### 10.3 Simulation of the Schrödinger equation

The interactive figure 10.2 show the solution of the one-dimensional Schrödinger equation for a particle in an infinitely deep rectangular potential wall, whose width can be adjusted with the slider. The square of the absolute value of the complex wave function $Math content$ gives the probability density for the particle at position $Math content$. It is normalized to $Math content$, which means, that the particle can be found inside of the box with certainty irrespective of the spatial distribution.

The two curves in Fig.10.2 show the real part of the wave function (probability amplitude) $Math content$ in red and the imaginary part in blue

In Fig10.3 a second presentation mode mode that is popular in quantum mechanics is used, for which the absolute value of the wave function $Math content$ (square root of the probability density) is shown as envelope. Inside of this the phase angle $Math content$ is indicated via colour shading.

The phase angle $Math content$ is indicated via the following colours:

• blue $Math content$,
• golden yellow $Math content$,
• rose coloured $Math content$,
• green $Math content$

This simulation allows the choice among many examples of potential wells, in which quantum particles can move. It was developed by the pioneer of the OSP program, Wolfgang Christian and slightly simplified by us. The description pages in Fig.11.2 contain detailed hints about theory and usage.