Schrodinger Equation for a Half Spin Electron ¨ in a Time Dependent Magnetic Field

Qiong-Gui Lin

We study the valence electron of an alkaline atom or a general charged particle with arbitrary spin and with magnetic moment moving in a rotating magnetic field. By using a time-dependent unitary transformation, the Schr\"odinger equation with the time-dependent Hamiltonian can be reduced to a Schr\"odinger-like equation with a time-independent effective Hamiltonian. Eigenstates of the effective Hamiltonian correspond to cyclic solutions of the original Schr\"odinger equation. The nonadiabatic geometric phase of a cyclic solution can be expressed in terms of the expectation value of the component of the total angular momentum along the rotating axis, regardless of whether the solution is explicitly available. For the alkaline atomic electron and a strong magnetic field, the eigenvalue problem of the effective Hamiltonian is completely solved, and the geometric phase turns out to be a linear combination of two solid angles. For a weak magnetic field, the same problem is solved partly. For a general charged particle, the problem is solved approximately in a slowly rotating magnetic field, and the geometric phases are also calculated.

D. Laroze, R. Rivera

William Hodge, S. V. Migirditch, W. C. Kerr

This paper analyzes how the existence of electron spin changes the equation for the probability current density in the quantum-mechanical continuity equation. A spinful electron moving in a potential energy field experiences the spin-orbit interaction, and that additional term in the time-dependent Schrödinger equation places an additional spin-dependent term in the probability current density. Further, making an analogy with classical magnetostatics hints that there may be an additional magnetization current contribution. This contribution seems not to be derivable from a non-relativistic time-dependent Schrödinger equation, but there is a procedure described in the quantum mechanics textbook by Landau and Lifschitz to obtain it. We utilize and extend their procedure to obtain this magnetization term, which also gives a second derivation of the spin-orbit term. We conclude with an evaluation of these terms for the ground state of the hydrogen atom with spin-orbit interaction. The magnetization contribution is generally the larger one, except very near an atomic nucleus.

N. J. Ionesco‐Pallas

Susan S. D’Amato

Takuji Koiso, M. Muraguchi, Kyozaburo Takeda +1 more

We theoretically studied the time-developing ballistic phenomena of a single-electron confined in a half-ellipse infinite-potential wall by solving the time-dependent Schrödinger equation numerically. We also solved the corresponding Newton equation in order to compare the classical results with the quantum ones, and extracted the quantum features. The ellipse-shaped potential wall completely reflects an electron and causes the focusing ratio of unity in the classical limit. The dispersion of the wave packet of an electron, however, weakens this characteristic nature, and reduces the focusing ratio from unity. Because the dispersion also lets an electron arrive at the collector indistinctly, we define the effective arrival time by finding inflections in the time-dependent profile of the probability density at the collector. Based on the second-derivation technique, we further determine the quantum arrival time (QAT) at which the intrusion of the wave packet occurs dominantly. The comparison of this QAT with the classical arrival time (CAT) determines whether the corresponding ballistic propagation can be discussed on the basis of the quantum consideration or the classical prediction. We further studied how the change in the half-ellipse potential wall shape affects the ballistic phenomena through the change in the ellipticity γ, the system size L and the dispersion degree σ of the wave packet. Using the ellipse-shaped infinite-potential wall, the application of the magnetic field causes irrational cyclotron motion assisted by the ellipse potential, in addition to the rational cyclotron motions. The numerical solution of the time-dependent Schrödinger equation determines the unique cyclotron motion whose peculiarity is caused by the dispersion of the wave packet and is rarely predicted by the classical limit.