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7.7.1 Normalized Hydrogenic Bound States

Function: double gsl_sf_hydrogenicR_1 (double Z, double r)

Function: int gsl_sf_hydrogenicR_1_e (double Z, double r, gsl_sf_result * result)

These routines compute the lowest-order normalized hydrogenic bound state radial wavefunction R_1 := 2Z \sqrt{Z} \exp(-Z r).

Function: double gsl_sf_hydrogenicR (int n, int l, double Z, double r)

Function: int gsl_sf_hydrogenicR_e (int n, int l, double Z, double r, gsl_sf_result * result)

These routines compute the n-th normalized hydrogenic bound state radial wavefunction,

R_n := 2 (Z^{3/2}/n^2) \sqrt{(n-l-1)!/(n+l)!} \exp(-Z r/n) (2Zr/n)^l
          L^{2l+1}_{n-l-1}(2Zr/n).  

where L^a_b(x) is the generalized Laguerre polynomial (see section 7.22 Laguerre Functions). The normalization is chosen such that the wavefunction \psi is given by \psi(n,l,r) = R_n Y_{lm}.