Comparison of the Expectation Value of Electron Momentum

Study Case Comparison of the Expectation Value of Electron Momentum in Tritium Atoms with Variations in Principal Quantum Number and Momentum Interval With Principal Quantum Number n ≤ 3 Michael Kurniawan (2018123456) University of Cincinnati, 2600 Clifton Ave, Cincinnati, OH 45221, United States kurniaml@mail.uc.edu Abstract This study aims to analyze the expectation value of electron momentum in tritium atoms using a numerical approach. Through computational simulations, the expectation value of momentum was obtained with variations in principal quantum number and different momentum intervals. The calculation results show that, for the same momentum interval limit, the expectation value of electron momentum tends to decrease as the principal quantum number (n) increases. This phenomenon indicates that the higher the energy level of electrons in tritium atoms, the smaller the momentum expectation value. Convercelly, when the momentum interval increases, the momentum expectation value also increases. By comparing each n value at the same interval, it can be seen that the larger the principal quantum number, the smaller the momentum expectation value. These results provide a deeper understanding of the relationship between the principal quantum number, momentum interval, and momentum expectation value in the tritium atomic system. Keywords: Expectation Value; Momentum; Tritium; Quantum INTRODUCTION In the world of physics, classical mechanics and quantum mechanics provide two different perspectives on particle behavior. Classical mechanics allows for deterministic prediction of particle trajectories, while quantum mechanics emphasizes the probabilistic nature of particle motion, which requires in-depth mathematical and MICHAEL KURNIAWAN 1 Study Case physical interpretation. The development of modern quantum theory has enabled scientists to comprehensively explain the relationship between wave and particle properties [1]. Quantum mechanical theory plays an important role in understanding atomic and subatomic phenomena. The simplest atomic system that is often used as the basis for learning this theory is the hydrogen atom. Because it has only one electron, the hydrogen atom is an ideal model for testing Schrödinger's equation analytically [2]. The two main isotopes of hydrogen, deuterium and tritium, have quantum characteristics similar to those of ordinary hydrogen. Therefore, the equations and principles of physics that apply to hydrogen can also be applied to these two isotopes [3]. Deuterium (²H or D) and tritium (³H or T) are isotopes of hydrogen that have one electron but differ in the number of neutrons. Deuterium contains one neutron, while tritium has two. In the context of nuclear energy, deuterium and tritium play an important role as the main fuels in thermal nuclear fusion reactions. Deuterium can be obtained from seawater, while tritium is usually produced through reactions between neutrons and lithium in nuclear reactors [4]. Tritium itcellf is a radioactive isotope of hydrogen with a half-life of approximately 12.3 years and a maximum beta particle energy of 18.6 keV [5]. In nature, tritium is formed in very small amounts due to the interaction of cosmic rays with atmospheric nitrogen. Artificially, tritium can be produced through the irradiation of lithium in a nuclear reactor. In technology, tritium is used as nuclear fusion fuel, a component in military watches, an energy source for betavoltaic batteries (Nano Tritium Battery), and radiometric tracing for groundwater studies [6]. Research developments in nuclear and quantum physics in the United States continue to progress rapidly, in line with increasing energy needs and radionuclide exploration. One important aspect of quantum studies is the analysis of atomic systems in momentum space (p-space) in addition to position space (r-space). Through Fourier transformation, wave functions in position space can be converted into functions in momentum space [7]. The wave function of a tritium atom in momentum space can be derived from the wave function in position space with a one-dimensional Fourier transformation as follows: 𝜓(𝑝) = MICHAEL KURNIAWAN 1 √2𝜋ℏ ∫ 𝜓(𝑥)𝑒 −𝑖𝑝𝑥/ℏ 𝑑𝑥 2 Study Case 𝜓(𝑥) = 1 √2𝜋ℏ ∫ 𝜓(𝑝)𝑒 𝑖𝑝𝑥/ℏ 𝑑𝑝 If the wave function in position space ψ(r) is expressed as: 𝜓𝑛,𝑙,𝑚 (𝑟) = √ 1 −𝑟/𝑎 0 (𝑟/𝑎 )𝑙 𝐿2𝑙+1 (2𝑟/𝑎 ) 0 0 𝑛−𝑙−1 3𝑒 𝜋𝑎0 then the wave function in