The authors felt inspired by the well known fact that certain most
elementary Schroedinger equations (i.e., the second-order ordinary
differential equations) and Dirac equations (i.e., systems of
first-order ordinary differential equations) often happen to be
intimately interrelated. Thus, the authors's recalled the well known
supersymmetric approach to the former equations and asked themselves
if this could not help in solving the latter equations. They found
that up to the entirely trivial ground-state cases, it cannot.
Nevertheless, they did not get discouraged and they wrote this
absolutely useless paper describing the failure of their not too
well designed attempt, anyhow.
MR2821759 Panahi, H.; Bakhshi, Z. Dirac equation and ground state of
solvable potentials: supersymmetry method. Internat. J. Theoret.
Phys. 50 (2011), no. 9, 2811--2818. 81Q60
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Excerpta:
Dirac Equation and Ground State of Solvable Potentials:
Supersymmetry Method H. Panahi · Z. Bakhshi Received: 22 December
2010 / Accepted: 30 March 2011 / Published online: 15 April 2011 ©
Springer Science+Business Media, LLC 2011 Abstract The supersymmetry
in non-relativistic quantum mechanics is applied as an algebraic
method to obtain the solutions of the Dirac equation with spherical
symmetry electromagnetic potentials. We show that some of the
superpotentials related to ground state of the solvable potentials
in non-relativistic quantum mechanics can be used for studying of
the Dirac equation. Keywords Dirac equation · Solvable potentials ·
Supersymmetric quantum mechanics
[22] and etc. In this work, in Sect. 2, we give a brief introduction
for the approach which transform the Dirac equation with spherically
symmetric potentials to two Schrödinger equations for upper and
lower spinor fields. We factorize these obtained Hamiltonians in
terms of two incretion and annihilation differential operators and
then try to solve them by SUSY QM method. According to concepts of
SUSY in non-relativistic quantum mechanics, the solvable potentials
can be stated in terms of superpotentials by a Riccati equation and
so we try to relate the well-known superpotentials to our obtained
potentials in Dirac equation. It is seen that by inducing one
assumption over electrostatic potential and gauge field of Dirac
equation, a large class of solvable potentials can be used for
solving Dirac equation. We show that the relativistic energy of all
of these potentials are constant and the spinor fields of them are
obtained by ground state of the solvable potentials.We complete our
calculation for two examples in Sect. 3 and we give a table for
other potentials. In Sect. 4, the paper ends with a brief
conclusion.
We have presented an idea for connecting the ground state of the
exactly solvable potentials in non-relativistic quantum mechanics to
the solution of the radial Dirac equation with spherical symmetry
electromagnetic potentials. By using the supersymmetry method in
nonrelativistic quantum mechanics for each superpotential of the
solvable potentials, we have obtained the corresponding
electrostatic potential and gauge field for radial Dirac. Then, we
have shown that the spinor fields of the obtained electromagnetic
potentials can be calculated from the ground state of solvable
potentials and the relativistic energy for all of them is constant
too.