DiracHydrogen Namespace Reference

Detailed Description

Exact relativistic hydrogen-like (Dirac-Coulomb) wavefunctions.

Analytic solutions of the single-particle Dirac equation for a pointlike nuclear charge Z (the Dirac-Coulomb problem). The orbitals have the form

\[ \psi_{n\kappa m}(\vb{r}) = \frac{1}{r} \begin{pmatrix} f_{n\kappa}(r)\,\Omega_{\kappa m}(\hat n) \\ i\,g_{n\kappa}(r)\,\Omega_{-\kappa,m}(\hat n) \end{pmatrix}, \]

with f the large (upper) and g the small (lower) radial component. Expressions follow H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One- and Two-Electron Atoms (1977).

The optional lepton mass mass defaults to 1 (atomic units). The full relativistic energy is \( E = mass\,c^2 + \en_{n\kappa} = mass/\alpha^2 + \en_{n\kappa} \), and the radial functions scale with mass as \( f_{mass}(r) = \sqrt{mass}\,f_1(mass\,r) \).

Non-relativistic limit

Passing \( \alpha \le 0 \) selects the non-relativistic (Schrodinger) limit directly, rather than evaluating the Dirac expressions at a tiny \( \alpha \) (which is numerically unstable). In this limit the large component reduces to the non-relativistic radial function (f = P_nl), the small component vanishes (g = 0), and the energy is \( \en_n = -mass\,Z^2/(2n^2) \).

Note

Uses some numerically unstable functions (Gamma functions and confluent hypergeometric functions), so may be inaccurate for extreme inputs. For reasonable inputs (Zeff = 5, up to n ~ 10) tested good to at least parts in 10^12.

Functions
double	enk (double n, int kappa, double zeff, double alpha, double mass=1.0)
	Single-particle energy, without rest mass: enk.
double	Enk (double n, int kappa, double zeff, double alpha, double mass=1.0)
	Full energy, including rest mass: Enk = mass/alpha^2 + enk.
double	gamma (int kappa, double zeff, double alpha)
	Relativistic factor gamma = Sqrt[kappa^2 - (Z*alpha)^2].
double	f (double r, double n, int kappa, double zeff, double alpha, double mass=1.0)
	Upper (large) radial component, f(r)
double	g (double r, double n, int kappa, double zeff, double alpha, double mass=1.0)
	Lower (small) radial component, g(r)
double	P_nl (double r, double n, int l, double zeff, double mass=1.0)
	Nonrelativistic bound-state radial function: P_nl(r) = r * R_nl(r)
double	gfratio (double r, int kappa, double zeff, double alpha, double en, double mass=1.0)
	Ratio g/f at radius r.

Function Documentation

enk()

double DiracHydrogen::enk	(	double	n,
		int	kappa,
		double	zeff,
		double	alpha,
		double	mass = 1.0
	)

Single-particle energy, without rest mass: enk.

Bound-state Dirac-Coulomb energy, excluding the rest-mass term:

\[ \en_{n\kappa} = mass\,c^2\left[ \left(1 + \frac{(Z\alpha)^2}{(\gamma + n - |\kappa|)^2}\right)^{-1/2} - 1 \right], \]

evaluated in a numerically stable form (avoids cancellation between the \( O(c^2) \) terms). For \( \alpha \le 0 \) returns the non-relativistic energy \( \en_n = -mass\,Z^2/(2n^2) \).

Parameters

n	Principal quantum number (may be non-integer, effective n)
kappa	Dirac quantum number (ignored in the non-relativistic limit)
zeff	Effective nuclear charge, Z
alpha	Fine-structure constant; alpha <= 0 gives non-relativistic
mass	Lepton mass, in units of the electron mass (default 1)

Returns

enk (atomic units)

Note

alpha <= 0 selects the non-relativistic limit

Enk()

double DiracHydrogen::Enk	(	double	n,
		int	kappa,
		double	zeff,
		double	alpha,
		double	mass = 1.0
	)

Full energy, including rest mass: Enk = mass/alpha^2 + enk.

\[ E_{n\kappa} = mass\,c^2 + \en_{n\kappa} = mass/\alpha^2 + \en_{n\kappa}. \]

