### 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 ${\left|\psi \left(x\right)\right|}^{2}$
gives the probability density for the particle at position
$x$. It is
normalized to $1$,
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) $\psi \left(x\right)$
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
$\left|\psi \right|$
(square root of the probability density) is shown as envelope. Inside of this the phase
angle $\alpha =arctan\left(\frac{Im\psi}{Re\psi}\right)$
is indicated via colour shading.

The phase angle $\alpha $
is indicated via the following colours:

- blue $\alpha =0\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\text{or}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}2\pi \phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\left(\text{thatmeans.}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\psi \phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\text{positive}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\text{real}\right)$,
- golden yellow $\alpha =\pi \phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\left(\psi \phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\text{realnegative}\right)$,
- rose coloured $\alpha =\pi \u22152\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\left(\psi \phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\text{positiveimaginary}\right)$,
- green $\alpha =3\pi \u22152\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\left(\psi \phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\text{negativeimaginary}\right)$

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.