The nature of, and notations and common vocabulary for, the eigenvalues and eigenfunctions of self-adjoint second order differential operators is overviewed in §1.18. This material is employed below with little additional discussion. While non-normalizable continuum, or scattering, states are mentioned, with appropriate references in what follows, focus is on the $L^2$ eigenfunctions corresponding to the point, or discrete, spectrum, and representing bound rather than scattering states, these former being expressed in terms of OP’s or EOP’s.

The fundamental quantum Schrödinger operator, also called the Hamiltonian, $\mathscr{H}$, is a second order differential operator of the form

18.39.7		$$\mathscr{H}=-\frac{{\mathrm{\hslash }}^2}{2m}\frac{{\partial }^2}{{\partial x}^2}+V(x),$$
$x\in (a,b)$ $(ℝ)$,
ⓘ Symbols: $\in$: element of, $(a,b)$: open interval, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$, $\partial x$: partial differential of $x$, $ℝ$: real line, $m$: nonnegative integer and $x$: real variable Referenced by: §18.39(i), §18.39(ii), Erratum (V1.2.0) §18.39 Permalink: http://dlmf.nist.gov/18.39.E7 Encodings: TeX, pMML, png See also: Annotations for §18.39(i), §18.39(i), §18.39 and Ch.18

where $x$ is a spatial coordinate, $m$ the mass of the particle with potential energy $V(x)$, $\mathrm{\hslash }=h/(2\pi )$ is the reduced Planck’s constant, and $(a,b)$ a finite or infinite interval.

Here the term $-\frac{{\mathrm{\hslash }}^2}{2m}\frac{{\partial }^2}{{\partial x}^2}$ represents the quantum kinetic energy of a single particle of mass $m$, and $V(x)$ its potential energy. As in classical dynamics this sum is the total energy of the one particle system. The properties of $V(x)$ determine whether the spectrum, this being the set of eigenvalues of $\mathscr{H}$, is discrete, continuous, or mixed, see §1.18. Below we consider two potentials with analytically known eigenfunctions and eigenvalues where the spectrum is entirely point, or discrete, with all eigenfunctions being $L^2$ and forming a complete set. Also presented are the analytic solutions for the $L^2$, bound state, eigenfunctions and eigenvalues of the Morse oscillator which also has analytically known non-normalizable continuum eigenfunctions, thus providing an example of a mixed spectrum.

However, in the remainder of this section will will assume that the spectrum is discrete, and that the eigenfunctions of $\mathscr{H}$ form a discrete, normed, and complete basis for a Hilbert space.

These eigenfunctions are the orthonormal eigenfunctions of the time-independent Schrödinger equation

18.39.8		$$\mathscr{H}{\psi }_n(x)=\left(-\frac{{\mathrm{\hslash }}^2}{2m}\frac{d^2}{{dx}^2}+V(x)\right){\psi }_n(x)={ϵ}_n{\psi }_n(x),$$
$n=0,1,2,\mathrm{\dots },$
ⓘ Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable Referenced by: (18.39.17), (18.39.18), §18.39(i), §18.39(i), §18.39(i), §18.39(ii) Permalink: http://dlmf.nist.gov/18.39.E8 Encodings: TeX, pMML, png See also: Annotations for §18.39(i), §18.39(i), §18.39 and Ch.18

which in one dimensional systems are typically non-degenerate, namely there is only a single eigenfunction corresponding to each ${ϵ}_n$, $n\ge 0$. The solutions of (18.39.8) are subject to boundary conditions at $a$ and $b$. The ${ϵ}_n$ are the observable energies of the system, and an increasing function of $n$. ${ϵ}_0$ is referred to as the ground state, all others, $n=1,2,\mathrm{\dots },$ in order of increasing energy being excited states. These eigenfunctions are quantum wave-functions whose absolute values squared give the probability density of finding the single particle at hand at position $x$ in the $n$th eigenstate, namely that probability is $P(x\to x+\mathrm{\Delta }x)$ = ${|{\psi }_n(x)|}^2\mathrm{\Delta }(x)$, $\mathrm{\Delta }(x)$ being a localized interval on the $x$-axis.

