**Quantum Wave in a Box** **Version**: 1.0.2**Language**: english**Developer**: Michel Ramillon contact developer*© 2015-2017 Michel Ramillon*

In Quantum Mechanics the **one-dimensional Schrödinger equation** is a fundamental academic and also 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 \(V(x)\). But very few solutions can be derived with just paper and pencil.

**Quantum Wave in a Box**

**Quantum Wave in a Box**

**is a software for ***iPhone* and *iPad*

*iPhone*and

*iPad*

**designed to solve numerically**

**the one-dimensional Schrödinger equation**

**$$\quad i\frac { \partial }{ \partial t } \psi (x,t)\quad =\quad \underbrace { \left( -\frac { 1 }{ 2m } \frac { { \partial }^{ 2 } }{ { \partial x }^{ 2 } } \quad +\quad V(x) \right) }_{ H\quad operator } \quad \psi (x,t)\quad$$**

*$$in\quad atomic\quad units\quad (ħ = 1,\quad m_e=1,\quad e = 1)$$.*

After** input of the potential energy \(V(x)\) as an RPN expression**, the software computes (for a potential with no imaginary part) eigenvalues and eigenfunctions of the **hamiltonian operator H**. A finite elements method is used for the spatial variable* x* over the interval representing 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 using quick diagonalization routines. **The solution \(ψ(x,t)\) of the time-dependent Schrödinger equation** is then computed **in matrix form** as

$$ψ(x,t) = e^{-iHt} ψ_{0}(x)$$

where

$$ψ_{0}(x) = \left(\frac{1}{2 \pi \alpha}\right)^{\frac{1}{4}}e^{\frac{-{\left(x-x_{0}\right)}^{2}}{4 \alpha}}e^{ik_{0}\left(x-x_{0}\right)}$$

is a gaussian wave-packet at initial time \(t = 0\). Parameter \(x_{0}\) is the center of the wave-packet, \(\alpha = \sigma^{2}\) is related to the standard deviation \(\sigma\) and \(k_{0} = mv_{0}\) is the wave-vector.

Evolution in time of a gaussian wave-packet can be watched in animation for a large range of initial parameters.

### Examples of animations running on *iPad* for various potentials V(x)

In the animations below, the **yellow curve** represents the real part of the solution ψ(x,t) of Schrödinger’s equation, the** white curve** represents the density of probability of presence|ψ|² and the **blue curve** the free wave calculated from its exact analytical formula. The** red curve** stands for the potential energy V(x).

**Available for iPad and iPhone on the App Store**

- 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. If 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 displays gaussian wave-packet ψ(x,t) evolving with real-time evaluation of average velocity, kinetic energy and total energy.
- Toggle between clockwise and counterclockwise 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 solution for any time value t when animation is slower in cases where N is 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 of any part of the curves and watch how ψ(x,t) evolves locally.

**Privacy Policy**

We as developer of the *Quantum Wave in a Box* App have absolutely no access to the user’s personal data during the installation or the use of the App. January 8th, 2023.