Special functions in daily use

“Special functions” is an intimidating name for something mundane: they are just the solutions of a few common differential equations, computed once and for all by our predecessors, numbered and tabulated. After separating variables in the Schrödinger equation for the standard potentials, the leftover ordinary differential equations are the same handful over and over — and each corresponds to one family of special functions. This section is organised family by family: which equation it solves, how it is defined, its recurrence relations, what the first few look like, and which chapter of the main text uses it.

First, a master index:

Family	Physical setting	Appears in
Hermite $H_n$	Harmonic-oscillator eigenstates	2.9
Legendre $P_\ell$ and spherical harmonics $Y_\ell^m$	Angular momentum; the angular part of any central potential	5.1, 5.2
Laguerre $L_n^k$	Hydrogen radial wavefunctions	6.3
Spherical Bessel $j_\ell,n_\ell$	Free-particle radial solutions, 3D scattering	6.1, 6.2

Hermite polynomials

The problem they solve: the stationary equation of the harmonic oscillator. Substituting the dimensionless variable $\xi=\sqrt{m\omega/\hbar}\,x$ and peeling off the Gaussian decay factor $\psi=h(\xi)\,\ee^{-\xi^2/2}$, $h$ satisfies the Hermite equation

$$h''-2\xi h'+2n h=0\tag{A.3.1}$$

Only when $n$ is a non-negative integer is the solution a polynomial (otherwise the series never terminates and $\psi$ diverges) — this is exactly where the quantisation of the oscillator levels $E_n=(n+\tfrac12)\hbar\omega$ comes from.

Definition (Rodrigues formula — one line generates the whole family):

$$H_n(\xi)=(-1)^n\,\ee^{\xi^2}\frac{\dd^n}{\dd\xi^n}\ee^{-\xi^2}\tag{A.3.2}$$

Recurrence and derivative (numerics and hand calculation both rely on these two lines — no need to memorise explicit forms):

$$H_{n+1}=2\xi H_n-2nH_{n-1},\qquad H_n'=2nH_{n-1}\tag{A.3.3}$$

The first few:

$n$	$H_n(\xi)$
0	$1$
1	$2\xi$
2	$4\xi^2-2$
3	$8\xi^3-12\xi$
4	$16\xi^4-48\xi^2+12$

Orthogonality (with a Gaussian weight function):

$$\int_{-\infty}^{\infty}H_m H_n\,\ee^{-\xi^2}\dd\xi=2^n n!\sqrt{\pi}\,\delta_{mn}\tag{A.3.4}$$

From this follow the normalised oscillator eigenstates $\psi_n(x)=\left(\frac{m\omega}{\pi\hbar}\right)^{1/4}\frac{1}{\sqrt{2^n n!}}H_n(\xi)\,\ee^{-\xi^2/2}$. $H_n$ is a polynomial of degree $n$ with parity $(-1)^n$ and $n$ real zeros — correspondingly $\psi_n$ has $n$ nodes.

Legendre polynomials and spherical harmonics

The problem they solve: the angular equation obtained after separating variables in any central potential $V(r)$ — it does not involve $V$, so the angular solutions are universal. The $\theta$ equation (setting $u=\cos\theta$) is the Legendre equation

$$\frac{\dd}{\dd u}\!\left[(1-u^2)\frac{\dd P}{\dd u}\right]+\ell(\ell+1)P=0\tag{A.3.5}$$

Requiring the solution to stay finite at the poles $u=\pm1$ forces $\ell=0,1,2,\dots$ — the origin of angular-momentum quantisation.

Definition and recurrence:

$$P_\ell(u)=\frac{1}{2^\ell \ell!}\frac{\dd^\ell}{\dd u^\ell}(u^2-1)^\ell, \qquad (\ell+1)P_{\ell+1}=(2\ell+1)\,u\,P_\ell-\ell P_{\ell-1}\tag{A.3.6}$$

The first few: $P_0=1$, $P_1=u$, $P_2=\tfrac12(3u^2-1)$, $P_3=\tfrac12(5u^3-3u)$.

Adding the index $m$ gives the associated Legendre functions $P_\ell^m(u)=(1-u^2)^{|m|/2}\,\dd^{|m|}P_\ell/\dd u^{|m|}$; pairing these with the $\varphi$-direction factor $\ee^{\ii m\varphi}$ and a normalisation constant yields the spherical harmonics — the common eigenfunctions of the angular-momentum operators $\hat L^2,\hat L_z$:

$$\hat L^2Y_\ell^m=\ell(\ell+1)\hbar^2\,Y_\ell^m, \qquad \hat L_zY_\ell^m=m\hbar\,Y_\ell^m, \qquad m=-\ell,\dots,\ell\tag{A.3.7}$$

The most frequently used ones (section 5.2 has 3D pictures):

$Y_\ell^m$	Expression
$Y_0^0$	$\sqrt{1/4\pi}$
$Y_1^0$	$\sqrt{3/4\pi}\,\cos\theta$
$Y_1^{\pm1}$	$\mp\sqrt{3/8\pi}\,\sin\theta\,\ee^{\pm \ii\varphi}$
$Y_2^0$	$\sqrt{5/16\pi}\,(3\cos^2\theta-1)$

Orthonormality (integrated over the whole sphere):

$$\int_0^{2\pi}\!\!\int_0^{\pi}Y_{\ell'}^{m'*}\,Y_\ell^m\,\sin\theta\,\dd\theta\,\dd\varphi=\delta_{\ell\ell'}\delta_{mm'}\tag{A.3.8}$$

The $p_x,p_y,p_z$ orbitals of chemistry are just real linear combinations of $Y_1^{\pm1},Y_1^0$.