$\mathscr{H}$ also controls time evolution of the wave function $\mathrm{\Psi }(x,t)$ via the time-dependent Schrödinger equation,

18.39.9		$$\mathrm{i}\mathrm{\hslash }\frac{\partial \mathrm{\Psi }(x,t)}{\partial t}=\mathscr{H}\mathrm{\Psi }(x,t),$$
ⓘ Symbols: $\mathrm{i}$: imaginary unit, $(a,b)$: open interval, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$, $\partial x$: partial differential of $x$, $t$: real variable and $x$: real variable Referenced by: §18.39(i) Permalink: http://dlmf.nist.gov/18.39.E9 Encodings: TeX, pMML, png See also: Annotations for §18.39(i), §18.39(i), §18.39 and Ch.18

where $V(x)$ is assumed to be independent of time.

The solutions (18.39.8) are called the stationary states as separation of variables in (18.39.9) yields solutions of form

18.39.10		$$\mathrm{\Psi }(x,t)=\exp \left(-\mathrm{i}{ϵ}_{n}t/\mathrm{\hslash }\right){\psi }_n(x),$$
ⓘ Symbols: $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $(a,b)$: open interval, $t$: real variable, $n$: nonnegative integer and $x$: real variable Permalink: http://dlmf.nist.gov/18.39.E10 Encodings: TeX, pMML, png See also: Annotations for §18.39(i), §18.39(i), §18.39 and Ch.18

in which case the probability density is time-independent, as ${|\mathrm{\Psi }(x,t)|}^2=\mathrm{\Psi }(x,t)\overline{\mathrm{\Psi }(x,t)}={|{\psi }_n(x)|}^2$.

If $\mathrm{\Psi }(x,t=0)=\chi (x)$ is an arbitrary unit normalized function in the domain of $\mathscr{H}$ then, by self-adjointness,

18.39.11		$\chi (x)$	$={\displaystyle \sum _{n=0}^{\mathrm{\infty }}}{c}_n{\psi }_n(x),$
		$c_n$	$=⟨\chi ,{\psi }_n⟩,$
ⓘ Symbols: $⟨f,g⟩$: inner product over functions, $n$: nonnegative integer and $x$: real variable Permalink: http://dlmf.nist.gov/18.39.E11 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §18.39(i), §18.39(i), §18.39 and Ch.18

and

18.39.12		$$\mathrm{\Psi }(x,t)=\sum _{n=0}^{\mathrm{\infty }}{c}_n\exp \left(-\mathrm{i}{ϵ}_{n}t/\mathrm{\hslash }\right){\psi }_n(x),$$
ⓘ Symbols: $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $(a,b)$: open interval, $t$: real variable, $n$: nonnegative integer and $x$: real variable Permalink: http://dlmf.nist.gov/18.39.E12 Encodings: TeX, pMML, png See also: Annotations for §18.39(i), §18.39(i), §18.39 and Ch.18

which is the quantum superposition principle. Such a superposition yields continuous time evolution of the probability density ${|\mathrm{\Psi }(x,t)|}^2$.

For further details about the Schrödinger equation, including applications in physics and chemistry, see Gottfried and Yan (2004) and Pauling and Wilson (1985), respectively, among many others.

Here are three examples of solutions for (18.39.8) for explicit choices of $V(x)$ and with the ${\psi }_n(x)$ corresponding to the discrete spectrum. All are written in the same form as the product of three factors: the square root of a weight function $w(x)$, the corresponding OP or EOP, and constant factors ensuring unit normalization. Brief mention of non-unit normalized solutions in the case of mixed spectra appear, but as these solutions are not OP’s details appear elsewhere, as referenced.

