Quantum Wave in a Box vs Matrix

Schrödinger equation solver 1D. User defined potential V(x). Diagonalization of hamiltonian matrix. Animation showing evolution in time of a gaussian wave-packet. In Quantum Mechanics the one-dimensional Schrödinger equation is a fundamental academic though exciting subject of study for both students and teachers of Physics. 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 particular interaction field ? Quantum Wave in a Box does it ! For a large range of values of the quantum system parameters. Actually the originally continuous x-spatial differential problem is discretized over a finite interval (the Box) while time remains a continuous variable. The time-independent Schrödinger equation H ψ(x) = E ψ(x), represented by a set of linear equations, is solved by using quick diagonalization routines. The solution ψ(x,t) of the time-dependent Schrödinger equation is then computed as ψ(x,t) = exp(-iHt) ψ₀(x) where ψ₀(x) is a gaussian wave-packet at initial time t = 0. 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 (mass of electron = 1) - Quantum system defined by mass, interval [a, b] representing the Box and (real) potential energy V(x). - Spatially continuous problem discretized over [a, b] and time-independent Schrödinger equation represented by a system of N+1 linear 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: up to 4000 when computing eigenvalues and eigenvectors, up to 8000 when computing eigenvalues only. - Diagonalization of hamiltonian matrix H gives eigenvalues and eigenfunctions. When computing eigenvalues only, lowest energy levels of bound states (if any) with up to 10-digit precision. - Listing of energy levels and visualisation of eigenwave-functions. - 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 and out any part of the curves and watch how ψ(x,t) evolve locally.

The MatrixAnalysis is carefully designed to be a simple and elegant app. This app has tools to help you analyze matrices like finding eigenvalues and eigenvectors, diagonalization, QR, SLU, LU factorization. This app can also be used to do matrix operations like, finding inverse, determinant, dot product, cross product, and much more. You can also do elementwise operations and scalar operations. Want to do fast fourier transform (FFT) or inverse fourier transform (IFFT), this app is capable of IFFT Want to find an pseudo inverse of a non-square matrix, this app has you covered Want to know a rotation matrix for a given angle or rotate a vector by the given angle? This app has you covered. The expressions and the solutions are displayed in beautiful mathematical way. Once the expressions are solved, a beautifully written solutions are created that you can share with other or save it locally. It automatically saves your hard work locally on your device so you can come back and pick up from where you left off as your great effort will not be lost. There is also a Study page to help you with learning some important matrix vector formulas and some useful technique. This is an app for all student groups. From high school to college to university students. Be creative. Enjoy the app!