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The approximate solution to the Schrödinger equation of exotic doubly muonic helium-like systems has been obtained using a simple matrix method based on hydrogenic s-states. Each system considered consists of a positively charged nucleus surrounded by 2 negatively charged muons XZ+μ-μ- (2 Z 36). The present work aims to obtain approximate ground-state energies of the systems and to decompose the energies in terms of the basis states used. Here, the wave function was expressed as a linear combination of 15 eigenfunctions, each written as the product of two hydrogenic s-states. The elements of the Hamiltonian matrix were calculated and finally, the energy eigenvalue equation was numerically solved to obtain the ground-state energies of the systems with their corresponding eigenvectors. From the results, ground-state energies of all systems were in agreement with others from the literature, with percentage differences between 0.03% and 2.05%. The analysis of the probability amplitudes from the eigenvectors showed that the 1s1s state made the largest contribution to the ground state energies of the systems, approaching 90.99%, 96.98% and 98.54% for He2+μ-μ- , Li3+μ-μ- , and Be4+μ-μ- , respectively. Keywords: Doubly muonic helium-like systems, ground state energy, hydrogenic basis states, matrix mechanics, Schrödinger equation

This study aims at implementing a truncated matrix approach based on harmonic oscillator eigenfunctions to calculate energy eigenvalues of anharmonic oscillators containing quadratic, quartic, sextic, octic, and decic anharmonicities. The accuracy of the matrix method is also tested. Using this method, the wave functions of the anharmonic oscillators were written as a linear combination of some finite number of harmonic oscillator basis states. Results showed that calculation with 100 basis states generated accurate energies of oscillators with relatively small coupling constants, with computation time less than 1 minute. Including more basis states could result in more correct digits. For instance, using 300 harmonic oscillator basis states in a simple Mathematica code in about 8 minutes, highly accurate energies of the oscillators were obtained for relatively small coupling constants, with up to 15 correct digits. Reasonable accuracy was also found for much larger coupling constants with at least three correct digits for some low lying energies of the oscillators reported in this study. Some of our results contained more correct digits than other results reported in the literature.

A preliminary study has been carried out on the radial properties of the hydrogenic systems using the Laguerre polynomial. The aims of this study were to investigate radial properties of hydrogenic ions by evaluating the radial probability density, the most probable radial distance and the average distance of electron from the nuclei of hydrogenic ions. Based on the research results for Z=1 to Z=10 with the principal quantum number (n)=1 to 5, 150 radial function forms, 150 radial probability density, 150 most probable radial distance values and 150 electron average distance values were obtained. The most probable radial distance from the nucleus was compared with the Bohr model and showed the same results as the Bohr model.

Pendekatan Born-Oppenheimer diterapkan untuk menghitung tingkat energi vibrasi keadaan dasar molekul hidrogen. Persamaan Schrodinger untuk inti atom diselesaikan dengan menggunakan metode semi-klasik, di mana inti atom diasumsikan bergerak secara klasik dalam sumur potensial dan energi vibrasi ditentukan dengan menerapkan aturan kuantisasi kuantum. Potensial yang digunakan pada penelitian adalah potensial Morse. Dalam penelitian ini, tingkat energi vibrasi dihitung dengan metode numerik, yaitu metode integrasi Simpson dan metode regula falsi. 15 Tingkat energi vibrasi dari molekul H2 diperoleh dan dibandingkan dengan data hasil eksperimen. Perbandingan ini mengindikasikan pendekatan yang digunakan pada penelitian ini memberikan hasil yang sangat akurat pada tingkat energi vibrasi yang relatif rendah (0≤n≤4), dengan kesalahan kurang dari 0,7%, dan untuk 5≤n≤8 dengan kesalahan maksimum 7,3%. Keakuratan menurun ketika tingkat energi vibrasi meningkat. Secara khusus, untuk n = 13 dan n = 14, kesalahan meningkat secara signifikan, menunjukkan gagalnya pendekatan ini untuk tingkat energi vibrasi yang relatif tinggi, khususnya untuk dua tingkat energi ini. Born-Oppenheimer approximation was applied to calculate vibrational energy levels of ground state of Hydrogen molecule. The Schrodinger equation for the nuclei was solved using a semi-classical method, in which the nuclei are assumed to move classically in a potential well and the vibrational energies are determined by applying the quantum mechanical quantization rules. Potential used in this research was the Morse potential. Here, vibrational energy levels of the molecule were calculated using numerical methods, i.e. Simpson integration method and false position method. 15 Vibrational energy levels of the H2 molecule were obtained and compared to the corresponding results from experiments. The comparison indicated that the approximation used in this research yielded very accurate results for relatively low vibrational levels (0≤n≤4), with errors being less than 0.7% and for 5≤n≤8 with maximum of 7.3% errors. The accuracy decreased as the vibrational levels increased, as expected. In particular, for n=13 and n=14, errors significantly increased, indicating the breakdown of the approximation for relatively high vibrational levels, in particular for these two energy levels.

Calculation of He atomic energy levels using the first order perturbation theory taught in the Basic Quantum Mechanics course has led to relatively large errors. To improve its accuracy, several methods have been developed but most of them are too complicated to be understood by undergraduate students. The purposes of this study are to apply a simple matrix method in calculating some of the lowest energy levels of He atom (1s2, triplet 1s2s, and singlet 1s2s states) and to reduce errors obtained from calculations using the standard perturbation theory. The convergence of solutions as a function of the number of bases is also examined. The calculation is done analytically for 3 bases and computationally with the number of bases using MATHEMATICA. First, the 2-electron wave function of the Helium atom is written as the multiplication of two He+ ion wave functions, which are then expanded into finite dimension bases. These bases are used to calculate the elements of the Hamiltonian matrix, which are then substituted back to the energy eigenvalue equation to determine the energy values of the system. Based on the calculation results, the error obtained for the He ground state energy using 3 bases is 2.51 %, smaller than the errors of the standard perturbation theory (5.28 %). Despite the fact that the error is still relatively large from the analytical calculations for singlet-triplet 1s2s energy splitting of He atom, this error is successfully reduced significantly as more bases were used in the numerical calculations. In particular, for n = 25, the current calculation error for all states is much smaller than the errors obtained from calculations using standard perturbation theory. In conclusion, the analytical calculations for the energy eigenvalue equation for the 3 lowest states of the Helium atom using 3 bases have been carried out. It was also found in this study that increasing the number of bases in our numerical calculations has significantly reduced the errors obtained from the analytical calculations.

Sound Level Meter (SLM) is a tool used to measure the noise level for a moment. For improved performance, hence required a measure of noise level capable of displaying result automatically on the computer so that simplify user to observe and measure the noise. In this study, the design system of data acquisition consists of a MAX4466 sound sensor, Arduino UNO microcontroller, and computer to display the measurement result. The measurement results are displayed in the form of data and graphs. The display of software designed using the Delphi 7.0. The process of taking data in the room with a sound intensity of 44.6 dB. The value is the measurement result using the SLM tool, while the measurement results using the sound sensor performed every second during an interval of 30 seconds obtained an average of 44.19 dB. It can be seen that between the two results shows a relatively small difference, so it can be concluded that the design of this system is running well. Keywords: Noise, SLM, Sound sensor, Arduino Uno