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JURNAL PENDIDIKAN FISIKA DAN KEILMUAN (JPFK)

Radial Wave Function of Helium Ion in Momentum Space at N ≤ 4

Zhahrotun Nurroniah, Bambang Supriadi, Vandaria Dewi Cahyani, Nanda Salsabila Ayu, Kamilatul Imaniyah

Published Jul 8, 2023• 3 citations
Publisher
DOI
10.25273/jpfk.v9i1.16523

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Ions resulting from the ionization of helium atoms are known as helium ions. Atom of Helium The probability distribution of the particle locations and velocities inside a helium atom is referred to as probability. Using theoretical study methods, the goal of this research is to find the probability of an electron's position in a helium atom at the quantum number 3 ≤ n ≤ 4. This type of research is non-experimental research by developing previously existing theories. This type of research is non-experimental research by transforming the radial wave function of hydrogenic atoms in position space into a radial wave function in momentum space using the Fourier transformation, then including a number of Gegenbauer functions and using the probability equation of finding an electron in a Helium ion in momentum space. The results of this research provide information regarding the position and probability of the existence of electrons in the helium atom. The probability value for the Helium ion is obtained using the equation P(p) = which is used to indicate the probability of finding an electron in a helium atom is directly proportional to the principal quantum number (n) and the value of the electron's momentum. The larger the electron momentum interval, the greater the probability.

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In this paper, an analytical model of 2p orbital helium atom has been considered to quantify the values of the Rényi and Tsallis entropies along with their theoretical aspects. The normalized radial wave function used here (in atomic units) is obtained by solving the Schrödinger equation. The complete form of the coordinate space wave function is obtained by the use of the spherical harmonics. The momentum space wave function is obtained by taking the Fourier transform of the coordinate space wave function. The probability densities constituted with the respective coordinate and momentum space wave functions have been used to compute the numerical values of the Rényi and Tsallis entropies in the coordinate and momentum space for different values of the order β varying from 2 to 10. The computed values are presented in a tabular form. Further, it is mathematically demonstrated that in the limit of order β → 1, both the Rényi and Tsallis entropies lead to the Shannon entropy. Finally, an outlook of the present work has been summarized with some concluding remarks.

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Ions resulting from the ionization of helium atoms are known as helium ions. Atom of Helium The probability distribution of the particle locations and velocities inside a helium atom is referred to as probability. Using theoretical study methods, the goal of this research is to find the probability of an electron's position in a helium atom at the quantum number 3 ≤ n ≤ 4. This type of research is non-experimental research by developing previously existing theories. This type of research is non-experimental research by transforming the radial wave function of hydrogenic atoms in position space into a radial wave function in momentum space using the Fourier transformation, then including a number of Gegenbauer functions and using the probability equation of finding an electron in a Helium ion in momentum space. The results of this research provide information regarding the position and probability of the existence of electrons in the helium atom. The probability value for the Helium ion is obtained using the equation P(p) = which is used to indicate the probability of finding an electron in a helium atom is directly proportional to the principal quantum number (n) and the value of the electron's momentum. The larger the electron momentum interval, the greater the probability.
Lithium ion (3Li2+) is a light alkali metal which is reactive and flammable when exposed to water or air. Lithium becomes ionic when lithium metal loses two of its three valence electrons. The wave function is a complex quantity that shows the characteristics of the de Broglie wave. The wave function in question is the radial wave function in the particle, because to find the probability that the electron is in the particle, only the radial part of the wave function is needed. However, there is no literature that discusses the wave function in the momentum space in the case of hydrogenic ions. Therefore, this research will examine the wave function of Lithium ion (3Li2+) in the momentum space. This type of research in the preparation of this article uses non-experimental research. Based on the results of the research that has been done, the wave function of Lithium ion (3 Li2+) in the momentum space at n 3 is obtained. The wave function consists of a radial, angular (polar and azimuth) wave function, as well as the wave function of Lithium ion (3Li2+) as a whole. The fundamental difference between the wave function of Lithium ion (3Li2+) in the momentum space and the wave function of hydrogenic atoms in the other momentum space is the value of the radial function caused by the difference in the value of (z) or atomic number.
In this paper, an analytical model of 2p orbital helium atom has been considered to quantify the values of the Rényi and Tsallis entropies along with their theoretical aspects. The normalized radial wave function used here (in atomic units) is obtained by solving the Schrödinger equation. The complete form of the coordinate space wave function is obtained by the use of the spherical harmonics. The momentum space wave function is obtained by taking the Fourier transform of the coordinate space wave function. The probability densities constituted with the respective coordinate and momentum space wave functions have been used to compute the numerical values of the Rényi and Tsallis entropies in the coordinate and momentum space for different values of the order β varying from 2 to 10. The computed values are presented in a tabular form. Further, it is mathematically demonstrated that in the limit of order β → 1, both the Rényi and Tsallis entropies lead to the Shannon entropy. Finally, an outlook of the present work has been summarized with some concluding remarks.