argument a) The Harmonic Oscillator

18.39.13		$$V(x)=k{x}^2/2$$
ⓘ Symbols: $k$: nonnegative integer and $x$: real variable Referenced by: Figure 18.39.1, Figure 18.39.1, §18.39(ii) Permalink: http://dlmf.nist.gov/18.39.E13 Encodings: TeX, pMML, png See also: Annotations for §18.39(i), §18.39(i), §18.39 and Ch.18

defines the potential for a symmetric restoring force $-kx$ for displacements from $x=0$. Then $\omega =2\pi \nu =\sqrt{k/m}$ is the circular frequency of oscillation (with $\nu$ the ordinary frequency), independent of the amplitude of the oscillations. By Table 18.3.1#12 the normalized stationary states and corresponding eigenvalues are

18.39.14		${\psi }_n(x)$	$={\left(c{h}_n\right)}^{-1/2}{w}^{1/2}(cx)H_n\left(cx\right),$
		$c$	$={(mk)}^{1/4}{\mathrm{\hslash }}^{-1/2}$,
	$n=0,1,\mathrm{\dots }$,
ⓘ Symbols: $H_n\left(x\right)$: Hermite polynomial, $w(x)$: weight function, $k$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $h_n$ Referenced by: §18.39(i) Permalink: http://dlmf.nist.gov/18.39.E14 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §18.39(i), §18.39(i), §18.39 and Ch.18
18.39.15		${ϵ}_n$	$=\mathrm{\hslash }\sqrt{k/m}\left(n+\frac{1}{2}\right).$
ⓘ Symbols: $k$: nonnegative integer, $m$: nonnegative integer and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/18.39.E15 Encodings: TeX, pMML, png See also: Annotations for §18.39(i), §18.39(i), §18.39 and Ch.18

Here the $H_n\left(x\right)$ are Hermite polynomials, $w(x)={\mathrm{e}}^{-x^2}$, and $h_n=2^{n}n!\sqrt{\pi }$. With the normalization factor ${\left(c{h}_n\right)}^{-1/2}$ the ${\psi }_n$ are orthonormal in $L^2(ℝ,dx)$. The spectrum is entirely discrete as in §1.18(v).

argument b) The Morse Oscillator

18.39.16		$$V(x)=D{\mathrm{e}}^{-\alpha (x-x_e)}\left({\mathrm{e}}^{-\alpha (x-x_e)}-2\right)$$
ⓘ Symbols: $\mathrm{e}$: base of natural logarithm and $x$: real variable Source: Morse (1929) Referenced by: (18.39.17), (18.39.18) Permalink: http://dlmf.nist.gov/18.39.E16 Encodings: TeX, pMML, png See also: Annotations for §18.39(i), §18.39(i), §18.39 and Ch.18

allows anharmonic, or amplitude dependent, frequencies of oscillation about $x_e$, and also escape of the particle to $x=+\mathrm{\infty }$ with dissociation energy $D$. The orthonormal stationary states and corresponding eigenvalues are then of the form

18.39.17		${\psi }_n(x)$	$=\sqrt{{\displaystyle \frac{\alpha (2\lambda -2n-1)n!}{\mathrm{\Gamma }\left(2\lambda -n\right)}}}{\varphi }_n(\alpha (x-x_e);\lambda ),$
		$\lambda$	$={\displaystyle \frac{\sqrt{2mD}}{\alpha \mathrm{\hslash }}}$,
	$n=0,1,\mathrm{\dots },N=\lceil \lambda -\frac{3}{2}\rceil$,
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\lceil x\rceil$: ceiling of $x$, $!$: factorial (as in $n!$), $N$: positive integer, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable Proved: Nieto and Simmons (1979, §II)(proved) Proof sketch: Compare (18.39.8) with (18.39.16) substituted with (18.34.7_1), (18.34.7_2) and (18.34.7_3). Permalink: http://dlmf.nist.gov/18.39.E17 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §18.39(i), §18.39(i), §18.39 and Ch.18

and the corresponding eigenvalues are

18.39.18		$${ϵ}_n=-\frac{{(\alpha \mathrm{\hslash })}^2}{2m}{\left(\lambda -n-\frac{1}{2}\right)}^2.$$
ⓘ Symbols: $m$: nonnegative integer and $n$: nonnegative integer Proved: Nieto and Simmons (1979, §II)(proved) Proof sketch: Compare (18.39.8), with (18.39.16) substituted with (18.34.7_1) and (18.34.7_2). Referenced by: §18.39(i) Permalink: http://dlmf.nist.gov/18.39.E18 Encodings: TeX, pMML, png See also: Annotations for §18.39(i), §18.39(i), §18.39 and Ch.18

