DiracHydrogen.hpp

#pragma once
#include "qip/StrongType.hpp"
 
//! @brief Exact relativistic hydrogen-like (Coulomb) wavefuntions
/*!
@details in form phi = (1/r)(f O, ig O')
 
From: H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One-and Two-Electron
Atoms (Plenum, New York, 1977).
 
Note: Uses some numerically unstable functions, including Gamma functions and
confluent hypergeometric functions. So, for some inputs, may be numerically
unstable. For reasonable inputs (i.e., Zeff=5, up to n=~10), good to at least
parts in 10^12
 
*/
namespace DiracHydrogen {
 
//------------------------------------------------------------------------------
// Uses Strong Types:
enum class DiracTypes { DiracQN, AlphaFS, Zeff, PrincipalQN, RaB };
 
using DiracQN = qip::StrongType<DiracTypes::DiracQN, int>;
using AlphaFS = qip::StrongType<DiracTypes::AlphaFS, double>;
using Zeff = qip::StrongType<DiracTypes::Zeff, double>;
// double (allow eff):
using PrincipalQN = qip::StrongType<DiracTypes::PrincipalQN, double>;
using RaB = qip::StrongType<DiracTypes::RaB, double>;
 
//------------------------------------------------------------------------------
 
//! Relativistic factor gamma = Sqrt[k^2 - (aZ)^2]
double gamma(DiracQN k, Zeff z, AlphaFS a);
 
//! Energy, without rest mass
double enk(PrincipalQN n, DiracQN k, Zeff z, AlphaFS a);
 
//! Enk = enk + c^2
double Enk(PrincipalQN n, DiracQN k, Zeff z, AlphaFS a);
 
//! Upper radial component
double f(RaB r, PrincipalQN n, DiracQN k, Zeff z, AlphaFS a);
 
//! Lower (small) radial component
double g(RaB r, PrincipalQN n, DiracQN k, Zeff z, AlphaFS a);
 
double gfratio(double r, int k, double z, double a, double e, double m = 1.0);
 
} // namespace DiracHydrogen