Off-diagonal matrix elements < nl | r k | nl' > for hydrogen-like states: an exact correspondence relationship in terms of orthogonal polynomials and the WKB approximation

Abstract

A closed formula for and interpretation of exact quantum mechanical off-diagonal matrix elements of powers of r is given in terms of orthogonal polynomials. The compact formulae for r-k-2 and rk-1 are in exact correspondence to the semiclassical (diagonal) expectation values and are most suitable for the practical evaluation of the matrix elements. The expressions for < nl | r-k-2 | l'n > and < nl | rk-1 | l'n > consist only of the minimum of n-l or k+1 terms. An off-diagonal generalization of the well known diagonal inversion formula is derived and the matrix elements can thus be evaluated for arbitrary (complex) powers k. Also, an analytic continuation to an arbitrary effective n* is possible. It is thus established that, complementary to the O(2,1) approach by Armstrong (1970, 1971), orthogonality properties for purely radial quantities can be obtained without the introduction of a fictitious angular variable. Using the analytical results obtained, certain deficiencies of the semiclassical Coulomb (or WKB) approximation are discussed. The normalization error involved in the WKB approximation is identified analytically and an improved version of the semiclassical limit, a correspondence limit, is derived. Furthermore, it is established that, like the exact results, the WKB matrix elements obey a three-term recursion relation and an inversion formula. For low k some explicit formulae for exact matrix elements are tabulated.