Laguerre polynomials

The problem they solve: the hydrogen radial equation. After peeling off the long-range decay $\ee^{-r/na}$ and the short-range behaviour $r^\ell$, the remaining polynomial part satisfies the associated Laguerre equation. The series-termination condition delivers the principal quantum number $n$ — this is where the hydrogen levels $E_n=-13.6\,\text{eV}/n^2$ come from (full derivation in section 6.3).

Definition (Rodrigues formula; $L_q^k$ is a polynomial of degree $q$):

$$L_q(x)=\frac{\ee^{x}}{q!}\frac{\dd^q}{\dd x^q}\!\left(x^q\ee^{-x}\right), \qquad L_q^{k}(x)=(-1)^k\frac{\dd^k}{\dd x^k}L_{q+k}(x)\tag{A.3.9}$$

The first few: $L_0=1$, $L_1=1-x$, $L_2=1-2x+\tfrac{x^2}{2}$; $L_0^1=1$, $L_1^1=2-x$.

How the hydrogen radial wavefunction is assembled ($a$ is the Bohr radius):

$$R_{n\ell}(r)\;\propto\;\left(\frac{2r}{na}\right)^{\ell} L_{n-\ell-1}^{2\ell+1}\!\left(\frac{2r}{na}\right)\ee^{-r/na}\tag{A.3.10}$$

The polynomial part has $n-\ell-1$ zeros — exactly the number of radial nodes.

Spherical Bessel functions

The problem they solve: the radial equation in regions of zero (or constant) potential. After setting $u=rR$, the radial equation becomes $R''+\frac{2}{r}R'+\left[k^2-\frac{\ell(\ell+1)}{r^2}\right]R=0$, whose solutions are the spherical Bessel functions $j_\ell(kr)$ (regular at the origin) and the spherical Neumann functions $n_\ell(kr)$ (divergent at the origin).

Low-order explicit expressions (all trigonometric functions over powers — hand-computable):

$\ell$	$j_\ell(x)$	$n_\ell(x)$
0	$\dfrac{\sin x}{x}$	$-\dfrac{\cos x}{x}$
1	$\dfrac{\sin x}{x^2}-\dfrac{\cos x}{x}$	$-\dfrac{\cos x}{x^2}-\dfrac{\sin x}{x}$

Recurrence ($f_\ell$ stands for either $j_\ell$ or $n_\ell$ — same recurrence for both):

$$f_{\ell+1}(x)=\frac{2\ell+1}{x}f_\ell(x)-f_{\ell-1}(x)\tag{A.3.11}$$

Behaviour at the two ends (deciding which solution to keep hinges entirely on these two lines):

$$j_\ell(x)\xrightarrow{x\to0}\frac{x^\ell}{(2\ell+1)!!},\qquad j_\ell(x)\xrightarrow{x\to\infty}\frac{\sin(x-\ell\pi/2)}{x}\tag{A.3.12}$$

$n_\ell$ diverges at the origin like $x^{-\ell-1}$, so a region containing the origin can only use $j_\ell$; the sinusoidal oscillation at large distance (phase shifted by $\ell\pi/2$) is the reference zero for the “phase shift” concept in the partial-wave treatment of 3D scattering.

∎Key formulas

• Hermite recurrence

  $$H_{n+1}=2\xi H_n-2nH_{n-1},\qquad H_0=1,\ H_1=2\xi$$

  Harmonic oscillator; ψₙ has n nodes and parity (−1)ⁿ

• Legendre recurrence

  $$(\ell+1)P_{\ell+1}=(2\ell+1)uP_\ell-\ell P_{\ell-1}$$

  Angular equation of central potentials; finiteness at the poles ⇒ integer ℓ

• Spherical-harmonic eigenequations

  $$\hat L^2Y_\ell^m=\ell(\ell+1)\hbar^2Y_\ell^m,\quad \hat L_zY_\ell^m=m\hbar Y_\ell^m$$

  The angular solution shared by every central potential

• Hydrogen radial solution

  $$R_{n\ell}\propto(2r/na)^{\ell}\,L_{n-\ell-1}^{2\ell+1}(2r/na)\,\ee^{-r/na}$$

  Number of radial nodes = n − ℓ − 1

• Spherical Bessel asymptotics

  $$j_\ell\xrightarrow{x\to0}\frac{x^\ell}{(2\ell+1)!!},\qquad j_\ell\xrightarrow{x\to\infty}\frac{\sin(x-\ell\pi/2)}{x}$$

  Regions containing the origin discard nℓ; the far-field phase is the zero point for partial-wave phase shifts

?Self-check3 questions

1. 1.

   How many nodes does the third excited state ψ₃ of the harmonic oscillator have, and what is its parity?

   • 3 nodes, odd parity
   • 3 nodes, even parity
   • 2 nodes, odd parity
   • 4 nodes, even parity

   Submit
2. 2.

   Why must the radial solution in a region containing r = 0 use jℓ alone, with no admixture of nℓ?

   • nℓ diverges at the origin, making the wavefunction non-normalisable or failing the equation
   • nℓ is not a solution of the radial equation
   • jℓ has lower energy
   • It is a mere convention — either choice would do

   Submit
3. 3.

   How many radial nodes does the hydrogen 3p state (n = 3, ℓ = 1) have?

   • 1
   • 2
   • 3
   • 0

   Submit