The functions ${\varphi }_n$ are expressed in terms of Romanovski–Bessel polynomials, or Laguerre polynomials by (18.34.7_1). The finite system of functions ${\psi }_n$ is orthonormal in $L^2(ℝ,dx)$, see (18.34.7_3). The spectrum is mixed as in §1.18(viii), with the discrete eigenvalues given by (18.39.18) and the continuous eigenvalues by ${(\alpha \mathrm{\hslash }\gamma )}^2/(2m)$ ($\gamma \ge 0$) with corresponding eigenfunctions ${\mathrm{e}}^{\alpha (x-x_e)/2}{W}_{\lambda ,\mathrm{i}\gamma }\left(2\lambda {\mathrm{e}}^{-\alpha (x-x_e)}\right)$ expressed in terms of Whittaker functions (13.14.3). The corresponding eigenfunction transform is a generalization of the Kontorovich–Lebedev transform §10.43(v), see Faraut (1982, §IV).

c) A Rational SUSY Potential argument

The Schrödinger equation with potential

18.39.19		$$V(x)=\frac{1}{2}{x}^2+\frac{3}{2}+\frac{8(4x^2-2)}{{\left(4x^2+2\right)}^2},$$
ⓘ Symbols: $x$: real variable Referenced by: Figure 18.39.1, Figure 18.39.1, §18.39(i) Permalink: http://dlmf.nist.gov/18.39.E19 Encodings: TeX, pMML, png See also: Annotations for §18.39(i), §18.39(i), §18.39 and Ch.18

and $\mathrm{\hslash }=k=m=1$, has eigenfunctions

18.39.20		$${\widehat{\psi }}_{n+3}(x)=\sqrt{w(x)}{\widehat{H}}_{n+3}\left(x\right),$$
$n=-3,0,1,2,\mathrm{\dots }$,
ⓘ Symbols: ${\widehat{H}}_n\left(x\right)$: exceptional Hermite polynomial, $w(x)$: weight function, $n$: nonnegative integer and $x$: real variable Permalink: http://dlmf.nist.gov/18.39.E20 Encodings: TeX, pMML, png See also: Annotations for §18.39(i), §18.39(i), §18.39 and Ch.18

and eigenvalues $n+3$, with $n$ as above, with $w(x)$ the weight function of (18.36.10), and ${\widehat{H}}_{n+3}\left(x\right)$ a type III Hermite EOP defined by (18.36.8) and (18.36.9). There is no need for a normalization constant here, as appropriate constants already appear in §18.36(vi). The spectrum is entirely discrete as in §1.18(v).

An important, and perhaps unexpected, feature of the EOP’s is now pointed out by noting that for 1D Schrödinger operators, or equivalent Sturm-Liouville ODEs, having discrete spectra with $L^2$ eigenfunctions vanishing at the end points, in this case $\pm \mathrm{\infty }$ see Simon (2005c, Theorem 3.3, p. 35), such eigenfunctions satisfy the Sturm oscillation theorem. Namely the $k$th eigenfunction, listed in order of increasing eigenvalues, starting at $k=0$, has precisely $k$ nodes, as real zeros of wave-functions away from boundaries are often referred to. This seems odd at first glance as ${\widehat{H}}_{n+3}\left(x\right)$ is a polynomial of order $n+3$ for $n=0,1,2,\mathrm{\dots }$, seemingly suggesting that for $n=0$, this being the first excited state, i.e., $k=1$, there might be 3 nodes, rather than Sturm’s 1. This is illustrated in Figure 18.39.1 where the first and fourth excited state eigenfunctions of the Schrödinger operator with the rationally extended harmonic potential, of (18.39.19), are shown, and compared with the first and fourth excited states of the harmonic oscillator eigenfunctions of (18.39.14) of paragraph a), above. Both satisfy Sturm’s theorem. Thus the two missing quantum numbers corresponding to EOP’s of order $1$ and $2$ of the type III Hermite EOP’s are offset in the node counts by the fact that all excited state eigenfunctions also have two missing real zeros. Kuijlaars and Milson (2015, §1) refer to these, in this case complex zeros, as exceptional, as opposed to regular, zeros of the EOP’s, these latter belonging to the (real) orthogonality integration range.

