Time-dependent formulation of the linear unitary transformation and the time evolution of general time-dependent quadratic Hamiltonian systems
The author investigates the time evolution of quantum systems which is described by the following time-dependent Hamiltonian: \[ H(t)=B_+(t)J_++B_-(t)J_-+B_0(t)J_0+B_1(t)e_1+B_2(t)e_2+B_3(t)e_3, \] where \(B_j(t), (j=\pm,0,1,2,3)\) are nonsingular functions of time, the \(J_j(j=\pm,0)\) and \(e_j(j=1,2,3)\) are the generators of the \(su(1,1)\) and Heisenberg-Weyl algebra \(h_3\), respectively. In section 2 he develops a time-depending closed-form formulation of the linear transformation for harmonic-oscillator annihilation and creation operators by means of a Lie algebraic approach. It is shown that the differential equation satisfied by the unitary operator performing the transformation possesses the two-photon Lie algebraic structure and reveals the interrelationship between the unitary operator and the transformation. In section 3, by combining the above formulation with the time evolution given the Hamiltonian, another method to solve the time evolution equation of time dependent quadratic Hamiltonian systems is established. In the last section 4 the developed methods are verified for the motion of a charged particle with variable mass in a time-dependent electric field.
scientific article; zbMATH DE number 2092927