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Eclética Química

Print version ISSN 0100-4670On-line version ISSN 1678-4618

Eclet. Quím. vol.22  São Paulo  1997 

Solving Schrödinger equation for two dimensional potentials using supersymmetry*





ABSTRACT: The formalism of supersymmetric Quantum Mechanics can be extended to arbitrary dimensions. We introduce this formalism and explore its utility to solve the Schrödinger equation for a bidimensinal potential. This potential can be applied in several systems in physical and chemistry context , for instance, it can be used to study benzene molecule.
KEYWORDS: Quantum Mechanics; supersymmetry, bidimensional systems, Schrödinger equation.




Supersymmetric Quantum Mechanics (SQM) in one dimension has been used in several contexts to analyze the Schrödinger equation. For example, in analytical solutions for determination of shape invariant potentials,6 in the study of partially solvable potentials,9 to obtain isoespectral potentials8 and used in WKB approximation1 and in Variational Method.5

Recently, a formalism of SQM for two or more dimensions was introduced and this formalism can be used, for example, to solve the Scrodinger equation.4

In this work, the two dimensional realization for the superalgebra is presented and applies to obtain the solution (eigenvalues and eigenfunctions) for Schrödinger equation for the general potential:

06eq1.gif (905 bytes)(1)

where a1, a2, b1, b2 and c are parameters. With special values of these parameters one can obtain, for example, Hartmann potential or Coulomb plus an Akaranov-Bohm potential. A different approach, based in so (2,1) dynamic algebra, to study the solutions of this potential is presented in ref 2. In particular, the Hartmann potential has been applied to study the benzene molecule.7


SQM in two-dimensions

In SQM we have the nilpotent operators Q and Q+ that satisfy the algebra

{Q, Q+} = H; Q+ = Q+2 = 0                     (2)

The standard realization for the operators is a matrix of rank two.1,5,6,8-10
However, the superalgebra (2) can also be realized by4

Q = dl-xs-and Q+ = dl+ xs+                       (3)

where:06eq3b.gif (853 bytes)

a± and b± are bosonic operators which may be written as

06eq4.gif (752 bytes)(4)

W(x) and Z(y) are the superpotential for the coordinates x and y, respectively

In this two dimensional (2D) realization Q and Q+ are 4 x 4 matrices; and the super Hamiltonian is

06eq5.gif (1724 bytes)(5)

Restricting the approach to 2D systems which are separable into two 1D Hamiltonians, the original 2D Hamiltonian is written as

06eq6.gif (307 bytes)(6)

This formalism allows the immediate extension of the results obtained for 1D systems into two dimensions. Specifically, it is possible to construct a family of Hamiltonians10

06eq7.gif (304 bytes)(7)

and obtain the relation among their members:

06eq8.gif (562 bytes)(8)

Thus, the n-th member of the super-family is related with the (n+1)-th state of the original Hamiltonian. The ground state of each member yo(n) is given in terms of the superpotential:

06eq9.gif (742 bytes)(9)

Then, if it is possible to construct the superfamily, one can obtain the spectrum and eigenfunctions from the original problem by the relations in (8). In the next section, the potential (1) is studied by this formalism.


Solving a 2D Schrödinger equation

In parabolic rotational coordinates (x , h , j ) [x = x h cos j , y = x h sin j and Z = (x2 + h2)/2] we can write the Hamiltonian to potential (1) in two 1D equations:2

06eq10.gif (1425 bytes)(10)

06eq11.gif (1454 bytes)(11)


e1 + e2 + c = 0                        (12)

and the eigenfunctions in the new coordinates are

y(x , h , j ) = (xh)1/2 ca (x)cb(h)eimj                    (13)

Proceeding as in the 1D case, the superfamily can be constructed and dln± determined:

06eq14.gif (2025 bytes)(14)

where h1, h2 = 0, 1, 2... From eq. (6) and (7) we get

06eq15.gif (476 bytes)(15)


06eq15b.gif (1680 bytes)

Using the constraint (12) we get

06eq16.gif (2183 bytes)(16)

for A1 # A2, and

06eq17.gif (828 bytes)(17)

for A1 = A2 = A (n1 = n2 = 2n, n=0, ± 1, ± 2...).

These results are in agreement with those in ref.2 The eigenfunctions are obtained in the usual form from the relations (8) and (9):

06eq18.gif (436 bytes)(18)

where co(n) are given by

06eq19.gif (2705 bytes)(19)

Then, the eigenfunctions for the original problem are y (x , h , j ) = (x h )1/2 ca (x ) cb (h ) eimj , where ca(x ) and cb (h) are the components of cn(1).



A non-usual formalism for SQM and a constructive method to solve Schrodinger equation for 2D potentials is presented. The solution for the potential given in (1) is explicitly determined and the results are the same as obtained from so(2,1) dynamic algebra.2

In principle, all results in SQM for 1D systems can be extended for 2D ones. Up to now the formalism presented here is used only to solve Schrodinger equation4 and to determine isoespectral potentials.3 However, in principle, all results in SQM for 1D systems (for instance, ref.1,5,6,8,9) can be extended for two dimensions.

Finally, we note that the approach presented can be directly extended to higher dimensional systems.



The author acknowledges the partial support by Fapesp and CNPq (Brazil).



DRIGO FILHO, E. Resolvendo a equação de Schrödinger para potenciais bidimensionais usando supersimetria. Ecl. Quím. (São Paulo), v.22, p.67-73, 1997.

RESUMO: O formalismo da mecânica quântica supersimétrica pode ser estendido para dimensões arbitrarias. Nós introduzimos este formalismo e exploramos sua utilidade na resolução da equação de Schrödinger para um potencial bidimensional. Este potencial possuí varias aplicações em física e química, por exemplo, no estudo da molécula de benzeno.
PALAVRAS-CHAVE: Mecânica quântica; supersimetria; sistemas bidimensionais; equação de Schrödinger.




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Recebido em 4.12.1996
Aceito em 21.3.1997.



* This works was part of the communication presented at the VIII International Conference of Symmetry in Physics (1995), Duba/Russia.

** Departamento de Física - Instituto de Biociências, Letras e Ciências Exatas - UNESP - 15054-000 - São José do Rio Preto - SP - Brasil.

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