Motion of wave packets in a central force field

A.B. Kukanov, N. D. Naumov, T'hai K'uatig

Sergio M. De Nicola, Renato Fedele, Dušan Jovanović +2 more

The tomography of a single quantum particle (i.e., a quantum wave packet) in an accelerated frame is studied. We write the Schrödinger equation in a moving reference frame in which acceleration is uniform in space and an arbitrary function of time. Then, we reduce such a problem to the study of spatiotemporal evolution of the wave packet in an inertial frame in the presence of a homogeneous force field but with an arbitrary time dependence. We demonstrate the existence of a Gaussian wave packet solution, for which the position and momentum uncertainties are unaffected by the uniform force field. This implies that, similar to in the case of a force-free motion, the uncertainty product is unaffected by acceleration. In addition, according to the Ehrenfest theorem, the wave packet centroid moves according to classic Newton's law of a particle experiencing the effects of uniform acceleration. Furthermore, as in free motion, the wave packet exhibits a diffraction spread in the configuration space but not in momentum space. Then, using Radon transform, we determine the quantum tomogram of the Gaussian state evolution in the accelerated frame. Finally, we characterize the wave packet evolution in the accelerated frame in terms of optical and simplectic tomogram evolution in the related tomographic space.

George H. Duffey

Ter Haar

A collection of problems of varying difficulty is presented, most of which are provided with detailed solutions. Included in the coverage are one dimensional motion, the tunnel effect, commutation and Heisenberg relations, spreading of wave packets, angular momentum, spin, the central field of force, motion of particles in a magnetic field, atoms, molecules, scattering, creation and annihilation operators, the density matrix, and relativistic wave equations. (JFP)

K. T. Hecht

Hideo Tsuru

A motion of a Gaussian wave packet of a single electron in a uniform magnetic field is obtained by solving the time dependent Schrödinger equation analytically. The wave packet moves changing its width and central position periodically. The period of the change of the width is the same as that of the central position. The interpretation of these periodic changes in terms of the Green's function which can be derived by the path integration formulation is also discussed.