## Theory and simulation

## Algebraic Formulation Quantum Dynamics

The investigation of single magnetic atoms and magnetic clusters with a spin-polarized scanning tunneling microscope (SP-STM) raises the necessity of a quantum mechanical description of spin systems. The components of a spin, whose expectation values can be measured with an SP-STM, can be represented as elements of a local -algebra .

## Monte Carlo simulation

A Monte Carlo simulation is a stochastic method, which relies on repeated random experiments. This method is mainly used, if it is impossible or infeasible to compute an exact result. The calculated observables are statistical averages, whose variance strongly depends on the number of performed random samplings.

## Spin dynamic simulation

Life means change and change in magnetism means dynamics. However, spin dynamics is a great challenge in physics caused by the underlying nonlinear differential equations. These equations cannot be solved analytical in general.

## Mean field theory (MFT)

The mean field theory or mean field approximation is one of the most famous and on the other hand one of the most underestimated theories in solid science.