Reply to Comment on Relativistic shape invariant potentials



A. D. Alhaidari

Physics Department, King Fahd University of Petroleum & Minerals, Box 5047,

Dhahran 31261, Saudi Arabia

E-mail: haidari@mailaps.org



The points raised in the Comment are addressed and except for one error, which will

be corrected, the conclusion is that all of our findings are accurate.



PACS numbers: 03.65.Pm, 03.65.Ge





The Hamiltonian that resulted in the radial equation (1) of our Paper [1] is not the

minimum coupling Hamiltonian H shown on the second page of the Paper but the one

!
! !
!
obtained from it by replacing the two off-diagonal terms A with i A ,

respectively. Consequently, our interpretation of (V, rW ) as the electromagnetic

potential and the statement that W(r) is a gauge field are not correct. Likewise, calling

equation (3) in the Paper, or any other derived from it, as the gauge fixing condition is

not accurate. This has to be replaced everywhere by the term constraint. Nevertheless,

aside from an error which is corrected below, all developments based on, and findings

subsequent to equation (1) still stand independent of that interpretation.



One of the main contributions in our Paper is the choice of constraint, which

resulted in Schr^dinger-like equation for the upper spinor component. This makes the

solution of the relativistic problem easily obtainable by correspondence with well-

known exactly solvable nonrelativistic problems. As such, we find enough justification

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for making that particular choice of constraint as given by equation (4). Furthermore,

the angle parameter in the unitary transformation (2) is proportional to as shown

explicitly on page 9830 when taking the nonrelativistic limit of the Dirac-Rosen-M^rse

I potential (i.e., tan 2 ). Therefore, this transformation does reduce to the identity

in the nonrelativistic limit ( 0 ).



The error, which was pointed out in the Comment [2] of assigning the

inadmissible value = 0 in the cases where V(r) = 0, was hastily made to eliminate the

centrifugal barrier 2
( +1) r and the term 2W r simultaneously from equation (11)

so that we end up with the super-potential 2
W -W . This mistake, which will now be

corrected, affects only the Dirac-Rosen-M^rse II, Dirac-Scarf, and Dirac-P^schl-Teller

problems. Eliminating these two terms can be achieved properly by replacing the

potential function W(r) given in the Paper for each of the three problems by W (r) - r ,

where is now arbitrary. That is, in equations (10) and (11) of the Paper and in the

Table we substitute the following potential function for the corresponding problem:



Dirac-Rosen-M^rse II: W (r) = F coth(r) - G csch(r) - r

Dirac-Scarf: W (r) = F tanh(r) + G sech(r) - r

Dirac-P^schl-Teller: W (r) = F tanh(r) - G coth(r) - r



where F, G, and are the potential parameters defined in the paper. This gives the same

differential equations for the spinor components and reproduces the same solutions

(energy spectrum and wave functions) as those given in the Paper for each of the three

problems.



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REFERENCES



[1] A. D. Alhaidari, J. Phys. A 34, 9827 (2001); ibid. 35, 3143 (2002).

[2] A. N. Vaidya and R. L. Rodrigues,  v1







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