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Fine Chemical Technologies

Движение электрона в почти кулоновском поле

Д. В. Калиновский, Е. С. Савин

Published Apr 28, 2013• 0 citations
OpenAlex

Abstract

will be added. One can explain the nature of the term by examining the relativistic case. Below the formulas for the radial and angular parts of the wave function were evaluated. The term for the energetic levels of the hydrogen atom for specific quantum numbers was given. We calculated the values of several specific radial functions for different quantum numbers, as well as the values of the radial function at the origin of coordinates and wide apart. A number of mean values for different powers of r was calculated. For the continuous spectrum the radial eigenfunctions were calculated.

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>H-Atom

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In this chapter we solve the radial TISE for the case of a radial potential, specifically the radial potential of an electron attracted to the proton which is the nucleus of a hydrogen atom. Again, we rely on solutions to a related equation solved by a French mathematician, this time named Laguerre, whose polynomials are, in this chapter, multiplied by a simple exponential function to produce the radial wave functions. These functions contain the principle quantum number, which corresponds to the row number in the periodic table. Multiplying these radial solutions by the spherical harmonics from Chapter 14 produces the full three-dimensional solutions for the hydrogen atom, also known as the atomic orbitals. The relationship between the quantum numbers n , l , and m arise naturally from the possible combinations of radial solutions with spherical harmonics.

CRC Press eBooks•Jul 12, 2023•0 citations
will be added. One can explain the nature of the term by examining the relativistic case. Below the formulas for the radial and angular parts of the wave function were evaluated. The term for the energetic levels of the hydrogen atom for specific quantum numbers was given. We calculated the values of several specific radial functions for different quantum numbers, as well as the values of the radial function at the origin of coordinates and wide apart. A number of mean values for different powers of r was calculated. For the continuous spectrum the radial eigenfunctions were calculated.
will be added. One can explain the nature of the term by examining the relativistic case. Below the formulas for the radial and angular parts of the wave function were evaluated. The term for the energetic levels of the hydrogen atom for specific quantum numbers was given. We calculated the values of several specific radial functions for different quantum numbers, as well as the values of the radial function at the origin of coordinates and wide apart. A number of mean values for different powers of r was calculated. For the continuous spectrum the radial eigenfunctions were calculated.
The radial functions for the discrete levels of Dirac's relativistic hydrogenic atom are expressed in terms of the generalized Laguerre polynomials which have known properties. This leads to an easy method of evaluating the normalization constants and the average values of rq{r}^{q}rq.
The simplest atom consisting of a proton and an electron is a hydrogen atom. The energy level and the probability of finding an electron in a hydrogen atom can be determined based on the function of radial wave. The function of radial wave of a hydrogen atom is influenced by the principal quantum number ( n ) and the orbital quantum number ( l ). Completion of the function of radial wave of a hydrogen atom in the principal quantum numbers( n ) 4 and 5 uses the time-independent Schrodinger equation approach in spherical coordinates, variable separation method and uses the associated Laguerre function. The normalized of the function of radial wave of a hydrogen atom R nl ( r ) at n = 4 produces four wave functions:R 40 ( r ); R 41 ( r ); R 42 ( r ); R 43 ( r ), whereas at n = 5 produces five wave functions:R 50 ( r ); R 51 ( r ); R 52 ( r ); R 53 ( r ); R 54 ( r ). The results of this research in the form of the functions of radial wave of hydrogen atom can be used to determine the behavior of electron in a hydrogen atom.
In this chapter we solve the radial TISE for the case of a radial potential, specifically the radial potential of an electron attracted to the proton which is the nucleus of a hydrogen atom. Again, we rely on solutions to a related equation solved by a French mathematician, this time named Laguerre, whose polynomials are, in this chapter, multiplied by a simple exponential function to produce the radial wave functions. These functions contain the principle quantum number, which corresponds to the row number in the periodic table. Multiplying these radial solutions by the spherical harmonics from Chapter 14 produces the full three-dimensional solutions for the hydrogen atom, also known as the atomic orbitals. The relationship between the quantum numbers n , l , and m arise naturally from the possible combinations of radial solutions with spherical harmonics.