The Energy Level Shifts, Wave Functions and the Probability Current Distributions for the Bound Scalar and Spinor Particles Moving in a Uniform Magnetic Field

摘要

We discuss the equations for the bound one-active electron states based onthe analytic solutions of the Schrodinger and Pauli equations for a uniformmagnetic field and a single attractive $\delta({\bf r})$-potential. It is varyimportant that ground electron states in the magnetic field differ essentiallyfrom the analogous state of spin-0 particles, whose binding energy wasintensively studied more than forty years ago. We show that binding energyequations for spin-1/2 particles can be obtained without using the language ofboundary conditions in the $\delta$-potential model developed in pioneeringworks. We use the obtained equations to calculate the energy leveldisplacements analytically and demonstrate nonlinear dependencies on fieldintensity. We show that the magnetic field indeed plays a stabilizing role inconsidered systems in a case of the weak intensity, but the opposite occurs inthe case of strong intensity. These properties may be important for realquantum mechanical fermionic systems in two and three dimensions. We alsoanalyze the exact solution of the Pauli equation for an electron moving in thepotential field determined by the three-dimensional $\delta$-well in thepresence of a strong magnetic field. We obtain asymptotic expressions for thissolution for different values of the problem parameters. In addition, weconsider electron probability currents and their dependence on the magneticfield. We show that including the spin in the framework of the nonrelativisticapproach allows correctly taking the effect of the magnetic field on theelectric current into account. The obtained dependencies of the currentdistribution, which is an experimentally observable quantity, can be manifesteddirectly in scattering processes, for example.