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.
This app has been updated by Apple to display the Apple Watch app icon.
- Language: english (previously appeared mistakenly as french).
- Fixes an issue encountered when trying to access Photos.
- Optimisation of successive graphic actions for repeated taps on HOME button.
Ratings and Reviews
Used to work great. Now crashes when loading and dumps back to home screen in one second.
Developer Response ,
Thanks for this feedback. When you update your device with a new release of iOS, the App may stop working. The code needs to be recompiled. This is what Apple does actually. I did download the App today and it works nice with the latest iOS 11.4. Even if it is still version 1.0.2 of Quantum Wave in a Box. I suggest you delete the App from your device then download it anew. Let me know if it solves the crashing problem and deserve maybe some additional stars.
The developer, Michel Ramillon, has not provided details about its privacy practices and handling of data to Apple.
No Details Provided
The developer will be required to provide privacy details when they submit their next app update.
- Michel Ramillon
- 14.8 MB
- Requires iOS 8.1 or later.
- Requires iPadOS 8.1 or later.
- iPod touch
- Requires iOS 8.1 or later.
- Requires macOS 11.0 or later and a Mac with Apple M1 chip or later.
- Age Rating
- © 2015-2017 Michel Ramillon
Up to six family members can use this app with Family Sharing enabled.