Raquel M. Lopez

Propagator of a charged particle with a spin in uniform magnetic and perpendicular electric fields

We construct an explicit solution of the Cauchy initial value problem for the time-dependent Schrödinger equation for a charged particle with a spin moving in a uniform magnetic field and a perpendicular electric field varying with time. The corresponding Green function (propagator) is given in terms of elementary functions and certain integrals of the fields with a characteristic function, which should be found as an analytic or numerical solution of the equation of motion for the classical oscillator with a time-dependent frequency. We discuss a particular solution of a related nonlinear Schrödinger equation and some special and limiting cases are outlined.

On a hidden symmetry of quantum harmonic oscillators

We consider a six-parameter family of the square integrable wave functions for the simple harmonic oscillator, which cannot be obtained by the standard separation of variables. They are given by the action of the corresponding maximal kinematical invariance group on the standard solutions. In addition, the phase space oscillations of the electron position and linear momentum probability distributions are computer animated and some possible applications are briefly discussed. A visualization of the Heisenberg uncertainty principle is presented.

Reconstructing the Schrödinger groups

We discuss the maximum kinematical invariance group of the quantum harmonic oscillator from the viewpoint of a Ermakov-type system. As an example, we consider a six-parameter family of the square integrable oscillator wave functions, which appears to be not obtainable by standard separation of variables. Finally, the invariance group of the generalized driven harmonic oscillator is shown to be isomorphic to the corresponding Schrodinger group of the free particle.