This research aims to develop computational visualization of the wave function and probability distribution of particles in a one dimensional infinite potential well. The wave function is obtained analytically from the time-independent Schrödinger equation and then normalized to give it probabilistic meaning. Numerical simulations are carried out using Python with the NumPy library for computation and Matplotlib for visualization, allowing manipulation of parameters such as box length and quantum number. The visualization results confirm the basic concepts of quantum mechanics, namely quantized energy levels, sinusoidal wave function forms with a specific number of nodes, and probability distributions that have several peaks and are always positive. The simulation also shows the correspondence between wave-function nodes and zero-probability points, as well as the increasing complexity of oscillations and variations in probability distributions as the quantum number increases. These findings are consistent with theoretical predictions and show that the computational approach is effective in visualizing abstract concepts in quantum mechanics, thereby clarifying the relationship between mathematical formulation and its physical interpretation.
This research aims to develop computational visualization of the wave function and probability distribution of particles in a one dimensional infinite potential well. The wave function is obtained analytically from the time-independent Schrödinger equation and then normalized to give it probabilistic meaning. Numerical simulations are carried out using Python with the NumPy library for computation and Matplotlib for visualization, allowing manipulation of parameters such as box length and quantum number. The visualization results confirm the basic concepts of quantum mechanics, namely quantized energy levels, sinusoidal wave function forms with a specific number of nodes, and probability distributions that have several peaks and are always positive. The simulation also shows the correspondence between wave-function nodes and zero-probability points, as well as the increasing complexity of oscillations and variations in probability distributions as the quantum number increases. These findings are consistent with theoretical predictions and show that the computational approach is effective in visualizing abstract concepts in quantum mechanics, thereby clarifying the relationship between mathematical formulation and its physical interpretation.
Harmonic oscillator is a part of quantum mechanics which is commonly used as an approximation model to solve cases in physics that are considered complicated. The harmonic oscillator has a function as an energy source, namely as a description of anatomic particles. This research focuses on solving the Schrödinger equation for particles under the influence of a harmonic oscillator potential. This study aims to visualize the wave function and particle probability under the influence of a harmonic oscillator potential. The wave function is obtained by solving the Schrödinger equation. The Schrödinger equation can be solved using a second order differential equation that satisfies the solution of the hermite polynomial. The nature of the first hermite polynomial is the relationship between polynomials that have adjacent orders, the second is the derivative of hermite polynomials and the third is the orthogonality of hermite polynomials. The solution to the Schrödinger equation is in the form of a wave function which is then developed to determine the probability of particles in a harmonic oscillator which is then visualized using Matlab programming. The wave function and particle probability visualization is obtained by plotting the wave function and probability using Matlab programming. Osilator harmonik menjadi salah satu bagian dari mekanika kuantum yang biasa digunakan sebagai model pendekatan untuk menyelesaikan kasus-kasus dalam fisika yang dianggap rumit. Osilator harmonik memiliki fungsi dalam sumber energi yaitu sebagai pendeskripsi partikel anatomik. Penelitian ini memfokuskan bahasan mengenai penyelesaian persamaan Schrödinger untuk partikel dibawah pengaruh potensial osilator harmonik. Penelitian ini bertujuan memvisualisasikan fungsi gelombang dan probabilitas partikel dalam pengaruh potensial osilator harmonik. Fungsi gelombang diperoleh dengan penyelesaian persamaan Schrödinger. Persamaan Schrödinger tersebut dapat diselesaikan menggunakan penyelesaian persamaan diferensial orde 2 yang memenuhi solusi polinomial hermitte. Sifat dari polinom hermitte yang pertama adalah hubungan antar polinom yang memiliki orde berdekatan, kedua adalah derivatif polinom hermitte dan ketiga adalah keortogonalan polinom hermitte. Penyelesaian persamaan Schrödinger tersebut berupa fungsi gelombang yang kemudian dikembangkan untuk mengetahui probabilitas pada partikel dalam osilator harmonik yang kemudian dilakukan visualisasi menggunakan pemrograman Matlab. Visualisasi fungsi gelombang dan probabilitas partikel diperoleh dengan melakukan ploting terhadap fungsi gelombang dan probabilitas menggunakan pemrograman Matlab.
The wave function is a fundamental concept in quantum mechanics, but its abstract nature makes it difficult for students to understand its behavior. Furthermore, it is the probabilistic nature of particles that makes direct observation of quantum wave dynamics impossible. Therefore, this research investigates the eigenstates, superposition, and the time-dependent probability density of a particle within an infinite potential well through theoretical analysis and computational visualization based on Streamlit. The research method combines analysis with mathematical calculations to obtain the eigenstates, calculate |Ψ|², the expected position value, and the phase evolution of the superposition state. The visualizations illustrate the relationship between the wave function and quantum numbers, as well as interference patterns and time-dependent behavior. The visualization results show how the method represents theoretical concepts related to the probabilistic nature of particles. This research shows that theoretical analysis combined with interactive visualization provides an interactive representation that may facilitate students’ understanding of quantum mechanics concepts, a potential that still requires empirical validation through classroom-based studies. Unlike existing closed-source quantum applets, the Quantum Probability Explorer is fully open-source and directly couples the analytical solution of the Schrödinger equation with real-time interactive visualization of eigenstates and superposition, which constitutes the novelty of this study.
