Exact time dependence of solutions to the time-dependent Schrödinger equation

Abstract

Solutions of the Schrödinger equation with an exact time dependence are derived as eigenfunctions of dynamical invariants which are constructed from time-independent operators using time-dependent unitary transformations. Exact solutions and a closed form expression for the corresponding time evolution operator are found for a wide range of time-dependent Hamiltonians in d dimensions, including non-Hermitean -symmetric Hamiltonians. Hamiltonians are constructed using time-dependent unitary spatial transformations comprising dilatations, translations and rotations and solutions are found in several forms: as eigenfunctions of a quadratic invariant, as coherent state eigenfunctions of boson operators, as plane wave solutions from which the general solution is obtained as an integral transform by means of the Fourier transform, and as distributional solutions for which the initial wavefunction is the Dirac δ-function. For the isotropic harmonic oscillator in d dimensions radial solutions are found which extend known results for d = 1, including Barut–Girardello and Perelomov coherent states (i.e., vector coherent states), which are shown to be related to eigenfunctions of the quadratic invariant by the ζ-transformation. This transformation, which leaves the Ermakov equation invariant, implements SU(1, 1) transformations on linear dynamical invariants. coherent states are derived also for the time-dependent linear potential. Exact solutions are found for Hamiltonians with electromagnetic interactions in which the time-dependent magnetic and electric fields are not necessarily spatially uniform. As an example, it is shown how to find exact solutions of the time-dependent Schrödinger equation for the Dirac magnetic monopole in the presence of time-dependent magnetic and electric fields of a specified form.

Shortcuts to adiabaticity for non-Hermitian systems

“…This may be generalized to non-Hermitian Hamiltonians [38][39][40]. The invariants for P T -symmetric Hamiltonians have also been studied by Lohe [41]. We shall assume that for a Hamiltonian …”

“…Such a path would be formed by a succession of ellipses slowly varying from the initial to the final ones. 1 The linear invariants of the classical harmonic oscillator [41,42] could be obtained from ˆ |I| by using instead of Eqs. (59) or (66) one of the eigenvectors of I times the corresponding Lewis-Riesenfeld phase factor.…”

Generation of atomic NOON states via shortcuts to adiabatic passage

“…We first give a brief description of Lewis-Riesenfeld invariants theory [45,46]. Let us consider a quantum system evolving with a time-dependent Hamiltonian…”

Fast transport of Bose–Einstein condensates

“…Our starting point is the GPE for potentials whose Schrödinger (linear) dynamics admit a quadratic invariant in momentum [8,[14][15][16][17],…”