317 Prog. Theor. Phys. Vol. 94, No. 2, August 1995, Letters 80(2, 1), Supersymmetry and D-Dimensional Radial Schrodinger Equation B. BAGCHI and P. K. Roy* Department of Applied Mathematics, University of Calcutta, 92 APC Road Calcutta 700 009 *Department of Physics, Haldia Government College, Haldia 721 657, Midnapore, WB (Received May 16, 1995) We point out a connection between the underlying function of the representations of the 50(2, 1) algebra and the superpotential of SUSY quantum mechanics. We also write down the expressions of the generators of 50(2, 1) in D·dimensions. In a recent paper Barut, Beker and Rador 0 have explored a class of realization of S0(2, 1) algebra within the framework of dynamical 0(4, 2) to study the radial Schrodinger equation (1) where VE(r)= t(l + 1)/r 2 + V(r) represents the effective potential. Since it is well-accepted [Refs. 2), 3) and references therein] that superpotential resides in the radial equation in an obvious manner, a natural question arises whether a link could be set up connecting supersymmetry (SUSY) with S0(2, 1). The purpose of this paper is to establish this by expressing the superpotential in terms of the underlying function of S0(2, 1). Our results turn out to be generalizable to systems of D-dimensions also. The S0(2, 1) algebra defined by the generators (To, I4, T) satisfies the commuta- tion relations [I4, T]=- i10, [ T, 10]= il4, [10, I4]= iT. (2) The Casimir operator C 2 is given by 10 2 - !4 2 - T 2 • In terms of an arbitrary function G( r) the algebra (2) admits the following representations: 10=(G/G' 2 )K 2 + C 2 /G- GG"' /2G' 3 +3GG" 2 /4G' 4 + G/4, I4=(G/G' 2 )K 2 + C 2 /G- GG"' /2G' 3 +3GG" 2 /4G' 4 - G/4, T=(G/G')K-iGG"/2G' 2 , (3) where K =- i(d/dr + 1/r ). Taking the eigenvalue of C 2 to be (1/2)r((1/2)r-1) and the spectrum 10 given by v+(1/2)r, v=O, 1, 2···, Barut et al. 0 have compared 10 with (1) to arrive at the following relation: G"' /2G' -3G" 2 /4G' 2 - r( r-1 )c' 2 /G 2 +( v+ f )c' 2 /G- G' 2 /4 =- VE(r)+En,z, (4) Downloaded from https://academic.oup.com/ptp/article-abstract/94/2/317/1851485 by guest on 17 June 2020