# §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 Ch.33

(33.22.1) becomes

 33.22.3 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}x}^{2}}+\left({\sf k}^{2}-\frac{2Z}{x}-% \frac{\ell(\ell+1)}{x^{2}}\right)w=0.$ ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: derivative, $\ell$: nonnegative integer, $x$: real variable, $Z_{k}$: charges and $\mathsf{k}$: scaling Referenced by: §33.22(iii) Permalink: http://dlmf.nist.gov/33.22.E3 Encodings: TeX, pMML, png See also: Annotations for §33.22(i), §33.22 and Ch.33

## §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 Ch.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: $\mathrm{i}$: imaginary unit, $Z_{k}$: 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 Ch.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: $\mathrm{i}$: imaginary unit, $\rho$: 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 Ch.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: $\mathrm{i}$: imaginary unit, $\rho$: 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 Ch.33
 33.22.10 $\displaystyle r$ $\displaystyle=\kappa z/2,$ $\displaystyle\epsilon$ $\displaystyle=-1/\kappa^{2},$ $Z$ from $\mathrm{i}{\sf k}$. ⓘ Symbols: $\mathrm{i}$: imaginary unit, $r$: 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 Ch.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 Ch.33
 33.22.12 $\displaystyle\kappa$ $\displaystyle=\pm(-\epsilon)^{-1/2},$ $\displaystyle z$ $\displaystyle=2r/\kappa,$ $\mathrm{i}{\sf k}$ from $Z$. ⓘ Symbols: $\mathrm{i}$: imaginary unit, $r$: 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 Ch.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).