The Energy Level Shifts, Wave Functions and the Probability Current Distributions for the Bound Scalar and Spinor Particles Moving in a Uniform Magnetic Field
We discuss the equations for the bound one-active electron states based on the analytic solutions of the Schrodinger and Pauli equations for a uniform magnetic field and a single attractive $\delta({\bf r})$-potential. It is vary important that ground electron states in the magnetic field differ essentially from the analogous state of spin-0 particles, whose binding energy was intensively studied more than forty years ago. We show that binding energy equations for spin-1/2 particles can be obtained without using the language of boundary conditions in the $\delta$-potential model developed in pioneering works. We use the obtained equations to calculate the energy level displacements analytically and demonstrate nonlinear dependencies on field intensity. We show that the magnetic field indeed plays a stabilizing role in considered systems in a case of the weak intensity, but the opposite occurs in the case of strong intensity. These properties may be important for real quantum mechanical fermionic systems in two and three dimensions. We also analyze the exact solution of the Pauli equation for an electron moving in the potential field determined by the three-dimensional $\delta$-well in the presence of a strong magnetic field. We obtain asymptotic expressions for this solution for different values of the problem parameters. In addition, we consider electron probability currents and their dependence on the magnetic field. We show that including the spin in the framework of the nonrelativistic approach allows correctly taking the effect of the magnetic field on the electric current into account. The obtained dependencies of the current distribution, which is an experimentally observable quantity, can be manifested directly in scattering processes, for example.