| @cindex Coulomb wave functions |
| @cindex hydrogen atom |
| |
| The prototypes of the Coulomb functions are declared in the header file |
| @file{gsl_sf_coulomb.h}. Both bound state and scattering solutions are |
| available. |
| |
| @menu |
| * Normalized Hydrogenic Bound States:: |
| * Coulomb Wave Functions:: |
| * Coulomb Wave Function Normalization Constant:: |
| @end menu |
| |
| @node Normalized Hydrogenic Bound States |
| @subsection Normalized Hydrogenic Bound States |
| |
| @deftypefun double gsl_sf_hydrogenicR_1 (double @var{Z}, double @var{r}) |
| @deftypefunx int gsl_sf_hydrogenicR_1_e (double @var{Z}, double @var{r}, gsl_sf_result * @var{result}) |
| These routines compute the lowest-order normalized hydrogenic bound |
| state radial wavefunction @c{$R_1 := 2Z \sqrt{Z} \exp(-Z r)$} |
| @math{R_1 := 2Z \sqrt@{Z@} \exp(-Z r)}. |
| @end deftypefun |
| |
| @deftypefun double gsl_sf_hydrogenicR (int @var{n}, int @var{l}, double @var{Z}, double @var{r}) |
| @deftypefunx int gsl_sf_hydrogenicR_e (int @var{n}, int @var{l}, double @var{Z}, double @var{r}, gsl_sf_result * @var{result}) |
| These routines compute the @var{n}-th normalized hydrogenic bound state |
| radial wavefunction, |
| @comment |
| @tex |
| \beforedisplay |
| $$ |
| R_n := {2 Z^{3/2} \over n^2} \left({2Z r \over n}\right)^l \sqrt{(n-l-1)! \over (n+l)!} \exp(-Z r/n) L^{2l+1}_{n-l-1}(2Z r / n). |
| $$ |
| \afterdisplay |
| @end tex |
| @ifinfo |
| |
| @example |
| 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). |
| @end example |
| |
| @end ifinfo |
| @noindent |
| where @math{L^a_b(x)} is the generalized Laguerre polynomial (@pxref{Laguerre Functions}). |
| The normalization is chosen such that the wavefunction @math{\psi} is |
| given by |
| @c{$\psi(n,l,r) = R_n Y_{lm}$} |
| @math{\psi(n,l,r) = R_n Y_@{lm@}}. |
| @end deftypefun |
| |
| @node Coulomb Wave Functions |
| @subsection Coulomb Wave Functions |
| |
| The Coulomb wave functions @math{F_L(\eta,x)}, @math{G_L(\eta,x)} are |
| described in Abramowitz & Stegun, Chapter 14. Because there can be a |
| large dynamic range of values for these functions, overflows are handled |
| gracefully. If an overflow occurs, @code{GSL_EOVRFLW} is signalled and |
| exponent(s) are returned through the modifiable parameters @var{exp_F}, |
| @var{exp_G}. The full solution can be reconstructed from the following |
| relations, |
| @tex |
| \beforedisplay |
| $$ |
| \eqalign{ |
| F_L(\eta,x) &= fc[k_L] * \exp(exp_F)\cr |
| G_L(\eta,x) &= gc[k_L] * \exp(exp_G)\cr |
| \cr |
| F_L'(\eta,x) &= fcp[k_L] * \exp(exp_F)\cr |
| G_L'(\eta,x) &= gcp[k_L] * \exp(exp_G) |
| } |
| $$ |
| \afterdisplay |
| @end tex |
| @ifinfo |
| |
| @example |
| F_L(eta,x) = fc[k_L] * exp(exp_F) |
| G_L(eta,x) = gc[k_L] * exp(exp_G) |
| |
| F_L'(eta,x) = fcp[k_L] * exp(exp_F) |
| G_L'(eta,x) = gcp[k_L] * exp(exp_G) |
| @end example |
| |
| @end ifinfo |
| @noindent |
| |
| @deftypefun int gsl_sf_coulomb_wave_FG_e (double @var{eta}, double @var{x}, double @var{L_F}, int @var{k}, gsl_sf_result * @var{F}, gsl_sf_result * @var{Fp}, gsl_sf_result * @var{G}, gsl_sf_result * @var{Gp}, double * @var{exp_F}, double * @var{exp_G}) |
| This function computes the Coulomb wave functions @math{F_L(\eta,x)}, |
| @c{$G_{L-k}(\eta,x)$} |
| @math{G_@{L-k@}(\eta,x)} and their derivatives |
| @math{F'_L(\eta,x)}, |
| @c{$G'_{L-k}(\eta,x)$} |
| @math{G'_@{L-k@}(\eta,x)} |
| with respect to @math{x}. The parameters are restricted to @math{L, |
