Exact solution of a quantum forced time-dependent harmonic oscillator

The Schrodinger equation is used to exactly evaluate the propagator, wave function, energy expectation values, uncertainty values, and coherent state for a harmonic oscillator with a time dependent frequency and an external driving time dependent force. These quantities represent the solution of the classical equation of motion for the time dependent harmonic oscillator

The Wave Function and the Minimum Uncertainty Function of the Time Dependent Harmonic Oscillator(New Developments in Statistical Physics Similarities in Diversities,YITP Workshop)

この論文は国立情報学研究所の電子図書館事業により電子化されました。The time dependent harmonic oscillator is solved explicitly for quantum mechanics by the operator method with an auxiliary condition as the classical solution. Two classical invariant quantities which determine whether or not the system is bound are derived by the classical equation of motion. We obtain the invariant operator from one classical invariant quantity. Its eigenfunction is related to the solution of Schrodinger equation of the system and its eigenvalue is related to another classical quantity. The wave function is evaluated exactly by the eigenfunction of the invariant operator but it is not the eigenfunction of the Hamiltonian of the system. The uncertainty which calculates with the wave function is not a minimum one. We will confirm that the function which holds minimum uncertainty is a eigenfunction of the Hamiltonian

The wave function and minimum uncertainty function of the bound quadratic Hamiltonian system

The bound quadratic Hamiltonian system is analyzed explicitly on the basis of quantum mechanics. We have derived the invariant quantity with an auxiliary equation as the classical equation of motion. With the use of this invariant it can be determined whether or not the system is bound. In bound system we have evaluated the exact eigenfunction and minimum uncertainty function through unitary transformation