This lecture discusses the Algebraic Bethe Ansatz method for solving integrable models, particularly the XXX spin-1/2 chain. The method is based on the quantum inverse scattering method, which involves the Lax operator and the R-matrix. The key idea is to use the commutation relations of the Lax operator and the R-matrix to derive the Bethe Ansatz equations, which determine the eigenvalues of the Hamiltonian. The Bethe Ansatz equations are algebraic equations that describe the spectrum of the model in the thermodynamic limit. The method is applied to the XXX spin-1/2 chain, which is a quantum spin model with spin-1/2 particles on a lattice. The Bethe Ansatz equations are derived by considering the action of the Lax operator on a reference state and using the commutation relations of the Lax operator and the R-matrix. The equations are then solved to find the eigenvalues of the Hamiltonian, which describe the physical spectrum of the model. The method is extended to other integrable models, including the Sine-Gordon equation and the Nonlinear Schrödinger equation. The Bethe Ansatz method is a powerful tool for solving integrable models and has been widely used in both theoretical and applied physics.This lecture discusses the Algebraic Bethe Ansatz method for solving integrable models, particularly the XXX spin-1/2 chain. The method is based on the quantum inverse scattering method, which involves the Lax operator and the R-matrix. The key idea is to use the commutation relations of the Lax operator and the R-matrix to derive the Bethe Ansatz equations, which determine the eigenvalues of the Hamiltonian. The Bethe Ansatz equations are algebraic equations that describe the spectrum of the model in the thermodynamic limit. The method is applied to the XXX spin-1/2 chain, which is a quantum spin model with spin-1/2 particles on a lattice. The Bethe Ansatz equations are derived by considering the action of the Lax operator on a reference state and using the commutation relations of the Lax operator and the R-matrix. The equations are then solved to find the eigenvalues of the Hamiltonian, which describe the physical spectrum of the model. The method is extended to other integrable models, including the Sine-Gordon equation and the Nonlinear Schrödinger equation. The Bethe Ansatz method is a powerful tool for solving integrable models and has been widely used in both theoretical and applied physics.