§33.22 Particle Scattering and Atomic and Molecular Spectra

§33.22(i) Schrödinger Equation

With $e$ denoting here the elementary charge, the Coulomb potential between two point particles with charges $Z_{1}e,Z_{2}e$ and masses $m_{1},m_{2}$ separated by a distance $s$ is $V(s)=Z_{1}Z_{2}e^{2}/(4\pi\varepsilon_{0}s)=Z_{1}Z_{2}\alpha\hbar c/s$, where $Z_{j}$ are atomic numbers, $\varepsilon_{0}$ is the electric constant, $\alpha$ is the fine structure constant, and $\hbar$ is the reduced Planck’s constant. The reduced mass is $m=m_{1}m_{2}/(m_{1}+m_{2})$, and at energy of relative motion $E$ with relative orbital angular momentum $\ell\hbar$, the Schrödinger equation for the radial wave function $w(s)$ is given by

 33.22.1 $\left(-\frac{\hbar^{2}}{2m}\left(\frac{{\mathrm{d}}^{2}}{{\mathrm{d}s}^{2}}-% \frac{\ell(\ell+1)}{s^{2}}\right)+\frac{Z_{1}Z_{2}\alpha\hbar c}{s}\right)w=Ew,$

With the substitutions

 33.22.2 $\displaystyle{\sf k}$ $\displaystyle=(2mE/\hbar^{2})^{1/2},$ $\displaystyle Z$ $\displaystyle=mZ_{1}Z_{2}\alpha c/\hbar,$ $\displaystyle x$ $\displaystyle=s,$ ⓘ Defines: $\mathsf{k}$: scaling (locally) Symbols: $x$: real variable, $Z_{k}$: charges, $m_{k}$: masses, $s$: separation and $E$: energy Permalink: http://dlmf.nist.gov/33.22.E2 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for 33.22(i), 33.22 and 33

(33.22.1) becomes

 33.22.3 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}x}^{2}}+\left({\sf k}^{2}-\frac{2Z}{x}-% \frac{\ell(\ell+1)}{x^{2}}\right)w=0.$

§33.22(ii) Definitions of Variables

${\sf k}$ Scaling

The ${\sf k}$-scaled variables $\rho$ and $\eta$ of §33.2 are given by

 33.22.4 $\displaystyle\rho$ $\displaystyle=s(2mE/\hbar^{2})^{1/2},$ $\displaystyle\eta$ $\displaystyle=Z_{1}Z_{2}\alpha c(m/(2E))^{1/2}.$

At positive energies $E>0$, $\rho\geq 0$, and:

Positive-energy functions correspond to processes such as Rutherford scattering and Coulomb excitation of nuclei (Alder et al. (1956)), and atomic photo-ionization and electron-ion collisions (Bethe and Salpeter (1977)).

At negative energies $E<0$ and both $\rho$ and $\eta$ are purely imaginary. The negative-energy functions are widely used in the description of atomic and molecular spectra; see Bethe and Salpeter (1977), Seaton (1983), and Aymar et al. (1996). In these applications, the $Z$-scaled variables $r$ and $\epsilon$ are more convenient.

$Z$ Scaling

The $Z$-scaled variables $r$ and $\epsilon$ of §33.14 are given by

 33.22.5 $\displaystyle r$ $\displaystyle=-Z_{1}Z_{2}(mc\alpha/\hbar)s,$ $\displaystyle\epsilon$ $\displaystyle=E/(Z_{1}^{2}Z_{2}^{2}m{c^{2}}{\alpha^{2}}/2).$ ⓘ Symbols: $r$: real variable, $\epsilon$: real parameter, $Z_{k}$: charges, $m_{k}$: masses, $s$: separation and $E$: energy Referenced by: §33.22(ii) Permalink: http://dlmf.nist.gov/33.22.E5 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 33.22(ii), 33.22(ii), 33.22 and 33

For $Z_{1}Z_{2}=-1$ and $m=m_{e}$, the electron mass, the scaling factors in (33.22.5) reduce to the Bohr radius, $a_{0}=\hbar/(m_{e}c\alpha)$, and to a multiple of the Rydberg constant,

$R_{\infty}=m_{e}c{\alpha^{2}}/(2\hbar)$.

$\mathrm{i}{\sf k}$ Scaling

The $\mathrm{i}{\sf k}$-scaled variables $z$ and $\kappa$ of §13.2 are given by

 33.22.6 $\displaystyle z$ $\displaystyle=2\mathrm{i}s(2mE/\hbar^{2})^{1/2},$ $\displaystyle\kappa$ $\displaystyle=\mathrm{i}Z_{1}Z_{2}\alpha c(m/(2E))^{1/2}.$ ⓘ Defines: $z$: variable (locally) and $\kappa$: variable (locally) Symbols: $Z_{k}$: charges, $m_{k}$: masses, $s$: separation and $E$: energy Permalink: http://dlmf.nist.gov/33.22.E6 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 33.22(ii), 33.22(ii), 33.22 and 33

Customary variables are $(\epsilon,r)$ in atomic physics and $(\eta,\rho)$ in atomic and nuclear physics. Both variable sets may be used for attractive and repulsive potentials: the $(\epsilon,r)$ set cannot be used for a zero potential because this would imply $r=0$ for all $s$, and the $(\eta,\rho)$ set cannot be used for zero energy $E$ because this would imply $\rho=0$ always.

