Supersymmetric Quantum Mechanics and Large-NExpansions

Abstract

Supersymmetric quantum mechanics is formulated for spherically symmetric potentials in TV spatial dimensions. It is seen that the supersymmetric partner potential of a given potential can be effectively treated as being in N + 2 dimensions. This fact is exploited in calculations using the shifted \/N expansion. Also, the violation of the no-degeneracy theorem in one dimension by the Coulomb potential is seen as a consequence of this result.PACS numbers: ll.30.Pb, 03.65.Ge Given any one-dimensional potential, supersymmetric quantum mechanics provides a simple recipe for generating a partner potential with the same energy eigenvalues (except for the ground state). 1,2 Often, for many physical problems, it is profitable to deal with partner potentials. This possibility has been used by several people for finding classes of analytically solvable potentials, 3 evaluating the eigenvalues of a bistable potential, 4 studying atomic systems, 5 and improving the WKB approximation. 6 Recently, large-TV expansions, 7 " 10 particularly those involving shifted expansion parameters, 8 " 10 have proved to be very useful in the calculation of accurate eigenenergies for spherically symmetric potentials. The purpose of this paper is to show that the use of supersymmetric partner potentials can be exploited to improve further both the accuracy and the simplicity of large-A^ expansions. Typically, we find that just the leading term in a shifted large-N expansion using partner potentials yields all the energy levels correctly to three significant digits for essentially any threedimensional spherically symmetric potential of physical interest. Further accuracy is easily obtained from previously calculated higher-order terms. 9 We first review the ideas of supersymmetric quantum mechanics and set up the formalism for spherically symmetric potentials in N spatial dimensions. This will demonstrate that the supersymmetric partner of a given potential can be effectively treated as being in Af + 2 spatial dimensions. 11 This fact is responsible for substantially improving the convergence of large-A^ expansions.Several useful, illustrative potentials (Hulthen logarithmically screened Coulomb, quarkonium) are treated in order to demonstrate our approach. The Coulomb potential in one dimension is somewhat special since it is its own supersymmetric partner; the curious consequences of this (like intersecting energy levels and degeneracy in one dimension 12 ) are discussed.Assume that one has a potential V°(x) whose ground-state wave function \jj 0 (x) is known, and whose ground-state energy has been adjusted so that E 0 = 0. Then the Schrodinger equation for the ground state is (K = m = 1) and consequently ---+ V°(x) 2 dx 2 KX) <//(> = 0, H°-\ -d 2 { ftp] dx 2 $o J Define the operators ^ V2 d fto dx i// 0 This gives Q + Q~-= //°, 0-e+-/f i =-|^r+K i u), a) (2) (3) (4) 2 dx 2 where VHx) V°(x) dx *l>o */>P = -V°(x) + */>p */>p (5)If i//" is any eigenfunction of H° with eigenvalue E ny then Q~^n is an eigenfunction of H l with the same eigenvalue E...

The rotating Morse potential model for diatomic molecules in the tridiagonalJ-matrix representation: I. Bound states

“…For = 0, the radial Schrödinger equation with this potential has been solved exactly [1]. However, for > 0 only numerical solutions are possible where several approximation techniques have been proposed and extensively used with varying degrees of accuracy and stability [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19].…”

Parasupersymmetry in quantum mechanics

“…It is clear that although excellent results are obtained with the use of the shifted 1/N expansion for the original potential V H eff (r) in three dimensions, even faster convergence is obtained by using the supersymmetric partner potential, since we are now effectively working in five dimensions instead of three. Thus, SUSY has played an important role in making a very good expansion even better [77]. In fact, for many applications, considerable analytic simplification occurs since it is sufficient to just use the leading term in the shifted 1/N expansion for V 2 (r).…”

“…A slightly modified, physically motivated approach, called the "shifted large-N method" [196,197,198,199] incorporates exactly known analytic results into 1/N expansions, greatly enhancing their accuracy, simplicity and range of applicability. In this subsection, we will descibe how the rate of convergence of shifted 1/N expansions can be still further improved by using the ideas of SUSY QM [77].…”

“…Three of the notable ones are the 1/N expansion within SUSY QM [77], δ expansion for the superpotential [78] and a SUSY inspired WKB approximation (SWKB) in quantum mechanics for the case of unbroken SUSY [79,80,81]. It turns out that the SWKB approximation preserves the exact SUSY rela-tions between the energy eigenvalues as well as the scattering amplitudes of the partner potentials [82].…”

Witten index, axial anomaly, and Krein’s spectral shift function in supersymmetric quantum mechanics

“…13 ' 14 Indeed, the Hamiltonian defined as ( 1. 2) represents two Schrodinger operators, A *A and AA *, which are non-negative and which have the same spectrum, except perhaps for zero modes.…”