These are overviewed in §18.38(iii), and §18.36(vi), and typically involve OP’s or EOP’s.

In the case of a single electron, charge $-e$ and mass $m_e$, interacting with a fixed (infinite mass) nucleus of charge $+Ze$ at the co-ordinate origen, with the choice of SI units, $V(r)=-Z{e}^2/(4\pi {ϵ}_{0}r)$. This is Coulomb’s Law. The spectrum is mixed, as in §1.18(viii), the positive energy, non-$L^2$, scattering states are the subject of Chapter 33.

The non-relativistic Schrödinger equation describing a single, bound (negative energy) electron, in an $L^2$ eigenstate of energy $E$ is:

18.39.27		$$\left(\frac{-{\mathrm{\hslash }}^2}{2m_e}{\nabla }^2-\frac{Z{e}^2}{4\pi {ϵ}_{0}r}\right)\mathrm{\Psi }(r,\theta ,\varphi )=E\mathrm{\Psi }(r,\theta ,\varphi ).$$
ⓘ Symbols: $e$: elementary charge and $m_e$: mass of electron Referenced by: §18.39(ii) Permalink: http://dlmf.nist.gov/18.39.E27 Encodings: TeX, pMML, png See also: Annotations for §18.39(ii), §18.39(ii), §18.39 and Ch.18

In what follows the radial and spherical radial eigenfunctions corresponding to (18.39.27) are found in four different notations, with identical eigenvalues, all of which appear in the current and past mathematical and theoretical physics and chemistry literatures, regarding this central problem. Explicit normalization is given for the second, third, and fourth of these, paragraphs c) and d), below. All results are presented in Hartree atomic units, or a.u., i.e., $\mathrm{\hslash }=m_e=e^2=4\pi {ϵ}_0=1$, Mohr and Taylor (2005, Table XXX, p. 71), where the relationship of $a.u.$ to SI units is spelled out.

The functions ${\psi }_{p,l}(r)$ satisfy the equation,

18.39.28		$$\left(-\frac{1}{2}\frac{d^2}{{dr}^2}+\frac{l(l+1)}{2r^2}-\frac{Z}{r}\right){\psi }_{p,l}={ϵ}_{p,l}{\psi }_{p,l}(r),$$
ⓘ Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$ and $l$: nonnegative integer Referenced by: §18.39(ii), §18.39(ii), §18.39(iv), §18.39(iv) Permalink: http://dlmf.nist.gov/18.39.E28 Encodings: TeX, pMML, png See also: Annotations for §18.39(ii), §18.39(ii), §18.39 and Ch.18

with an infinite set of orthonormal $L^2$ eigenfunctions

18.39.29		$${\psi }_{p,l}(r)=\sqrt{\frac{Zp!}{(p+2l+1)!}}\frac{{\mathrm{e}}^{-{\rho }_p/2}{\rho }_p^{l+1}}{p+l+1}{L}_p^{(2l+1)}\left({\rho }_p\right),$$
$p=0,1,2,\mathrm{\dots }$; $l=0,1,2,\mathrm{\dots }$,
ⓘ Symbols: $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$) and $l$: nonnegative integer Referenced by: §18.39(ii), §18.39(ii), §18.39(ii) Permalink: http://dlmf.nist.gov/18.39.E29 Encodings: TeX, pMML, png See also: Annotations for §18.39(ii), §18.39(ii), §18.39 and Ch.18

