A New Kind of Shift Operators for Infinite Circular and Spherical Wells

Guo-Hua Sun,Shi-Hai Dong,J P Draayer,K D Launey,T Dytrych

doi:10.1155/2014/987376

Abstract

A new kind of shift operators for infinite circular and spherical wells is identified. These shift operators depend on all spatial variables of quantum systems and connect some eigenstates of confined systems of different radiiRsharing energy levels with a common eigenvalue. In circular well, the momentum operatorsP±=Px±iPyplay the role of shift operators. ThePxandPyoperators, the third projection of the orbital angular momentum operatorLz, and the HamiltonianHform a complete set of commuting operators with the SO(2) symmetry. In spherical well, the shift operators establish a novel relation betweenψlm(r)andψ(l±1)(m±1)(r).

Highlights

It is well-known that algebraic methods have become the subject of interest in different fields
We start with a brief review, with the purpose of introducing the general formalism, and we present the new shift operators that provide a novel insight into the problem
We have provided a brief review and a new insight on constructing the shift operators for infinite circular and spherical wells using the potential group approach

Summary

Introduction

It is well-known that algebraic methods have become the subject of interest in different fields. It should be pointed out that the ladder operators for those quantum systems depend on only one variable, that is, one-dimensional problem [8]. Motivated by the recently proposed factorization method [8], we generalize it to construct ladder operators that depend on all spatial variables. For this purpose, we derive shift operators for the circular [10,11,12,13] and spherical wells [14] directly from the normalized wave functions [15, 16].

Exact Solutions

Shift Operators

Conclusions