In Quantum Mechanics the one-dimensional Schrödinger equation is a fundamental academic though exciting subject of study for both physicists and amateurs. A solution of this differential equation represents the motion of a non-relativistic particle in a potential energy field V(x). But very few solutions can be derived with a paper and pencil.
Have you ever dreamed of an App which would solve this equation (numerically) for each input of V(x) ? Give you readily energy levels and wave-functions and let you see as an animation how evolves in time a gaussian wave-packet in this interaction field ?
Quantum Wave in a Box does it ! For a range of values of the quantum systems parameters.
Actually the originally continuous X spatial differential problem is discretized over a finite interval (the Box) while time remains a continuous variable. The set of time-dependent ordinary differential equations obtained is then solved using quick diagonalization routines.
You enter V(x) as RPN expression, set values of parameters and will get a solution in many cases within seconds !
- Atomic units used throughout.
- Quantum system (typically an electron) defined by mass, interval [a, b] representing the Box and (real) potential energy V(x).
- Continuous problem discretized over [a, b] and Schrödinger equation represented by a system of N+1 (ordinary differential) equations using a 3, 5 or 7 point stencil; N being the number of x-steps (Maximum value of N depends on device’s RAM).
- Diagonalization of hamiltonian matrix H for eigenvalues and eigenfunctions. When computing eigenvalues only, lowest energy levels of bound states (if any) with up to 10-digit precision.
- Animation shows gaussian wave-packet ψ(x,t) evolving with real-time evaluation of average velocity, kinetic energy and total energy.
- Toggle between clockwise and counter-clockwise evolution of ψ(x,t).
- Watch Real ψ, Imag ψ or probability density |ψ|².
- Change initial gaussian parameters of the wave-packet (position, group velocity, standard deviation), enter any time value, then tap refresh button to observe changes in curves without new diagonalization. This is particularly useful to get a (usually more precise) solution for any time value t when animation is slower in cases of N being large.
- Watch both solution ψ(x,t) and free wave-packet curves evolve together in time and separate when entering non-zero potential energy region.
- Zoom in/out any part of the curves.