| L-k > -1/2}, @math{x > 0} and integer @math{k}. Note that @math{L} |
| itself is not restricted to being an integer. The results are stored in |
| the parameters @var{F}, @var{G} for the function values and @var{Fp}, |
| @var{Gp} for the derivative values. If an overflow occurs, |
| @code{GSL_EOVRFLW} is returned and scaling exponents are stored in |
| the modifiable parameters @var{exp_F}, @var{exp_G}. |
| @end deftypefun |
| |
| @deftypefun int gsl_sf_coulomb_wave_F_array (double @var{L_min}, int @var{kmax}, double @var{eta}, double @var{x}, double @var{fc_array}[], double * @var{F_exponent}) |
| This function computes the Coulomb wave function @math{F_L(\eta,x)} for |
| @math{L = Lmin \dots Lmin + kmax}, storing the results in @var{fc_array}. |
| In the case of overflow the exponent is stored in @var{F_exponent}. |
| @end deftypefun |
| |
| @deftypefun int gsl_sf_coulomb_wave_FG_array (double @var{L_min}, int @var{kmax}, double @var{eta}, double @var{x}, double @var{fc_array}[], double @var{gc_array}[], double * @var{F_exponent}, double * @var{G_exponent}) |
| This function computes the functions @math{F_L(\eta,x)}, |
| @math{G_L(\eta,x)} for @math{L = Lmin \dots Lmin + kmax} storing the |
| results in @var{fc_array} and @var{gc_array}. In the case of overflow the |
| exponents are stored in @var{F_exponent} and @var{G_exponent}. |
| @end deftypefun |
| |
| @deftypefun int gsl_sf_coulomb_wave_FGp_array (double @var{L_min}, int @var{kmax}, double @var{eta}, double @var{x}, double @var{fc_array}[], double @var{fcp_array}[], double @var{gc_array}[], double @var{gcp_array}[], double * @var{F_exponent}, double * @var{G_exponent}) |
| This function computes the functions @math{F_L(\eta,x)}, |
| @math{G_L(\eta,x)} and their derivatives @math{F'_L(\eta,x)}, |
| @math{G'_L(\eta,x)} for @math{L = Lmin \dots Lmin + kmax} storing the |
| results in @var{fc_array}, @var{gc_array}, @var{fcp_array} and @var{gcp_array}. |
| In the case of overflow the exponents are stored in @var{F_exponent} |
| and @var{G_exponent}. |
| @end deftypefun |
| |
| @deftypefun int gsl_sf_coulomb_wave_sphF_array (double @var{L_min}, int @var{kmax}, double @var{eta}, double @var{x}, double @var{fc_array}[], double @var{F_exponent}[]) |
| This function computes the Coulomb wave function divided by the argument |
| @math{F_L(\eta, x)/x} for @math{L = Lmin \dots Lmin + kmax}, storing the |
| results in @var{fc_array}. In the case of overflow the exponent is |
| stored in @var{F_exponent}. This function reduces to spherical Bessel |
| functions in the limit @math{\eta \to 0}. |
| @end deftypefun |
| |
| @node Coulomb Wave Function Normalization Constant |
| @subsection Coulomb Wave Function Normalization Constant |
| |
| The Coulomb wave function normalization constant is defined in |
| Abramowitz 14.1.7. |
| |
| @deftypefun int gsl_sf_coulomb_CL_e (double @var{L}, double @var{eta}, gsl_sf_result * @var{result}) |
| This function computes the Coulomb wave function normalization constant |
| @math{C_L(\eta)} for @math{L > -1}. |
| @end deftypefun |
| |
| @deftypefun int gsl_sf_coulomb_CL_array (double @var{Lmin}, int @var{kmax}, double @var{eta}, double @var{cl}[]) |
| This function computes the Coulomb wave function normalization constant |
| @math{C_L(\eta)} for @math{L = Lmin \dots Lmin + kmax}, @math{Lmin > -1}. |
| @end deftypefun |
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