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Applied Physics Research

Spent Time of Orbiting Eletron Generating the Quantized Energy En in the Hydrogen Like Atom

Josip Z. Soln

Published Nov 6, 2024• 0 citations
Publisher

Abstract

Despite the large literature on hydrogen-like atoms, one cannot find any mention as to how long an electron spends in the orbit to generate the quanized radiation energy En. Here, the Heisenberg representation of operators is exteded so as to be able to deal also with imverse oprators. This is necessary when dealing with Coulomb potentials. This then allows to write down the general hydrogen-like atom in a compact form. With the help of operator algebra that includes also the inverse operators, one arrives at the differential equation of the electron trajectory with respect to time.The time solution is achieved through established coupled integral equations. While already established quantized radiation electron energy En is proportional to 1/n2, the newely derived quantized electron circular time tn(e) is proportional to n3 with n=1,2,3,..,and tn(e) being of the order of 10−16 sec in magnitude.

DOI
10.5539/apr.v16n2p155

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Despite the large literature on hydrogen-like atoms, one cannot find any mention as to how long an electron spends in the orbit to generate the quanized radiation energy En. Here, the Heisenberg representation of operators is exteded so as to be able to deal also with imverse oprators. This is necessary when dealing with Coulomb potentials. This then allows to write down the general hydrogen-like atom in a compact form. With the help of operator algebra that includes also the inverse operators, one arrives at the differential equation of the electron trajectory with respect to time.The time solution is achieved through established coupled integral equations. While already established quantized radiation electron energy En is proportional to 1/n2, the newely derived quantized electron circular time tn(e) is proportional to n3 with n=1,2,3,..,and tn(e) being of the order of 10−16 sec in magnitude.
The mere existence of stable atoms, and their arrangement into many-atom nanoscale structures, can only be explained by the laws of quantum mechanics. Indeed, the accumulated evidence regarding the internal structure of atoms motivated the formulation of quantum mechanics at the early twentieth century. Here we focus on single-electron (hydrogen-like) atoms. The dominance of the Coulomb central potential between the electron and nucleus, translates into a simple Schrödinger equation for their relative position distribution. Solving rigorously this equation reveals quantized energy levels, which explains the emission spectra of hydrogen-like atoms (Rydberg formula). The corresponding stationary probability density distributions for the electron around the nucleus are reminiscent of the Bohr model, where classical trajectories (orbits) are replaced by single-electron wave functions (orbitals). The degeneracies of different orbitals, associated with three quantum numbers, will later turn out to be critical for understanding many-electron atoms and the periodic table of the elements.
The purely algebraic technique associated with the creation and annihilation operators to resolve the radial equation of Hydrogen-like atoms (HLA) for generating the bound energy spectrum and the corresponding wave functions is suitable for many calculations in quantum physics. However, the analytical approach with series is extensively used based on wave mechanics theory in most of quantum textbooks. Indeed, much More complete than the old solution of Schrödinger's time-independent differential equation (TISE), one can simply earn all quantum information of a system by using the operational method. In addition to earlier two models, including the quantum harmonic oscillator and the total angular momentum, it can undoubtedly be a third fundamental model to solve Schrödinger's eigenvalue equation of the HLA systems in Hilbert space similar to the harmonic oscillator. We will illustrate how systematically making an appropriate groundwork to discover the coherent states can lead to providing the energy quantization and normalized radial wave functions attached to the matrix representation without additional assumptions.
This chapter begins with the simple solution of the Schrödinger equation for hydrogen atom for which the energy value has been derived. The Bohr radius of H atom has also been computed. After this, this chapter provides the generalised solution of the Schrödinger equation for H atom/hydrogen-like species in which F equation, R equation, and T equations have been derived in a simple and lucid way. This chapter also deals with the Laguerre differential equation, Laguerre polynomials, Rodrigues formula for the Laguerre polynomials, Laguerre associated equation and its solution, associated Laguerre polynomials, Rodrigues formula for the associated Laguerre polynomials, and solution of the radial equation. It also provides normalisation of the radial wave function and complete wave function; for the hydrogen atom, hydrogenic atomic orbital; electron density of 1 s electron; mathematical background of shape of orbitals; the most probable distance of electron from nucleus of hydrogen atom, and average distance of electron from the nucleus of hydrogen atom. The end of this chapter describes the references and adequate computational problems (solved problems and question on concepts).
The energy levels of hydrogen-like atoms are obtained from the phase-space quantization, one of the pillars of the old quantum theory, by three different methods - (i) direct integration, (ii) Sommerfeld's original method, and (iii) complex integration. The difficulties come from the imposition of elliptical orbits to the electron, resulting in a variable radial component of the linear momentum. Details of the calculation, which constitute a recurrent gap in textbooks that deal with phase-space quantization, are shown in depth in an accessible fashion for students of introductory quantum mechanics courses.