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JURNAL PENDIDIKAN FISIKA DAN KEILMUAN (JPFK)

Spectrum of kinetic energy of electrons of Helium ions in functions Radial Schrodinger waves on quantum numbers (n ≤ 3)

Dellya Natasha, Bambang Supriadi, Kevin Cahya Andilla Unwaru, Rery Irmawati, Friscela Yona Nagifea, Naziroh Faiqohtul Herlia, Tiara Angelita

Published Jul 8, 2024• 0 citations
Publisher

Abstract

The Schrodinger equation is an equation that describes the properties of a wave. The Schrodinger equation is solved using a variable separation method that produces a wave function. The wave function using the variable separation method is divided into 2 parts, namely the radial part and the angular part. The energy spectrum states the energy levels of electrons. Energy spectrum and wave function are related to each other. The energy level in the Schrodinger wave function in the form of kinetic energy is the energy for electrons to move from one point to another. The research aims to describe the kinetic energy spectrum of electrons in the radial wave function. The type of research used is basic research, namely the development of existing theories. The result obtained is a radial wave function with a certain quantum number n and an integrated Laplace constant that can produce electron kinetic energy, namely by integrating the wave function with each quantum number. The kinetic energy has its own value according to the quantum number. Kinetic energy in quantum numbers (n = 1,l = 0, Ek =171.18 eV ; n =2,l = 0, Ek =399.09 eV n =2,l = 1, Ek =152.16 eV n =3,l = 0, Ek =5096.81 eV n =3,l = 1, Ek =81142.43 eV n =3,l = 2, Ek =21.979 eV). The conclusion of this research is that the kinetic energy spectrum of electrons produces a directly proportional relationship to the quantum number n, where the greater the quantum number n, the greater the spectrum or kinetic energy level of the electron.

DOI
10.25273/jpfk.v10i1.19715

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The Schrodinger equation is an equation that describes the properties of a wave. The Schrodinger equation is solved using a variable separation method that produces a wave function. The wave function using the variable separation method is divided into 2 parts, namely the radial part and the angular part. The energy spectrum states the energy levels of electrons. Energy spectrum and wave function are related to each other. The energy level in the Schrodinger wave function in the form of kinetic energy is the energy for electrons to move from one point to another. The research aims to describe the kinetic energy spectrum of electrons in the radial wave function. The type of research used is basic research, namely the development of existing theories. The result obtained is a radial wave function with a certain quantum number n and an integrated Laplace constant that can produce electron kinetic energy, namely by integrating the wave function with each quantum number. The kinetic energy has its own value according to the quantum number. Kinetic energy in quantum numbers (n = 1,l = 0, Ek =171.18 eV ; n =2,l = 0, Ek =399.09 eV n =2,l = 1, Ek =152.16 eV n =3,l = 0, Ek =5096.81 eV n =3,l = 1, Ek =81142.43 eV n =3,l = 2, Ek =21.979 eV). The conclusion of this research is that the kinetic energy spectrum of electrons produces a directly proportional relationship to the quantum number n, where the greater the quantum number n, the greater the spectrum or kinetic energy level of the electron.
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.
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.
The purpose of this study is to find the expected value of the intrinsic kinetic energy of deuterium atoms in radial and total wave functions at quantum numbersn≤3 . The method used in this study is non-experimental research in the field of theoretical-mathematical physics concerning hydrogen atoms. The results of the analysis of the intrinsic kinetic energy expectation value of deuterium atoms calculated through the radial wave functionR_(n,l) (r) show that the energy value decreases significantly as the orbital quantum numberl increases for each fixed value ofn . Meanwhile, the calculation of the intrinsic kinetic energy expectation value of deuterium atoms using the total wave function ψ_(n,l,m) (r,θ,φ) shows that the intrinsic kinetic energy expectation value obtained in the1s state is〈K〉=ℏ^2/(2m〖a_0〗^2 ) , in the2s state, the〈K〉=ℏ^2/(8m〖a_0〗^2 ) value, the 2p state value〈K〉=ℏ^2/(24m〖a_0〗^2 ) , the 3s state value〈K〉=ℏ^2/(18m〖a_0〗^2 ) , the 3p state value 〈K〉=〖5ℏ〗^2/(162m〖a_0〗^2 ) , and the 3d state value 〈K〉=ℏ^2/(90m〖a_0〗^2 ) have the same results as the calculations using the radial wave function. This indicates that the expectation value originates from the radial component, while the contributions from the polar angle(θ) and azimuth(φ) show small variations in the total kinetic energy expectation. Thus, the intrinsic energy expectation of the electron can be obtained through the calculation of the radial wave functionR_(n,l) (r) or the total wave functionψ_(n,l,m) (r,θ,φ).
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.
During the solution of the Schrodinger equation for the hydrogen atom by separation of variables, we come across three quantum numbers, namely the azimuthal quantum number l , the magnetic quantum number m l , and the principal quantum number n . The quantum number l arises as a separation constant during the breaking apart of radial and angular segments of the Schrodinger equation and the quantum number m l occurs while separating the azimuthal component from the polar component of the angular part. The quantum number n is necessary to render the series for the non-asymptotic part of the radial wave function convergent. The three quantum numbers are interdependent because after selecting n , the selection of l is restricted to allowed values and similarly after choosing l , limited values are permitted for m l . The fourth quantum number m s representing electron spin is not a direct consequence of solving the Schrodinger equation. The detailed solution procedure of the Schrodinger equation including the normalization of the wave function, restrictions on the values of quantum numbers, and allied analysis are explained.