Recurrence relation for hydrogen-atom expectation values

Recurrence relation for hydrogen-atom expectation values 

The expectation values of various powers of the radial variable r for a hydrogen-like atom with quantum numbers n and I are given by equation (6.69) 

To simplify the notation, we define the real function M(r) by u = rRni r) and denote the first and second derivatives of ii(r) by u and u . Equation (H.l) then takes the form 

Before beginning the direct derivation of equation (H.2), we first derive a useful 

The integrated part vanishes because R(r) 0 exponentially as r cxj and u(r) 0 as r — 0. Expanding the derivative within the integral on the right-hand side, we have 

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