momentum space is obtained through Fourier transformation: 𝜙(𝑝) = √ 8𝑎05 𝑝𝑙 𝑎02 𝑝2 − 1 𝑙+1 𝐶 ( ) 𝜋 (1 + 𝑎02 𝑝2 )𝑙+2 𝑛−𝑙−1 𝑎02 𝑝2 + 1 where a₀ is the Bohr radius, and C is the Gegenbauer polynomial [8]. In momentum space, tritium atoms with principal quantum numbers n ≤ 3 have 14 different wave functions, consisting of radial and angular functions. These functions represent the probabilistic behavior of electrons in atomic systems, although they do not directly indicate observable physical meaning. To obtain more physically meaningful values, a mathematical approach is used through the calculation of expectation values, which represent the average results of repeated momentum measurements [9]. The expectation value of the electron momentum ⟨p⟩ in a tritium atom can be calculated using the following formulation: ⟨𝑝⟩ = ∫ 𝜓 ∗ (𝑝) 𝑝 𝜓(𝑝) 𝑑𝑝 This formula shows that the expectation value of momentum depends on the principal quantum number (n) and the angular momentum quantum number (l). Thus, this study focuses on examining the behavior of the expected value of electron momentum in tritium atoms for n ≤ 3 using the momentum space approach. RESEARCH METHOD This study uses a non-experimental or theoretical approach, which relies on literature studies as the main source of information and data. The purpose of this method is to develop mathematical models and quantum physics theories that have been proposed by previous researchers. In this study, the independent variable is the principal quantum number (n ≤ 3), while the dependent variable is the expectation value of electron momentum in tritium atoms in momentum space. The research stages were carried out through several systematic steps as follows: MICHAEL KURNIAWAN 3 Study Case 1. Literature Study At this stage, references were collected from various international scientific journals, quantum mechanics textbooks, and recent research reports relevant to the expected value of electron momentum in tritium atoms. The main references were used to strengthen the theoretical basis and support the development of mathematical models in this study. 2. Model Validation and Theoretical Simulation Validation was carried out by comparing the numerical calculation results with the theory and data from previous research. The validation study refers to the results published by Pratikha et al. [1], Damayanti et al. [2], Utami et al. [3], and Supriadi et al. [4]. This comparison aims to ensure that the mathematical model used is consistent and in accordance with widely accepted quantum theory. 3. Mathematical Model Development and Data Collection At this stage, data was obtained from numerical calculations based on the developed mathematical model. The model was constructed to determine the expected value of the momentum of tritium atoms with various variations in principal quantum numbers (n) and momentum ranges (p). This process was carried out using a scientific computing-based numerical simulation approach. 4. Data Analysis and Physical Interpretation After the momentum expectation values are obtained for each variation of the principal quantum number, an analysis is performed on the relationship between n and the momentum expectation values. The analysis includes interpreting the calculation results in the context of quantum physics, comparing the numerical trends with the applicable theory. The results of this analysis provide an overview of the behavior of electrons in tritium atoms in momentum space, as well as how an increase in the principal quantum number affects the average momentum value. MICHAEL KURNIAWAN 4 Study Case Figure 1. Research flowchart for calculating the expected momentum value of electrons in tritium atoms in momentum space. RESULTS AND DISCUSSION This study produced the expected momentum value of electrons in tritium atoms calculated through a numerical approach based on wave functions in momentum space. In quantum mechanics, the concept of expected value is clocelly related to probability distribution. Probability distribution describes the possibility of a random variable (in this case, electron momentum) being within a certain range of values. Expectation value, on the other hand, represents the average value of momentum measurements if the experiment is repeated under identical conditions. The expectation value of electron momentum in tritium atoms is calculated using the following mathematical relationship: ∞ ⟨𝑝⟩ = ∫ 𝜓 ∗ (𝑝) 𝑝 𝜓(𝑝) 