Quantum Wave in a Box 1.0.3

The one-dimensional Schrödinger equation solver offers a powerful tool for exploring the complexities of Quantum Mechanics, particularly in an academic setting. It allows both students and educators to engage deeply with the fundamentals of quantum behavior. This numerical solution provides an invaluable resource in deriving motion representations for non-relativistic particles in defined potential energy fields, V(x), a task often impractical through manual calculations.

Users can input their own potential expressions and obtain results almost instantaneously, including energy levels and wave-functions. The accompanying animation visualizes the temporal evolution of a Gaussian wave-packet within the specified interaction field, highlighting the dynamic nature of quantum systems.

Key features of the solver include:

• Utilization of atomic units, normalizing the mass of the electron to 1.
• Definition of quantum systems through mass and finite intervals, [a, b], representing the Box along with the real potential energy V(x).
• Discretization of the continuous spatial problem over [a, b], while preserving time as a continuous variable. The time-independent Schrödinger equation is transformed into a series of linear equations using various point stencils (3, 5, or 7 points). The maximum discretization steps (N) depend on device RAM, accommodating up to 4000 in eigenvalue and eigenvector computations and 8000 for eigenvalues alone.
• Efficient diagonalization of the Hamiltonian matrix H yields eigenvalues and eigenfunctions with remarkable precision, particularly for bound states.
• A comprehensive listing of energy levels alongside visualizations of corresponding eigenwave-functions.
• An engaging animation that depicts the evolving Gaussian wave-packet ψ(x,t), with real-time assessments of average velocity, kinetic energy, and total energy.
• An option to toggle between clockwise and counter-clockwise wave evolution.
• Advanced visualization capabilities allowing observers to monitor Real ψ, Imag ψ, or probability density |ψ|².
• Adjustable initial parameters for the Gaussian wave-packet (position, group velocity, standard deviation), enabling quick refreshes without re-diagonalization to study specific time values with increased precision.
• The ability to observe joint evolution between the solution ψ(x,t) and free wave-packet curves, particularly as they navigate non-zero potential energy regions.
• Zoom functionality for localized examination of curve behavior over time.

This solver serves as an essential platform for both enhancing educational experiences in Quantum Mechanics and facilitating advanced studies through user-defined inputs and visual analytics.