And Sergei

We evaluate the matrix elements 〈r〉 for the n-dimensional harmonic oscillator in terms of the dual Hahn polynomials and derive a corresponding three-term recurrence relation and a Pasternack-type reflection relation. A short review of similar results for nonrelativistic hydrogen atom is also given. 1. An Introduction The purpose of this note is to present a simple evaluation of the expectation values 〈r〉 for the ndimensional harmonic oscillator in terms of the dual Hahn polynomials by direct integration. Other methods of solving similar problems in elementary quantum mechanics appeal to general principles and involve the Hellmann–Feynman theorem (see [6], [7], [8] and references given there, textbooks [27], [32], [11], [40]), commutation relation ([23], [20], [42], [43]), and dynamical groups ([3], [4], [9], [10], [16], [37], [41]). Our approach allows to study these expectation values with the help of the advanced theory of classical polynomials [2], [25], and [35]. We recall also that the first-order of the time-independent perturbation theory equates the energy correction to the expectation value of the perturbing potential. Thus expressions of the form 〈r〉 gain utmost importance. The plan of the paper is as follows. We review results on expectation values 〈r〉 for the nonrelativistic hydrogenlike wave functions first and then extend them, in a similar fashion, to the case of n-dimensional harmonic oscillator. An attempt to collect the available literature is made. 2. Expectation Values 〈r〉 for Coulomb Problems The problem of evaluation of matrix elements 〈r〉 between nonrelativistic bound-state hydrogenlike wave functions has a long history in quantum mechanics. An incomplete list of references includes [1], [3], [8], [11], [12], [13], [14], [15], [16], [20], [23], [27], [28], [29], [32], [33], [37], [38], [39], [40], [45], [49], [50], [54], and [56] and references therein. Although different methods were used in order to evaluate these matrix elements, one of possible forms of the answer seems has been missing until recently. In Ref. [49] the mean values for states of definite energy