There is also a notion of quantum integrable systems. In the quantum setting, functions on phase space must be replaced by self-adjoint operators on a Hilbert space, and the notion of Poisson commuting functions replaced by commuting operators.
Since there is no clear definition of independence of operators, except for special classes, the definition of integrable system, in the quantum sense, is not yet agreed upon. The working definition that is mostly used is that there is a maximal set of commuting operators, including the Hamiltonian, and a semiclassical limit in which these operators have symbols that are independent Poisson commuting functions on the phase space.
Quantum integrable systems can be explicitly solved by Bethe Ansatz or Quantum inverse scattering method. Examples are Lieb-Liniger Model, Hubbard model and Heisenberg model (quantum).
Read more about this topic: Integrable System
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