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Solutions to the time-dependent Schrodinger equation in the continuous spectrum case

By M. Maamache and Y. Saadi


We generalize the Lewis-Riesenfeld technique of solving the time-dependent Schrodinger equation to cases where the invariant has continuous eigenvalues. An explicit formula for a generalized Lewis-Riesenfeld phase is derived in terms of the eigenstates of the invariant. As an illustration the generalized phase is calculated for a particle in a time-dependent linear potential. PACS: 03.65.Ca, 03.65.Vf The study of time dependent quantum systems has attracted considerable interest in the litterature. The origin of this development was no doubt the discovery of an exact invariant by Lewis and Riesenfeld [1]. The work of Lewis and Riesenfeld and others assumes that the eigenvalue spectrum for the invariant I is discrete. Let us recall that the general method to introduce the Lewis and Riesenfeld theory, valid whatever the time dependence ( of the) parameters, considers invariant operators. For a system specified by a time-dependent Hamiltonian H X (t) , and a corresponding evolution operator U (t), an invariant is an operator I(t) such that or dI d

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