§33.22(iii) Conversions Between Variables

 33.22.7 $\displaystyle r$ $\displaystyle=-\eta\rho,$ $\displaystyle\epsilon$ $\displaystyle=1/\eta^{2},$ $Z$ from ${\sf k}$.
 33.22.8 $\displaystyle z$ $\displaystyle=2\mathrm{i}\rho,$ $\displaystyle\kappa$ $\displaystyle=\mathrm{i}\eta,$ $\mathrm{i}{\sf k}$ from ${\sf k}$. ⓘ Symbols: $\rho$: nonnegative real variable, $\eta$: real parameter and $\mathsf{k}$: scaling Permalink: http://dlmf.nist.gov/33.22.E8 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 33.22(iii), 33.22 and 33
 33.22.9 $\displaystyle\rho$ $\displaystyle=z/(2\mathrm{i}),$ $\displaystyle\eta$ $\displaystyle=\kappa/\mathrm{i},$ ${\sf k}$ from $\mathrm{i}{\sf k}.$ ⓘ Symbols: $\rho$: nonnegative real variable, $\eta$: real parameter and $\mathsf{k}$: scaling Permalink: http://dlmf.nist.gov/33.22.E9 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 33.22(iii), 33.22 and 33
 33.22.10 $\displaystyle r$ $\displaystyle=\kappa z/2,$ $\displaystyle\epsilon$ $\displaystyle=-1/\kappa^{2},$ $Z$ from $\mathrm{i}{\sf k}$. ⓘ Symbols: $r$: real variable, $\epsilon$: real parameter, $Z_{k}$: charges and $\mathsf{k}$: scaling Permalink: http://dlmf.nist.gov/33.22.E10 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 33.22(iii), 33.22 and 33
 33.22.11 $\displaystyle\eta$ $\displaystyle=\pm\epsilon^{-1/2},$ $\displaystyle\rho$ $\displaystyle=-r/\eta,$ ${\sf k}$ from $Z$. ⓘ Symbols: $r$: real variable, $\rho$: nonnegative real variable, $\epsilon$: real parameter, $\eta$: real parameter, $Z_{k}$: charges and $\mathsf{k}$: scaling Referenced by: §33.22(iii) Permalink: http://dlmf.nist.gov/33.22.E11 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 33.22(iii), 33.22 and 33
 33.22.12 $\displaystyle\kappa$ $\displaystyle=\pm(-\epsilon)^{-1/2},$ $\displaystyle z$ $\displaystyle=2r/\kappa,$ $\mathrm{i}{\sf k}$ from $Z$. ⓘ Symbols: $r$: real variable, $\epsilon$: real parameter, $Z_{k}$: charges and $\mathsf{k}$: scaling Referenced by: §33.22(iii) Permalink: http://dlmf.nist.gov/33.22.E12 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 33.22(iii), 33.22 and 33

Resolution of the ambiguous signs in (33.22.11), (33.22.12) depends on the sign of $Z/\mathsf{k}$ in (33.22.3). See also §§33.14(ii), 33.14(iii), 33.22(i), and 33.22(ii).

§33.22(iv) Klein–Gordon and Dirac Equations

The relativistic motion of spinless particles in a Coulomb field, as encountered in pionic atoms and pion-nucleon scattering (Backenstoss (1970)) is described by a Klein–Gordon equation equivalent to (33.2.1); see Barnett (1981a). The motion of a relativistic electron in a Coulomb field, which arises in the theory of the electronic structure of heavy elements (Johnson (2007)), is described by a Dirac equation. The solutions to this equation are closely related to the Coulomb functions; see Greiner et al. (1985).

§33.22(v) Asymptotic Solutions

The Coulomb solutions of the Schrödinger and Klein–Gordon equations are almost always used in the external region, outside the range of any non-Coulomb forces or couplings.

For scattering problems, the interior solution is then matched to a linear combination of a pair of Coulomb functions, $F_{\ell}\left(\eta,\rho\right)$ and $G_{\ell}\left(\eta,\rho\right)$, or $f\left(\epsilon,\ell;r\right)$ and $h\left(\epsilon,\ell;r\right)$, to determine the scattering $S$-matrix and also the correct normalization of the interior wave solutions; see Bloch et al. (1951).

For bound-state problems only the exponentially decaying solution is required, usually taken to be the Whittaker function $W_{-\eta,\ell+\frac{1}{2}}\left(2\rho\right)$. The functions $\phi_{n,\ell}(r)$ defined by (33.14.14) are the hydrogenic bound states in attractive Coulomb potentials; their polynomial components are often called associated Laguerre functions; see Christy and Duck (1961) and Bethe and Salpeter (1977).

§33.22(vi) Solutions Inside the Turning Point

The penetrability of repulsive Coulomb potential barriers is normally expressed in terms of the quantity $\rho/({F_{\ell}^{2}}\left(\eta,\rho\right)+{G_{\ell}^{2}}\left(\eta,\rho\right))$ (Mott and Massey (1956, pp. 63–65)). The WKBJ approximations of §33.23(vii) may also be used to estimate the penetrability.

§33.22(vii) Complex Variables and Parameters

The Coulomb functions given in this chapter are most commonly evaluated for real values of $\rho$, $r$, $\eta$, $\epsilon$ and nonnegative integer values of $\ell$, but they may be continued analytically to complex arguments and order $\ell$ as indicated in §33.13.

Examples of applications to noninteger and/or complex variables are as follows.

• Scattering at complex energies. See for example McDonald and Nuttall (1969).

• Searches for resonances as poles of the $S$-matrix in the complex half-plane $\Im\sf{k}<0$. See for example Csótó and Hale (1997).

• Regge poles at complex values of $\ell$. See for example Takemasa et al. (1979).

• Eigenstates using complex-rotated coordinates $r\to re^{\mathrm{i}\theta}$, so that resonances have square-integrable eigenfunctions. See for example Halley et al. (1993).

• Solution of relativistic Coulomb equations. See for example Cooper et al. (1979) and Barnett (1981b).

• Gravitational radiation. See for example Berti and Cardoso (2006).

For further examples see Humblet (1984).