Geometric phases for wave packets in a uniform magnetic field

Thai Quang

Mohamed Nazih Rhimi, R. El-Bahi

S. Seshadri, S. Lakshmibala, V. Balakrishnan

William Zimmermann

A discussion is given of the construction of wave packets to describe the motion of a charged particle in a uniform magnetic field. Examples are given of the construction of wave packets from eigenstates of the Hamiltonian for both the Landau gauge in Cartesian coordinates and the cylindrically symmetric gauge in cylindrical coordinates.

H. Y. Kim, Jerome H. Weiner

For the case of uniform magnetic field and a potential quadratic in its coordinates an electron state originally represented by a Gaussian wave packet remains Gaussian in its subsequent evolution. Under these conditions the state is completely characterized by the first-and second-order quantum means, that is, quantities such as $〈{q}_{i}〉$, $〈{p}_{i}〉$ and $〈{q}_{i}{p}_{j}〉$, $〈{p}_{i}{p}_{j}〉$, respectively. A tensor-product relationship is shown to exist between the problem of the determination of the second- and first-order means which facilitates the solution. Two problems are treated in detail: (i) an isotropic wave packet in an isotropic potential well and (ii) the dynamics of a wave packet on a quadratic saddle point. In the former case, the wave packet "spins" and its radius pulsates as its mean follows the classical trajectory. In the second the magnetic field effect on tunneling through such barriers is studied and found to be small at attainable field strengths unless the effective electronic mass is very small. A method for the numerical solution for the time-dependent Schr\"odinger equation, termed the particle method and based on the hydrodynamic analogy to that equation, is here extended to include magnetic fields. It is applied to the two problems just described and found to agree well with the analytical solutions.

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.