeneous field in a certain region is derived from this normalization coefficient. We illustrate the derivation of physical quantities using normalization coefficients through selected examples. The derivation provides an interesting application of delta function normalization, and thus can serve as a good textbook example of this normalization method, which used to play a limited role in quantum mechanics.

Shan Gao

In quantum mechanics, the physical state of an electron is described by a wave function. According to the standard probability interpretation, the wave function of an electron is probability amplitude, and its modulus square gives the probability density of finding the electron in a certain position in space. In this article, we show that this central assumption of quantum mechanics may have an ontological extension. It is argued that microscopic particles such as electrons are indeed particles, but their motion is not continuous but discontinuous and random. On this view, the modulus square of the wave function not only gives the probability density of the particles being found in certain locations, but also gives the probability density of the particles being there. In other words, the wave function in quantum mechanics can be regarded as a representation of the state of random discontinuous motion of particles, and at a deeper level, it may represent the dispositional property of the particles that determines their random discontinuous motion.

V.V. Babikov