Quantenintegrabilität

Quantenintegrabilität refers to the quantum mechanical analogue of classical integrability. In classical mechanics, an integrable system is one that possesses a sufficient number of conserved quantities (integrals of motion) to allow for an explicit determination of its time evolution. These conserved quantities often take the form of commuting functions of the system's phase space variables.

In quantum mechanics, the concept is extended to systems where there exists a complete set of commuting, time-independent operators whose eigenvalues determine the system's energy levels and other observable properties. The existence of such a set of operators implies that the quantum system is also exactly solvable, meaning its spectrum and eigenstates can be found analytically.

Key characteristics of quantum integrable systems include the presence of a Yang-Baxter equation, which plays a crucial role in constructing the conserved quantities and understanding the underlying algebraic structure. These systems are often found in one-dimensional models in condensed matter physics and quantum field theory, such as the one-dimensional Bose gas or certain spin chains.

The study of quantum integrability is important for understanding exactly solvable models, which serve as benchmarks for more complex, non-integrable systems. While many physical systems are not integrable, the insights gained from integrable models can inform approximate treatments and numerical methods for their study. Integrability is a powerful concept that simplifies the analysis of complex quantum phenomena.