Quantum States in Hilbert Space

Abstract

In analogy to 3-dimensional vectors in Euclidean space quantum states are defined by high dimensional vectors in Hilbert space. Eigenvectors of an operator are introduced as measurement results (quantum states) of the corresponding observable. Hereby the eigenvalues are the possible measured numbers of the out-come of the experiment. Beside the discussion of various representations of quantum states, the Dirac notation of states and basics of commutator algebra are presented. As an example for application the oscillator model and its analytical solution by commutator algebra is described.