$p$ here being the order of the Laguerre polynomial, $L_p^{(2l+1)}$ of Table 18.8.1, line 11, and $l$ the angular momentum quantum number, and where

18.39.30		$${\rho }_p=\frac{2Zr}{p+l+1},$$
ⓘ Symbols: $l$: nonnegative integer Referenced by: §18.39(ii) Permalink: http://dlmf.nist.gov/18.39.E30 Encodings: TeX, pMML, png See also: Annotations for §18.39(ii), §18.39(ii), §18.39 and Ch.18

with eigenvalues

18.39.31		$${ϵ}_{p,l}=\frac{-Z^2}{2{\left(p+l+1\right)}^2}.$$
ⓘ Symbols: $l$: nonnegative integer Referenced by: §18.39(iv), §18.39(ii), §18.39(ii) Permalink: http://dlmf.nist.gov/18.39.E31 Encodings: TeX, pMML, png See also: Annotations for §18.39(ii), §18.39(ii), §18.39 and Ch.18

See Titchmarsh (1962a, pp. 99–100).

Orthogonality, with measure $dr$ for $r\in [0,\mathrm{\infty })$, for fixed $l$

18.39.32		$${\int }_0^{\mathrm{\infty }}{\psi }_{p,l}(r){\psi }_{q,l}(r)dr={\delta }_{p,q},$$
ⓘ Symbols: ${\delta }_{j,k}$: Kronecker delta, $dx$: differential of $x$, $\int$: integral, $q$: real variable and $l$: nonnegative integer Permalink: http://dlmf.nist.gov/18.39.E32 Encodings: TeX, pMML, png See also: Annotations for §18.39(ii), §18.39(ii), §18.39 and Ch.18

noting that the ${\psi }_{p,l}(r)$ are real, follows from the fact that the Schrödinger operator of (18.39.28) is self-adjoint, or from the direct derivation of Dunkl (2003). This is not the orthogonality of Table 18.8.1, as the co-ordinate arguments depend, independently on $p$ and $q$. This indicates that the Laguerre polynomials appearing in (18.39.29) are not classical OP’s, and in fact, even though infinite in number for fixed $l$, do not form a complete set. Namely for fixed $l$ the infinite set labeled by $p$ describe only the $L^2$ bound states for that single $l$, omitting the continuum briefly mentioned below, and which is the subject of Chapter 33, and so an unusual example of the mixed spectra of §1.18(viii).

The fact that both the eigenvalues of (18.39.31) and the scaling of the $r$ co-ordinate in the eigenfunctions, (18.39.30), depend on the sum $p+l+1$ leads to the substitution

18.39.33		$$n=p+l+1,$$
ⓘ Symbols: $l$: nonnegative integer and $n$: nonnegative integer Referenced by: §18.39(ii) Permalink: http://dlmf.nist.gov/18.39.E33 Encodings: TeX, pMML, png See also: Annotations for §18.39(ii), §18.39(ii), §18.39 and Ch.18

where $n$ is now the Bohr Principle Quantum Number.

Physical scientists use the $n$ of Bohr as, to $0$th and $1$st order, it describes the structure and organization of the Periodic Table of the Chemical Elements of which the Hydrogen atom is only the first.

This follows from the fact that the eigenvalues (18.39.31), ${ϵ}_{n,l}={ϵ}_n=-Z^2/(2n^2)$ depend only on the single quantum number $n$, the Bohr Principal quantum number, $n=1,2,\mathrm{\dots }$, and depend explicitly neither on $l$ nor $m_l$. Thus the overall degeneracy of the solutions (18.39.29) (the number of independent eigenfunctions corresponding to a single eigenvalue (18.39.31) for all values of $l$ and $m_l$) consistent with each $n$ is $n^2$, which controls the lengths of the rows in the Periodic Table. Interactions between electrons, in many electron atoms, breaks this degeneracy as a function of $l$, but $n$ still dominates.

