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Acta Sinica Quantum Optica

Double-Wave Description of a Three-Dimensional Hydrogen Atom and a Three-Dimensional Similar Hydrogen Atom

Ma Ai-Qun

Published Jan 1, 2003• 0 citations

Abstract

In this paper, double-wave quantum theory is applied to describe a thr ee-dimensional hydrogen atom and a three-dimensional similar hydrogen atom, the classical limit of this description is discussed. It is proved that double-wave function satisfies the normalization condition. Under the description of double- wav e functions, the expectation value of energy and angular momentum are E n a nd m, respectively. A t the same time, we calculated the average values of r, cos θ, sin θ , cos φ, sin φ and X, Y, Z, under the description of double-wave fu nction. The statistical average va lue for mechanics quantity  shows that single-wave function describes the states of the system of hydrogen atom and similar hydrogen atom. Only double-wave func tion can describe the states of single microcosmic body.

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•Apr 30, 2021•0 citations

In this paper, double-wave quantum theory is applied to describe a thr ee-dimensional hydrogen atom and a three-dimensional similar hydrogen atom, the classical limit of this description is discussed. It is proved that double-wave function satisfies the normalization condition. Under the description of double- wav e functions, the expectation value of energy and angular momentum are E n a nd m, respectively. A t the same time, we calculated the average values of r, cos θ, sin θ , cos φ, sin φ and X, Y, Z, under the description of double-wave fu nction. The statistical average va lue for mechanics quantity  shows that single-wave function describes the states of the system of hydrogen atom and similar hydrogen atom. Only double-wave func tion can describe the states of single microcosmic body.

Using the harmonic oscillator representation of the hydrogen atom and constructing the appropriate coherent state corresponding to the minimum uncertainty product, the classical limits of the two-dimensional and the three-dimensional hydrogen atom are examined. The method adopted is similar to that used by Bhaumik, Dutta Roy and Ghosh (1986). The authors deduce the classical limit by requiring that the expectation value (r) of the radial variable is large for the two-dimensional hydrogen atom. The resulting trajectory is an ellipse satisfying the conditions of the Kepler orbit in classical mechanics. In order to obtain the classical limit of the three-dimensional hydrogen atom, besides the condition of large (r), they make the uncertainty Delta r=((r 2 )-(r) 2 ) 1/2 a minimum with respect to certain parameters of the coherent state. Imposition of only the first condition leads to an elliptical orbit, which is not a Kepler orbit in general. Imposition of both conditions leads to an orbit identical with that of the two-dimensional hydrogen atom. However, the time evolution of the expectation values of position variables are consistent with the corresponding results of the Kepler problem only for small values of eccentricity, but not in general.

Abstract In this chapter, we go to three dimensions in space and look at the solution of the time independent Schrödinger equation for the hydrogen atom. The Hamiltonian is then the kinetic energy plus the potential energy due to the Coulomb coupling between the positively charged nucleus and the electron. We construct the angular momentum operator and find that the partial differential equation for the angular momentum eigenfunctions of the spherical coordinates θ,ϕ is the same as the angular part of the ∇2 operator in spherical coordinates. The angular momentum eigenfunctions are the spherical harmonics, with two quantum numbers, l and m, and the solution to the radial part of the Hamiltonian including the Coulomb potential are Laguerre polynomials with one quantum number, called the principle quantum number, n. The hydrogen wave function is the product of a Laguerre polynomial and a spherical harmonic with three quantum numbers. Since these are two- and three-dimensional functions for angular momentum and hydrogen respectively, they are best understood in a series of plots. The chapter concludes by giving the historical letter names to specific orbitals, since they continue to be used today.