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UMSimulations of quantum wave packet evolution are among the most requested topics by followers and students interested in quantum mechanics. These visualizations help bridge the gap between abstract mathematical formalism and physical intuition, making quantum phenomena such as tunneling, reflection, and transmission easier to understand. In this simulation, we study the time evolution of a Gaussian wave packet moving toward a potential barrier. As the wave packet approaches the barrier, part of the probability amplitude is reflected, while another portion may tunnel through the barrier — a purely quantum effect with no classical counterpart. This phenomenon illustrates one of the most fundamental principles of quantum mechanics: particles behave as waves and their dynamics are governed by probability amplitudes. The numerical method used here is based on direct time evolution via the Hamiltonian operator. First, we construct the discrete quantum Hamiltonian on a spatial lattice, including both kinetic and potential energy terms. Then, the time evolution operator is computed using the unitary propagator
$U = \exp(-i H dt)$ which is derived from the time-dependent Schrödinger equation. This operator is applied iteratively to the wave function, allowing us to track the quantum state as it evolves in time. This approach preserves unitarity and probability normalization, making it particularly robust for educational and visualization purposes. The simulation displays both the real part of the wave function and the probability density, along with dynamically calculated reflection and transmission coefficients. Beyond being visually engaging, this type of simulation provides deep insight into fundamental quantum behavior and serves as an excellent tool for teaching concepts such as wave-particle duality, tunneling, dispersion, and quantum interference. These simulations demonstrate how computational physics can transform abstract equations into dynamic and intuitive physical understanding.
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