A Lie Algebraic Approach to the Schr¨odinger Equation for Bound States of P¨oschl-Teller Potential

摘要

The application of Group theoretical techniques to physical problems has a long andfruitful history. Lie algebraic methods have been useful in the study of problems in physics eversince Lie algebras were introduced by M.Sophus Lie (1842-1899) at the end of the 19th century,especially after the development of quantum mechanics. This is because quantum mechanicsmakes use of commutators [x, Px] = i, which are the defining ingredients of Lie algebras. Thetheory of Lie groups and Lie algebras has become important not only in explaining the behaviourof various physical systems but also in constructing new physical theories. By identifying thesuitable Spectrum Generating Algebra (SGA) the problem of interest can be approached. ASpectrum Generating Algebra exists when the Hamiltonian H can be expressed in terms ofgenerators of the algebra. As a consequence the solution of the Schr¨ odinger equation thenbecomes an algebraic problem which can be attacked using the tools of group theory. Here inthis paper we derive the Schr¨ odinger equation for the bound states of P¨oschl-Teller potentialusing Lie algebra.