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Jurnal Fisika

Expectation of Intrinsic Kinetic Energy of Deuterium Atom Electrons at Principal Quantum Number n≤3

Windi Mu'afi, Bambang Supriadi, Firdha Kusuma Ayu Anggraeni, Nilai Mutia Dewi

Published Oct 11, 2026• 0 citations

Abstract

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).

DOI: 10.15294/jf.v16i2.58303

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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.

The intrinsic and dynamic kinetic energies and the potential energies of electron states in the hydrogen atom were determined using the operator formalism in Schrödinger’s nonrelativistic equation. Intrinsic energies were determined using the momentum operator, while for ℓ ≠ 0, the additional dynamic energies of the spinning fields were determined using the angular momentum operator. All 10 states up to the principal quantum number n = 3 and all 4 m states of n = 7, l = 3 were analyzed. The two forms of kinetic energy can only be explained with an electron field representation. All total kinetic and potential energies conformed with the well-known 1/n2 rule. Angular momentum analysis of the 2P1/2 state provided a field spinning rate; in addition, the dynamic kinetic energy of the spinning field determined by both operator analysis and explicit calculation based on the spinning rate gave the same energy results.

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 wave functions in momentum space for discrete states of hydrogenlike atoms are discussed in terms of both the early contributions to the subject and their present usefulness in atomic physics. The derivation of some of their important properties is outlined, in particular the expectation values of various powers of momentum and kinetic energy, and the kinetic energy distribution. As an application, the cross section for ionization of hydrogen in the metastable 2S state by electron impact is calculated with the aid of the binary-encounter approximation, which requires the use of momentum-space wave functions.

Information on electron behavior in a hydrogen atom, such as electron expectation, electron probability, electron position, electron energy spectrum, etc. can be obtained through Schrodinger equation.ions are atoms that lose so that they become hydrogenic atoms. This research aims to determine the expectation value of electron momentum of ion in momentum space at the main quantum number using Schrodinger equation approach. The method used in this research is literature study with non-experimental research type. The results obtained expectation value depends on the interval and the value of the quantum number. The larger the interval, the expectation value of electron momentum increases. While the greater the value of quantum numbers, the expectation value of electron momentum is getting smaller. Based on the analysis, the expectation value of electron momentum is quite varied from to . From this research, it is found that the expectation value of electron momentum is the same as the expectation value of electron position.