𝑑𝑝 0 The equation shows that the expected value of electron momentum depends on the principal quantum number (n) and the angular momentum quantum number (l). In general, an increase in n causes a decrease in the expected momentum value, while an increase in l produces the opposite effect, namely a decrease in the amplitude of the expected value because the momentum distribution becomes more dispersed. Meanwhile, the magnetic quantum number (m) does not directly affect the magnitude of momentum, but rather determines the orientation of the electron's momentum around the atomic nucleus. MICHAEL KURNIAWAN 5 Study Case This finding is consistent with the principle of quantum mechanics which states that position and momentum cannot be known simultaneously with absolute precision (Heisenberg Uncertainty Principle). Therefore, the position of electrons in tritium atoms can only be explained through probability distribution functions, not in the form of definite trajectories as in the classical Bohr model. Areas with high probability density indicate the most likely locations for the existence of electrons, even though the boundaries of these areas are blurred and experimentally uncertain. Table 1. Calculated Values of Electron Momentum Expectation in Tritium Atoms Based on Table 1, it can be seen that at the same momentum interval, the electron momentum expectation value decreases significantly as the principal quantum number (n) increases. For example, for Δp = 0.10–0.30 a.u., the value of ⟨p⟩ decreases from 0.184981 (n = 1) to 0.128945 (n = 3). This shows that the higher the electron energy level, the smaller the average momentum value. Furthermore, analysis of the variation in momentum interval (Δp) shows that an increase in the momentum range results in an increase in the expectation value. In other words, the larger the momentum interval used in the calculation, the greater the average momentum expectation value of the electron. This phenomenon is consistent with the characteristics of wave functions in momentum space, where an expansion of the momentum domain causes the probability distribution to increase at high momentum values. In the context of the quantum-modified Bohr atomic model, momentum observations are limited to certain energy levels (n ≤ 3) because electrons are most likely to be found in the vicinity of the atomic nucleus, where the probability of their MICHAEL KURNIAWAN 6 Study Case presence is highest. This limitation also reflects the experimental conditions in similar hydrogenic atomic systems. Quantum mechanically, the behavior of electrons in tritium atoms is influenced by four quantum numbers: • Principal quantum number (n): determines the energy level of the electron, n = 1, 2, 3. • Angular momentum quantum number (l): describes the shape of the electron orbital, where l = 0, 1, ..., n−1. • Magnetic quantum number (m): indicates the orientation of the orbital relative to an external magnetic field. • Spin quantum number (s): indicates the direction of the electron spin (+½ or −½). The numerical simulation results show that the expected value of the electron momentum is mainly influenced by n and l. The larger the value of n, the weaker the electron's binding to the nucleus, so that the momentum distribution shifts to smaller values. Convercelly, an increase in l changes the shape of the orbital to a more complex one and causes a decrease in the value of ⟨p⟩ due to a wider distribution of momentum probability. CONCLUSION Based on the simulation results and analysis, it can be concluded that the electron momentum expectation value in a tritium atom strongly depends on the principal quantum number (n) and the momentum interval (Δp). The higher the value of n, the smaller the electron momentum expectation value, which means that electrons with higher energy have a more dispersed momentum distribution. Convercelly, an increase in the momentum interval results in a larger expectation value, indicating that the expansion of the momentum domain increases the likelihood of electrons being in a high momentum state. Thus, this study reinforces the understanding that the relationship between the principal quantum number, momentum interval, and momentum expectation value can be used to explain the behavior of electrons in the tritium atomic system, while also enriching theoretical studies in modern quantum mechanics. MICHAEL KURNIAWAN 7 Study Case REFERENCE MICHAEL KURNIAWAN 8