  We give an introduction to some of the numerical aspects in quantum chaos.
The classical dynamics of two--dimensional area--preserving maps on the torus
is illustrated using the standard map and a perturbed cat map. The quantization
of area--preserving maps given by their generating function is discussed and
for the computation of the eigenvalues a computer program in Python is
presented. We illustrate the eigenvalue distribution for two types of perturbed
cat maps, one leading to COE and the other to CUE statistics. For the
eigenfunctions of quantum maps we study the distribution of the eigenvectors
and compare them with the corresponding random matrix distributions. The Husimi
representation allows for a direct comparison of the localization of the
eigenstates in phase space with the corresponding classical structures.
Examples for a perturbed cat map and the standard map with different parameters
are shown. Billiard systems and the corresponding quantum billiards are another
important class of systems (which are also relevant to applications, for
example in mesoscopic physics). We provide a detailed exposition of the
boundary integral method, which is one important method to determine the
eigenvalues and eigenfunctions of the Helmholtz equation. We discuss several
methods to determine the eigenvalues from the Fredholm equation and illustrate
them for the stadium billiard. The occurrence of spurious solutions is
discussed in detail and illustrated for the circular billiard, the stadium
billiard, and the annular sector billiard. We emphasize the role of the normal
derivative function to compute the normalization of eigenfunctions, momentum
representations or autocorrelation functions in a very efficient and direct
way. Some examples for these quantities are given and discussed.
