QUANTUM-MECHANICAL TREATMENT OF THE GENERAL TIME-DEPENDENT QUADRATIC HAMILTONIAN SYSTEM

We consider a general time-dependent quadratic Hamiltonian system with linear terms. In a bound system, the general solution of the classica l equation of motion can be represented by two real functions of time. From Hamilton's equation, we obtain the quadratic invariant quantity, and the quantum invariant operator has the same form as the classical invariant quantity whose canonical variables are replaced by quantum operators. By using the invariant operator, Re evaluate the Schrodinge r solutions, the uncertainty relations, and the coherent states, Ail t he above quantum mechanical quantities are expressed by two real funct ions of the classical solution.