Parameters

n	Principal quantum number (may be non-integer)
kappa	Dirac quantum number
zeff	Effective nuclear charge, Z
alpha	Fine-structure constant
mass	Lepton mass, in units of the electron mass (default 1)

Returns

Enk (atomic units)

Warning

Diverges as alpha -> 0 (rest energy mass/alpha^2); not meaningful in the non-relativistic limit.

gamma()

double DiracHydrogen::gamma	(	int	kappa,
		double	zeff,
		double	alpha
	)

Relativistic factor gamma = Sqrt[kappa^2 - (Z*alpha)^2].

\[ \gamma = \sqrt{\kappa^2 - (Z\alpha)^2}. \]

Parameters

kappa	Dirac quantum number
zeff	Effective nuclear charge, Z
alpha	Fine-structure constant

Returns

gamma

f()

double DiracHydrogen::f	(	double	r,
		double	n,
		int	kappa,
		double	zeff,
		double	alpha,
		double	mass = 1.0
	)

Upper (large) radial component, f(r)

Large (upper) component of the Dirac-Coulomb orbital (see the namespace description for the full expression). For \( \alpha \le 0 \) returns the non-relativistic large component, \( f(r) = P_{nl}(r) \) (see P_nl), with l = l(kappa).

Parameters

r	Radial coordinate (atomic units)
n	Principal quantum number (may be non-integer)
kappa	Dirac quantum number
zeff	Effective nuclear charge, Z
alpha	Fine-structure constant; alpha <= 0 gives non-relativistic
mass	Lepton mass, in units of the electron mass (default 1)

Returns

f(r)

Note

alpha <= 0 selects the non-relativistic limit: f = P_nl (and g = 0)

g()

double DiracHydrogen::g	(	double	r,
		double	n,
		int	kappa,
		double	zeff,
		double	alpha,
		double	mass = 1.0
	)

Lower (small) radial component, g(r)

Small (lower) component of the Dirac-Coulomb orbital (see the namespace description for the full expression). In the non-relativistic limit \( \alpha \le 0 \) the small component vanishes, g(r) = 0.

Parameters

r	Radial coordinate (atomic units)
n	Principal quantum number (may be non-integer)
kappa	Dirac quantum number
zeff	Effective nuclear charge, Z
alpha	Fine-structure constant; alpha <= 0 gives g = 0
mass	Lepton mass, in units of the electron mass (default 1)

Returns

g(r)

Note

alpha <= 0 selects the non-relativistic limit: g = 0

P_nl()

double DiracHydrogen::P_nl	(	double	r,
		double	n,
		int	l,
		double	zeff,
		double	mass = 1.0
	)

Nonrelativistic bound-state radial function: P_nl(r) = r * R_nl(r)

Hydrogen-like Schrodinger radial function, unit normalised: \( \int_0^\infty P_{nl}^2\,\d r = 1 \). Mass scales as \( P_{nl,mass}(r) = \sqrt{mass}\,P_{nl,1}(mass\,r) \).

Evaluated via the confluent hypergeometric form \( \rho^l e^{-\rho/2}\,{}_1F_1(l+1-n, 2l+2, \rho) \), \( \rho = 2Z\,mass\,r/n \), so n may be non-integer (effective n). For integer n this reduces to the usual associated Laguerre polynomial.

Parameters

r	Radial coordinate (atomic units)
n	Principal quantum number; may be non-integer (effective n)
l	Orbital angular momentum
zeff	Effective nuclear charge, Z
mass	Lepton mass, in units of the electron mass (default 1)

Returns

P_nl(r)

Warning

For non-integer n the series does not terminate and the function grows like \( e^{+Zr/n} \) at large r (not a normalisable bound state); it is only meaningful at finite r / near-integer n.

gfratio()

double DiracHydrogen::gfratio	(	double	r,
		int	kappa,
		double	zeff,
		double	alpha,
		double	en,
		double	mass = 1.0
	)

Ratio g/f at radius r.

Small-to-large component ratio at radius r, for a given energy en (without rest mass). Independent of the overall normalisation.

Parameters

r	Radial coordinate (atomic units)
kappa	Dirac quantum number
zeff	Effective nuclear charge, Z
alpha	Fine-structure constant
en	Single-particle energy (without rest mass)
mass	Lepton mass, in units of the electron mass (default 1)

Returns

g(r)/f(r)