Unified treatment of high-order perturbation theory for the Stark effect in a two- and three-dimensional hydrogen atom

We show how the so(2,1) Lie algebraic methods and symbolic computation can be used to obtain the energy expansion for a general parabolic state in both the two- (2D) and three-dimensional (3D) cases. It is shown by comparing the realizations of the so(2,1) generators that the resulting symbolic expressions for the energy corrections in the 3D case in terms of the parabolic quantum numbers also give the corresponding 2D expressions by a simple reinterpretation of the quantum numbers defining a parabolic state. The main 2D and 3D results reduce to algebraic expressions for the matrix elements of the operator V=${\mathit{X}}^2$=-1/2E${\xi }^2$, which suffice to provide polynomial expressions for the perturbative terms of any (n,${\mathit{n}}_1$,${\mathit{n}}_2$,m) level to any order j in the perturbation series.