Some recurrence relations among the radial matrix elements for the relativistic hydrogenic atoms

Abstract

The general calculation formula for the matrix elements 〈 n 1 ′ l 1 ′ | r k | n 2 ′ l 2 ′ 〉 is derived. We present a very lengthy and complicated general recurrence relation among the matrix elements of r k . Some useful recurrence relations among the matrix elements are studied in some special cases, which include the matrix elements of r k calculated in the same state | n ′ l ′ 〉 , the different states | n ′ l 1 ′ 〉 and | n ′ l 2 ′ 〉 , the k = | l 1 − l 2 | and k = l 1 + l 2 + 1 . On the basis of the integral properties of the product of two associated Laguerre polynomials multiplied by a weight factor, we obtain a new recurrence relation between the matrix elements 〈 n ′ l 1 ′ | r − k − 2 | n ′ l 2 ′ 〉 and 〈 n ′ l 1 ′ | r k − 1 | n ′ l 2 ′ 〉 . Another recurrence relation involving the matrix elements of logarithmic functions ln ( 2 r / α ′ n ′ ) ( α ′ = a 0 m c 2 / Z E n ′ ) is investigated.