This study presents a computational visualization of the stationary states of a particle confined in a one-dimensional infinite potential well, formulated entirely from analytical solutions and implemented in a cloud-based Python environment. Using Google Colab, wave functions, probability densities, and nodal structures were generated for several quantum numbers to illustrate fundamental quantum-mechanical characteristics, including energy quantization, spatial oscillatory behavior, and the emergence of classical correspondence at higher energy levels. The resulting visualizations offer a clear representation of the spatial distribution of quantum states, thereby supporting a conceptual understanding of the underlying analytical model. As a novelty, this study introduces a Colab-based analytical–numerical combination as a visualization medium, a format rarely explored in depth in quantum-well learning contexts. Beyond instructional applications, the modeled system accurately reflects the essential physical behavior of electrons in semiconductor quantum wells, where quantized subband energies and wave-function profiles significantly influence optical and electronic transitions. These findings demonstrate that analytically derived quantum models, when integrated with computational visualization, can effectively enhance conceptual comprehension while offering insight into the fundamental principles governing modern optoelectronic structures.
This paper discussed the digitization of some analytical solutions of the fundamental equation of non-relativistic quantum mechanics, the Schrödinger equation, in the Python programming environment. The main objective of the work is to automate the solution of several known analytically solvable problems of quantum mechanics and ensure their use in more complex problems. For this purpose, the Python programming environment was chosen, which, due to the large number of libraries and flexibility, is currently widely used in solving physical and mathematical problems. In the program we propose, which is available on the GitHub platform, it is solved easily. The problem of determining the probability of the spatial distribution of an electron in a hydrogen-like atom is discussed. The Schrödinger equation is presented in the spherical coordinate system. The wave function, describing the electron's state, quantized energy values, and probability density, is derived analytically. A recursive function was written in Python to handle the series coefficients defining the wave function. Utilizing relevant libraries, the resulting program constructs electron spatial distribution functions for different excitation levels. The paper then examines two one-dimensional tunneling scenarios and provides the tunneling coefficients in analytical form. A Python program was developed to plot the dependence of these coefficients on the particle's coordinate and energy. All the code is available on GitHub.
The Schrödinger equation governs the probabilistic behavior of quantum particles through the wave function. Microlocal singularities denote regions with significantly high probability density or abrupt changes therein. By visualizing the probability distribution in time and space, we discern regions with higher probability density, indicative of potential microlocal singularities. These regions probably correspond to areas with a greater probability of particle presence. Such analysis aligns with Theorem 1, predicting characteristics of microlocal singularities of wave functions. Furthermore, Theorem 2 postulates that semiclassical path integrals along these singularities contribute significantly to solving the Schrödinger equation. Interpreting the temporal evolution of the probability density in the probability distribution visualization reveals the propagation of the particle over time. Regions of high density mean likely presence of particles at specific times, aligning with the predictions of Theorem 2. Consequently, the analysis of the contribution of high-density regions to the temporal evolution of the wave function resembles semi-classical path integral calculations. Thus, our findings demonstrate that visualization of probability distributions obtained from the numerical resolution of the Schrödinger equation allows a comprehensive interpretation of the behavior of quantum particles, consistent with the theorems.
This study offers a comprehensive exploration of quantum mechanics by integrating theoretical analysis with simulation-based experimentation to investigate fundamental principles such as energy quantization, operator expectations, and wavefunction behavior. Utilizing a mixed-methods approach, the research employed numerical solutions to the time-independent Schrödinger equation and qualitative analysis of foundational quantum postulates to model and interpret the behavior of discrete quantum states. Nine detailed datasets were generated, each containing over 20 quantum states, capturing energy levels, expectation values of position and momentum, and probability density maxima. The analysis revealed consistent non-linear spacing in energy levels, symmetrical behavior in position and momentum distributions, and the emergence of wavefunction localization in high-probability states. Twelve complex visualizations—including line, bar, pie, scatter, and hybrid plots—further illustrated the dynamic relationships among observables. These results confirm core theoretical constructs such as the Heisenberg uncertainty principle and the correspondence between potential confinement and probability density peaks. Violin plots and scatter matrices identified subtle structural symmetries and degeneracies in state distributions, while hybrid line-bar visualizations highlighted state-dependent transitions in expectation values. Additionally, the study demonstrated the scalability and accuracy of simulation frameworks in modeling quantum systems and emphasized their pedagogical value in visualizing abstract quantum behaviors. Collectively, the results substantiate quantum mechanical theory while offering data-driven insights into operator correlations and system evolution. This research not only reinforces classical interpretations but also paves the way for further exploration in quantum computing, photonic systems, and educational simulations.