The solution, (18.39.29), of the spherical radial equation (18.39.28), now expressed in terms of the Bohr quantum number $n$, is

18.39.34		$${\psi }_{n,l}(r)=\frac{1}{n}\sqrt{\frac{Z(n-l-1)!}{(n+l)!}}{\mathrm{e}}^{-{\rho }_n/2}{\rho }_n^{l+1}{L}_{n-l-1}^{(2l+1)}\left({\rho }_n\right),$$
$n=1,2,\mathrm{\dots },l=0,1,\mathrm{\dots }n-1$,
ⓘ Symbols: $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $l$: nonnegative integer and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/18.39.E34 Encodings: TeX, pMML, png See also: Annotations for §18.39(ii), §18.39(ii), §18.39 and Ch.18

18.39.35		$${\rho }_n=\frac{2Zr}{n},{ϵ}_n=-\frac{Z^2}{2n^2},$$
ⓘ Symbols: $n$: nonnegative integer Referenced by: §18.39(ii) Permalink: http://dlmf.nist.gov/18.39.E35 Encodings: TeX, pMML, png See also: Annotations for §18.39(ii), §18.39(ii), §18.39 and Ch.18

thus recapitulating, for $Z=1$, line 11 of Table 18.8.1, now shown with explicit normalization for the measure $dr$. This is also the normalization and notation of Chapter 33 for $Z=1$ , and the notation of Weinberg (2013, Chapter 2). The radial Coulomb wave functions $R_{n,l}(r)$, solutions of

18.39.36		$$\left(\frac{-1}{2r^2}\frac{d}{dr}{r}^2\frac{d}{dr}+\frac{l(l+1)}{2r^2}-\frac{Z}{r}\right)R_{n,l}(r)={ϵ}_n{R}_{n,l}(r),$$
$n=1,2,\mathrm{\dots },l=0,1,\mathrm{\dots },n-1,$
ⓘ Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $l$: nonnegative integer and $n$: nonnegative integer Referenced by: §18.39(ii), §18.8 Permalink: http://dlmf.nist.gov/18.39.E36 Encodings: TeX, pMML, png See also: Annotations for §18.39(ii), §18.39(ii), §18.39 and Ch.18

with ${\rho }_n$ and ${ϵ}_n$ being those of (18.39.35), are then

18.39.37		$$R_{n,l}(r)=\frac{2}{n^2}\sqrt{\frac{Z^3(n-l-1)!}{(n+l)!}}{\mathrm{e}}^{-{\rho }_n/2}{\rho }_n^l{L}_{n-l-1}^{(2l+1)}\left({\rho }_n\right),$$
ⓘ Symbols: $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $l$: nonnegative integer and $n$: nonnegative integer Referenced by: §18.39(ii) Permalink: http://dlmf.nist.gov/18.39.E37 Encodings: TeX, pMML, png See also: Annotations for §18.39(ii), §18.39(ii), §18.39 and Ch.18

normalized with measure $r^{2}dr$, $r\in [0,\mathrm{\infty })$.

The same solutions as in paragraph c), above, appear frequently in the literature in terms of associated Laguerre polynomials, which are referred to here as associated Coulomb–Laguerre polynomials to avoid confusion with the more recent meaning of ‘associated’ of §18.30. The associated Coulomb–Laguerre polynomials are defined as

18.39.38		$${𝐋}_p^m(\rho )=\frac{d^m}{{d\rho }^m}{𝐋}_p^0(\rho ),$$
ⓘ Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$ and $m$: nonnegative integer Permalink: http://dlmf.nist.gov/18.39.E38 Encodings: TeX, pMML, png See also: Annotations for §18.39(ii), §18.39(ii), §18.39 and Ch.18

where

18.39.39		$${𝐋}_p^0(\rho )={\mathrm{e}}^{\rho }\frac{d^p}{{d\rho }^p}({\rho }^p{\mathrm{e}}^{-\rho }),$$
ⓘ Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$ and $\mathrm{e}$: base of natural logarithm Permalink: http://dlmf.nist.gov/18.39.E39 Encodings: TeX, pMML, png See also: Annotations for §18.39(ii), §18.39(ii), §18.39 and Ch.18

see Bethe and Salpeter (1957, p. 13), Pauling and Wilson (1985, pp. 130, 131); and noting that this differs from the Rodrigues formula of (18.5.5) for the Laguerre OP’s, in the omission of an $n!$ in the denominator.

