Two-dimensional solutions of the Dirac equation for the motion of an electron in a helical magnetic field

Quantum theory of a free electron in a helical wiggler magnetic field has been studied. In the previous studies the transverse momentum operators in the Dirac hamiltonian were ignored. In this note, the 2-D solutions of the Dirac equation, based on a hamiltonian which includes one of the transverse momentum operators (p_x ≠ 0, p_y = 0) and the wiggler field as an external potential is derived. It is shown that there are solutions in terms of the Mathieu functions of fractional order.     

New exact solutions of the Dirac equation

The search for new exact solutions to the Dirac and Klein-Gordon equations initiated in [1] is continued. New solutions are found for axisymmetric fields and one type of nonstationary field of special configuration. The basic notation and system of units of [1] are retained.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 10–16, April, 1980.     

Stationary solutions of the Dirac equation in the gravitational field of a charged black hole

A stationary solution of the Dirac equation in the metric of a Reissner-Nordström black hole has been found. Only one stationary regular state outside the black hole event horizon and only one stationary regular state below the Cauchy horizon are shown to exist. The normalization integral of the wave functions diverges on both horizons if the black hole is non-extremal. This means that the solution found can be only the asymptotic limit of a nonstationary solution. In contrast, in the case of an extremal black hole, the normalization integral is finite and the stationary regular solution is physically self-consistent. The existence of quantum levels below the Cauchy horizon can affect the final stage of Hawking black hole evaporation and opens up the fundamental possibility of investigating the internal structure of black holes using quantum tunneling between external and internal states.     

Modified Dirac equation with Lorentz invariance violation and its solutions for particles in an external magnetic field

The second-order modified Dirac equation leading to the modified dispersion relation due to the Lorentz invariance violation corrections is suggested. The equation is formulated in the 16-component first-order form. I have obtained the projection matrix extracting solutions of the equation with definite spin projections which can be considered as the density matrix for pure spin states. Exact solutions of the equation are found for particles in the external constant and uniform magnetic field. The synchrotron radiation radius within the novel modified Dirac equation is estimated.     

Dirac equation studies in the tunneling energy zone

We investigate the tunneling zone V₀-m<E<V₀+m for a one-dimensional potential in the Dirac equation. We find the appearance of superluminal transit times, akin to the Hartman effect. PACS 03.65.Pm; 03.65.Xp     

Approximate solution of the Dirac equation in an arbitrary,periodic electrostatic field

The Dirac equation for a relativistic electron in a periodic electric field, simulating the field of a crystal axis, is solved.Moscow Power Institute. Translated from Ivestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 45–47, July, 1995.     

New exact solutions of the Dirac equation. IV

This paper continues a series of articles [1–3] on new exact solutions of the Dirac equation; it contains solutions of the equation for four cases of crossed fields constituting combinations of the field of a plane, linearly polarized wave and a certain field depending on y and x₀?z in a complicated manner.     

New exact solutions of the Dirac equation. VII

This paper is a continuation of the studies begun in [1] to find exact solutions of the Dirac, Klein-Gordon, and Lorentz equations for an electron moving in new configurations of an external electromagnetic field.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 46–51, July, 1977.     

New exact solutions of the Dirac equation. IX

Exact solutions of the equations of motion of a charge in external electromagnetic fields of seven new types are investigated. The notation and system of units used in this article coincide with those used in [1].Translated from Izvestiya Vysshikh Uchebenykh Zavedenii, Fizika, No. 3, pp. 46–49, March, 1978.     

New exact solutions of the Dirac equation. V

Four types of external electromagnetic fields allowing an exact solution of the Dirac and Klein — Gordon equations are considered. The motion of the electron in such fields is of a specific character; however, from the mathematical point of view some of the problems reduce to cases already studied in earlier papers [1–4].Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 106–112, September, 1975.     

Some exact solutions of the Dirac equation in the Kasner space-time

The exact solutions of the covariant generalization of the Dirac equation are obtained in the Kasner space-time for two types of coordinates, cartesian coordinates and cylindrical ones. The tetrads for both cases are constructed on the basis of a global quasicartesian coordinate systems that allows one to get rid of coordinate effects connected with the rotation of the local frame under transition of the triad from one space-time point to another. The possibility of plane and cylindrical spinor waves in the Kasner space-time is proved.     

Exact solutions of the Kemmer equation for a Dirac oscillator

Exact solutions of Kemmer equation for charged, massive, spin-1 particles in the Dirac oscillator potential have been found. The eigensolutions of this potential have been calculated and discussed in both natural and unnatural parities.     

Makarov势的Dirac方程的束缚态解

给出了Makarov型标量势与矢量势相等条件下的Dirac方程的束缚态解. Dirac方程的角向方程用因子分解方法求解，在得出角向波函数的过程中，自然地得到了属于同一本征值的不同角向波函数间的递推操作. 径向束缚态波函数用合流超几何函数表示，束缚态的能量方程可由径向波函数满足的边界条件得到. 关键词： Makarov势 Dirac方程 束缚态 因子分解方法     

Dirac equation in the presence of a spatially periodic magnetic field

In the present article we separate variables in the Dirac equation in the presence of a spatially periodic magnetic field. We reduce the problem to solving a coupled system of first order differential equations. We show that the solution recently reported by Sen Gupta [7] is not correct.     

Spin symmetric solutions of Dirac equation with PSschl-Teller potential

The approximate analytical solutions of the Dirac equation with the Poeschl-Teller potential is presented for arbitrary spin-orbit quantum number κ within the framework of the spin symmetry concept. The energy eigenvalues and the corresponding two Dirac spinors are obtained approximately in closed forms. The limiting cases of the energy eigenvalues and the two Dirac spinors are briefly discussed.     

Invariance properties of the Dirac equation with external electro-magnetic field

In this paper, we attempt to obtain the nature of the external field such that the Dirac equation with external electro-magnetic field is invariant. The Poincaré group, which is the maximal symmetry group for field free case, is constrained by the presence of the external field. Introducing infinitesimal transformation ofx and ψ, we apply Lie’s extended group method to obtain the class of external field which admit of the invariance of the equation. It is important to note that the constraints for the existence of invariance are explicity on the electric and magnetic field, though only potentials explicity appears in the equation. Presented at the Sixth Chittagong Conference on Mathematical Physics, January 2001.     

Dirac equation exact solutions for generalized asymmetrical Hartmann potentials

In this work we solve the Dirac equation by constructing the exact bound state solutions for a mixing of vector and scalar generalized Hartmann potentials. This is done provided the vector potential is equal to or minus the scalar potential. The cases of some quasi-exactly solvable and Morse-like potentials are briefly commented.