_{1}

Schrödinger equation for pair of two massless Dirac particles when magnetic field is applied in Landau gauge is solved exactly. In this case, the separation of center of mass and relative motion is obtained. Landau quantization ε = ±
*B/ l *for pair of two Majorana fermions coupled via a Coulomb potential from massless chiral Dirac equation in cylindric coordinate is found. The root ambiguity in energy spectrum leads into Landau quantization for bielectron, when the states in which the one simultaneously exists are allowed. The tachyon solution with imaginary energy in Cooper problem (ε
^{2} < 0) is found. The continuum symmetry of Dirac equation allows perfect pairing between electron Fermi spheres when magnetic field is applied in Landau gauge creating a Cooper pair.

The graphene [

The graphene is the single graphite layer, i.e. two-dimensional graphite plane of thickness of single atom. The graphene lattice resembles a honeycomb lattice. The graphene lattice one can consider like into the composite of two triangular sublattices. In 1947, Wallace in “tight-binding” approximation consider a graphite which consist off the graphene blocks with taken into account the overlap only the nearest p-electrons.

The two-dimensional nature of graphene and the space and point symmetries of graphene acquire of the reason for the massless electron motion since lead into massless Dirac equation (Majorana fermions) [

The existing of the massless Dirac fermions in graphene was proved based on the unconventional quantum Hall effect. The reason of creation the integer Hall conductivity [

When the magnetic field is applied perpendicularly into graphene plane, the lowest (n = 0) Landau level has the energy

Calculation model of the graphene reflects continuum symmetry of QED_{2+1} including Lorentz group. SU(2) symmetry are shown to be found similar to chiral in the paper [

The energy bands for graphene was found using “tight-binding” approximation in the papers [

Calculate the quantized Landau energy as well as the wave function of the Majorana particles in cylindrical coordinate in magnetic field in Landau gauge. Enter the production and annihilation operators as following:

where

Hence for noninteracting Dirac particles we write the massless Dirac equation in the form:

The Schrödinger equation for the reduced energy can be rewritten in the form:

For graphene in vacuum the effective fine structure parameter

where

When magnetic field is applied perpendicularly into graphene plane in z axis along field distribution. The

vector potential in the gauge [

The Schrödinger Equation (6) with including the Coulomb potential Equation (5) one can rewritten in the form:

The solution Equation (7) with including the Coulomb potential Equation (5) can look for in the form:

Substituting the solution in Equation (7), one can find for the radial function the following equation

where

where

The solution of the Equation (10) can look for in the form:

To substitute the solution (11) in the Equation (10) it is necessarily to find as follows

Since

Substituting (12), (13), (14), (15) into the Equation (10) we find the equation for

Hence for

From the condition of finite of the wave function one can find the energy spectrum in the form:

where

where

Because the solution for the wave functions for the pair of two massless Dirac particles when magnetic field is applied in Landau gauge one can express via the product of the two identical wave functions one can conclude that in this case the separation of center of mass and relative motion is shown [

Entering the production and annihilation operators as following (1) and solving Schrödinger equation one can derive the known for quantum electrodynamics (QED) solution-the root ambiguity in energy spectrum

where

For graphene with strong Coulomb interaction the Bethe-Salpeter equation for the electron-hole bound state was solved and a tachyonic solution was found [

Calculation model of the graphene reflects continuum symmetry of QED_{2+1} including Lorentz group. SU(2) symmetry are shown to be found similar chiral in the paper [

In the paper [

In the paper [

In the paper [

In the paper [

The exciton Wannier equation for graphene was solved in the papers [

Schrödinger equation for pair of two massless Dirac particles when magnetic field is applied in Landau gauge is solved exactly. Landau quantization

potential from massless chiral Dirac equation in cylindric coordinate is found. In this case the separation of center of mass and relative motion is derived. The root ambiguity in energy spectrum leads into Landau quantization for bielectron, when the states in which the one simultaneously exists are allowed. The tachyon solution with imaginary energy in Cooper problem

From a algebraic manipulation one can find a following recurrence relations:

where

From a algebraic manipulation one can find a following integrals and recurrence relations which connect theirs:

Lyubov E.Lokot, (2015) Creation of Bielectron of Dirac Cone: The Tachyon Solution in Magnetic Field. Journal of Materials Science and Chemical Engineering,03,71-77. doi: 10.4236/msce.2015.38010