From (18.9.23) and (18.5.5) with Table 18.5.1, line 8:

18.39.40		$${𝐋}_{p+m}^m(\rho )={(-1)}^m(p+m)!L_p^{(m)}\left(\rho \right),$$
ⓘ Symbols: $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $!$: factorial (as in $n!$) and $m$: nonnegative integer Permalink: http://dlmf.nist.gov/18.39.E40 Encodings: TeX, pMML, png See also: Annotations for §18.39(ii), §18.39(ii), §18.39 and Ch.18

and thus replacing $p$ by $n-l-1$ as in Table 18.8.1, line 11, or as in (18.39.33),

18.39.41		$$L_{n-l-1}^{(2l+1)}\left({\rho }_n\right)=-{𝐋}_{n+l}^{2l+1}({\rho }_n)/(n+l)!,$$
ⓘ Symbols: $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $!$: factorial (as in $n!$), $l$: nonnegative integer and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/18.39.E41 Encodings: TeX, pMML, png See also: Annotations for §18.39(ii), §18.39(ii), §18.39 and Ch.18

(where the minus sign is often omitted, as it arises as an arbitrary phase when taking the square root of the real, positive, norm of the wave function), allowing equation (18.39.37) to be rewritten in terms of the associated Coulomb–Laguerre polynomials ${𝐋}_{n+l}^{2l+1}({\rho }_n)$.

18.39.42		$$R_{n,l}(r)=\frac{2}{n^2}\sqrt{\frac{Z^3(n-l-1)!}{{\left((n+l)!\right)}^3}}{\mathrm{e}}^{-{\rho }_n/2}{\rho }_n^l{𝐋}_{n+l}^{2l+1}({\rho }_n).$$
ⓘ Symbols: $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $l$: nonnegative integer and $n$: nonnegative integer Referenced by: §18.39(ii), §18.39(ii) Permalink: http://dlmf.nist.gov/18.39.E42 Encodings: TeX, pMML, png See also: Annotations for §18.39(ii), §18.39(ii), §18.39 and Ch.18

Derivations of (18.39.42) appear in Bethe and Salpeter (1957, pp. 12–20), and Pauling and Wilson (1985, Chapter V and Appendix VII), where the derivations are based on (18.39.36), and is also the notation of Piela (2014, §4.7), typifying the common use of the associated Coulomb–Laguerre polynomials in theoretical quantum chemistry. In all of these references these OP’s are simply referred to as the associated Laguerre OP’s.

Bound state solutions to the relativistic Dirac Equation, for this same problem of a single electron attracted by a nucleus with $Z$ protons, involve Laguerre polynomials of fractional index. A relativistic treatment becoming necessary as $Z$ becomes large as corrections to the non-relativistic Schrödinger picture are of approximate order ${\left(\alpha Z\right)}^2\sim {\left(Z/137\right)}^2$, $\alpha$ being the dimensionless fine structure constant $e^2/(4\pi {\epsilon }_0\mathrm{\hslash }c)$, where $c$ is the speed of light. See Martinez-y-Romero (2000).

The positive energy (scattering) eigenfunctions for the above Coulomb problem, with potential $V(r)=-Z{e}^2/r$ are discussed in Chapter 33 for each integer $l$. These, taken together with the infinite sets of bound states for each $l$, form complete sets. As the scattering eigenfunctions of Chapter 33, are not OP’s, their further discussion is deferred to §18.39(iv), where discretized representations of these scattering states are introduced, Laguerre and Pollaczek OP’s then playing a key role.

For the potential $V(r)=+Z{e}^2/r$, corresponding to interaction of particles with like charges, there are no bound states, the continuum scattering states form a complete set for each $l$, as discussed in Chapter 33, and their discretized versions in §18.39(iv).