Solution of Relativistic Feshbach-Villars Spin-1/2 Equations

We propose method for studying relativistic spin-1 / 2 particles by solving the corresponding Feshbach-Villars equation. We have found that the Feshbach-Villars spin-1 / 2 equations can be formulated as spin-coupled Feshbach-Villars spin-0 equations, that results in a Hamiltonian eigenvalue problem. We adopted an integral equation formalism. The potential operators are represented in a discrete Hilbert space basis and the relevant Green’s operator has been calculated by a matrix continued fraction.


Introduction
The basic equations of relativistic quantum mechanics, the Klein-Gordon and the Dirac equations are quite different in form and they also differ considerably from the non-relativistic Schrödinger equation.The Schrödinger equation is linear in time derivative and quadratic in spatial derivatives.On the other hand the Klein-Gordon equation is quadratic, while the Dirac equation is linear in both type of derivatives.
In 1958, by "squaring" the Dirac equation, Feynman and Gell-Mann proposed a variant that is quadratic in both types of variables [1].About the same time, Feshbach and Villars linearized the Klein-Gordon in the time variable [2].As a result, the Feshbach-Villars equations are linear in time and quadratic in spatial derivatives, just like the Schrödinger equation, but the wave function has two components.Much later in 1996, Robson and Staudte introduced the Feshbach-Villars linearized form of the Feynman-Gell-Mann equation [3,4].The Feshbach-Villars linearization of the relativistic equation results in a Hamiltonian that is not Hermitian in the usual sense, yet it possesses real eigenvalues.The Hamiltonian is multicomponent and explicitly exhibits the particle-antiparticle features.
Although the Feshbach-Villars formalism has some history, it has been used rather scarcely in actual calculations.The reason maybe is that the multicomponent non-Hermitian feature makes the solution rather cumbersome.Nevertheless, there are some notable exceptions [5][6][7][8].
The aim of this paper is to solve the Feshbach-Villars equation (FV1/2) for spin-1/2 particles.This work is the continuation of our previous one where we solved the spin-0 (FV0) equation [9].In Sections 2 and 3 we show the Feynman-Gell-Mann and Feshbach-Villars equations for spin-1/2 particles, respectively.In Section 4 we present the solution method as the extension of method in Ref. [9].The numerical power of the method is illustrated in Section 5 and conclusions are given in Section 6.

Feynman-Gell-Mann equation
Consider a relativistic particle with charge q in electromagnetic field A µ = (A, iΦ).The Dirac equation for the four-component wave function ψ reads where γ i are the 4 × 4 gamma matrices satisfying the anti-commutation relation Feynman and Gell-Mann "squared" this equation With a possible representation of gamma matrices, where σ 's are the 2 × 2 Pauli spin matrices, We thus have two identical two-component equations and can drop one of them to work with a two-component formalism.Instead of the four-component first order Dirac equation we can have, without loss of generality, the two-component second order Feynman-Gell-Mann equation.By "squaring" the Dirac equation, Feynman and Gell-Mann doubled the solution space, which is reduced again by keeping only one of Eq. ( 5).

Feshbach-Villars equation for spin-1/2 particles
In this work we consider a stationary field with B = 0 and assume that qΦ = V .Then one of Eqs.(5) becomes or The Feshbach-Villars linearization amounts of splitting the wave function into components such that This leads to the set of equations Reorganizing, we obtain a Schrödinger-like Feshbach-Villars equation for spin-1/2 particles with Hamiltonian With the help of Pauli and the unit matrices, that act in the Feshbach-Villars component space, we can write the Hamiltonian as We could also introduce a scalar interaction into the formalism by the substitution m → m + S/c 2 .This results in the Hamiltonian where U = S + S 2 /2mc 2 .

The solution method
Here we assume that both V and U are spherical potentials and thus depend only on radial variable r.Then the Hamiltonian forms a complete set of commuting observables with angular momentum operator J 2 and J z .The common eigenstates of J 2 and J z are the spin-orbit coupled angular momentum states where j = l + + 1/2 = l − − 1/2 and χ 1/2,m s denotes the spin states.For a spherical potential V the spin coupling term reads where e r is a unit vector in the radial direction.It has been shown in Refs.[10][11][12]] that e r σ Φ i.e. this term couples the basis states with different orbital angular momenta.
As a consequence, the angular momentum projected Hamiltonian becomes where It should be noted that H(±) FV 0 and H′(±) are still operators in radial variables.We can solve this problem in the way presented in Ref. [9].Assume that potentials U and V are combinations of long range and short range terms and write the eigenvalue problem in a Lippmann-Schwinger form.We have to put the long range potentials that behave asymptotically like 1/r for Coulomb, or like r or r 2 for confining systems, into the Green's operator.We can represent the short range potentials in a Hilbert space basis, in particular, in a Coulomb-Sturmian basis.The matrix elements of the short range potentials in Eqs. ( 20) and (21) can always be evaluated, at least numerically.The corresponding matrix elements of the long range Green's operator can be calculated as a matrix continued fraction.Then, the whole problem becomes a linear algebraic problem with a zero search of a determinant.

Numerical Illustrations
To illustrate the power of the method we take the same example we had in Ref. [9].We adopt units such that m = h = e 2 = 1 and c = 137.036and the potential is given by V (r) = 92/r − 240 exp(−r)/r + 320 exp(−4r)/r.( 22) In another example we consider a scalar confinement potential and a Coulomb vector potential, Table 2 and 3 show the non-relativistic and the relativistic FV0 and FV1/2 energies with l = 0 and l = 1.We take Coulomb V (r) = −1/r plus linear U(r) = r and quadratic U(r) = r 2 /2 confinement potentials, respectively.

Table 1
Bound and resonant state energies in potential of Eq. (22) for l = 0.