In this paper the one-dimensional Dirac equation with linear potential has been solved by the method of canonical transformation. The bound-state wavefunctions and the corresponding energy spectrum have been obtained for all bound states.
In a previous note (1), it was noted one could write a solution of the bound state Schrodinger equation as W(x)=Wo(x)Dn(x) where W(x) is the wavefunction and Wo(x), the ground state wavefunction. Inserting into the Schrodinger equation one may obtain the Schrodinger equation for Wo(x) and an equation coupling Wo(x) and Dn(x). In this note, we suggest such an approach may be applied to the one dimensional bound Dirac equations. This allows one to obtain decoupled equations for u1(x) and v1(x) where u(x)=uo(x)u1(x) and v(x)=v1(x)vo(x) are the full solutions. Thus, if one is able to obtain a solution for uo(x) and vo(x) by inspection, the decoupled equations allow one to find the general solution. We apply this approach to the case of a scalar potential g|x|. The Dirac problem for this potential has been solved in the literature (2) with decoupling occurring through the taking of spatial derivatives. Here we wish to suggest a general method.
An exact path-integral solution of the Dirac equation with a Coulomb potential is found. The energy-dependent Green's function of the second-order Dirac-Coulomb equation, expressed as a polar-coordinate path integral, is reduced to the exactly solvable path integral for an isotropic harmonic oscillator by a coordinate transformation combined with a local time rescaling. The Green's function of the linear Dirac equation is evaluated with the help of the Biedenharn transformation. The energy spectrum for the bound states is also obtained.
We study the one-dimensional non-Hermitian imaginary potential with a real energy spectrum in the framework of the position-dependent effective mass Dirac equation. The Dirac equation is mapped into the exactly solvable Schrödinger-like equation endowed with position-dependent effective mass that we present a new procedure to solve it. The point canonical transformation in non-relativistic quantum mechanics is applied as an algebraic method to obtain the mass function and then by using the obtained mass function, the imaginary potential can be obtained. The spinor wavefunctions for some of the obtained electrostatic potentials are given in terms of orthogonal polynomials. We also obtain the relativistic bound state spectrum for each case in terms of the bound state spectrum of the solvable potentials.
An approximation method, namely, the Extended Wronskian Determinant Approach, is suggested to study the one-dimensional Dirac equation. An integral equation, which can be solved by iterative procedure to find the wave functions, is established. We employ this approach to study the one-dimensional Dirac equation with one-well potential, and give the energy levels and wave functions up to the first order iterative approximation. For double-well potential, the energy levels up to the first order approximation are given.