Ricardo Cordero - Soto And

We evaluate the matrix elements 〈r〉 for the n-dimensional harmonic oscillator in terms of the dual Hahn polynomials and derive the corresponding three-term recurrence relation and the Pasternack-type reflection relation. A short review of similar results is also given. 1. An Introduction The purpose of this note is to present a simple evaluation of the expectation values 〈r〉 for the n-dimensional 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 [5], [6], [7] and references given there, textbooks [24], [30], [10], [38]), commutation relation ([21], [18], [40], [41]), and dynamical groups ([2], [3], [8], [9], [15], [35], [39]). Our approach allows to study these expectation values with the help of the advanced theory of classical polynomials [23], [33]. 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. 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 include [1], [2], [7], [10], [11], [12], [13], [14], [15], [18], [21], [24], [26], [27], [30], [31], [35], [36], [37], [38], [43], [46], [47], [51], and [53] 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. [46] the mean values for states of definite energy