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 ψ(x)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) ψ(x) in red and the imaginary part in blue

Figure 10.2: Animated solution ψ(x) of the Schrödinger equation for the development of an initial distribution (symmetric Gaussian) in a box. Shown are the real part in red and the imaginary part in blue. The probability density consists of the sum of squares of these two parts: ψ(x)2 = Reψ(x)2 + Imψ(x)2

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 ψ (square root of the probability density) is shown as envelope. Inside of this the phase angle α = arctan Imψ Reψ is indicated via colour shading.

The phase angle α is indicated via the following colours:

Figure 10.3: The function |ψ(x)| for Fig.10.2. The colour shading indicates the ratio of imaginary to real part. ψ(x) = probabilitydensity. In the blue inner region the real part dominates, while the imaginary part dominates in the red regions

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.