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Journal of Dalian Jiaotong University

Simple Calculation Method of Hydrogen Atom Characteristic Values

Huiping Liu

Published Jan 1, 2008• 0 citations
OpenAlex

Abstract

Based on classical equivalent model and Schrdinger equation,mathematical formulas of hydrogen energy and angular momentum magnitude as well as its component along with magnetic field direction in quantum mechanics are simply derived,the complicated method in which avoids the strict calculation of Schrdinger equations for its three characteristic values.

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Journal of Hebei Normal University•Jan 1, 2007•0 citations
Based on classical equivalent model and Schrdinger equation,mathematical formulas of hydrogen energy and angular momentum magnitude as well as its component along with magnetic field direction in quantum mechanics are simply derived,the complicated method in which avoids the strict calculation of Schrdinger equations for its three characteristic values.
Based on the previous works, the Schrdinger equation of the hydrogen-like atom is analytically solved further. Three sets of analytical solutions are obtained if the factor r~ -nis not neglected. The first solution is the same as the traditional radial wave function, the second diverges, and the last one is quite different from the traditional solution. On the consideration of the finite size of the nuclear, the third wave function does not diverge when r approaches to zero. Its radial wave function has the following characteristics: (1) the angular-momentum quantum number l must be greater than the principal quantum number n; (2) l must not be 0 or 1; (3) the electron-cloud distribution differs from the traditional one; (4) the electron is closer to the nuclear than that in traditional results. On the other hand, the validity of solutions needs to be verified experimentally.
An alternative method of deriving the values of the observables of atomic systems is presented. Rather than using operators and eigenvalues the local variables method uses the continuity equation together with current densities derived from wave functions that are solutions of the Dirac or Pauli equation. The method is applied to atomic hydrogen using the usual language of quantum mechanics rather than that of geometric algebra with which the method is often associated. The picture of the atom that emerges is one in which the electron density as a whole is rotating about a central axis. The results challenge some assumptions of conventional quantum mechanics. Electron spin is shown to be a property of the dynamical motion of the electron and not an intrinsic property of the electron, the ground state of hydrogen is shown to have an orbital angular momentum of ℏ, and excited states are shown to have angular momenta that are different from the eigenvalues of the usual quantum mechanical operators. The uncertainty relations are found not to be applicable to the orthogonal components of the angular momentum. No double electron spin gyromagnetic ratio is required to account for the observed magnetic moments, and the behavior of the atom in a magnetic field is described entirely in kinetic terms.