Special Mathematical Functions

Hermite polynomials

Note

What? Solution of the 1D Harmonic Oscillator
Where? Chapter 7: The Linear Harmonic Oscillator
Wave-function: $\psi(y)=h(y) e^{-y^{2} / 2}$ where $h(y)=\sum_{k=0}^{k_{max}} a_{k} y^{k}$
Recursion relation: $\frac{a_{k+2}}{a_{k}}=\frac{2 k+1-\varepsilon}{(k+2)(k+1)}$ ( $a_{k+2}=0 \text { for } k\ge n=\frac{\varepsilon-1}{2}$ )

Spherical harmonics

Note

What? Solution of the angular part of the Schrödinger equation of a central potential
Where? Chapter 9: Translational and Rotational Symmetry in the Two-Body Problem
Wave-function: $Y_{l, m}(\theta, \phi)=\frac{(-1)^{l}}{2^{l} l !} \sqrt{\frac{(2 l+1)(l+m) !}{4 \pi(l-m) !}} e^{i m \phi} \frac{1}{\sin ^{m} \theta} \frac{d^{l-m}}{d(\cos \theta)^{l-m}} \sin ^{2 l} \theta$
The first few solutions are:

$\begin{aligned} Y_{0,0}(\theta, \phi) &=\sqrt{\frac{1}{4 \pi}} \\ Y_{1,\pm 1}(\theta, \phi) &=\mp \sqrt{\frac{3}{8 \pi}} e^{\pm i \phi} \sin \theta \\ Y_{1,0}(\theta, \phi) &=\sqrt{\frac{3}{4 \pi}} \cos \theta \\ Y_{2,\pm 2}(\theta, \phi) &=\sqrt{\frac{15}{32 \pi}} e^{\pm 2 i \phi} \sin ^{2} \theta \\ Y_{2,\pm 1}(\theta, \phi) &=\mp \sqrt{\frac{15}{8 \pi}} e^{\pm i \phi} \sin \theta \cos \theta \\ Y_{2,0}(\theta, \phi) &=\sqrt{\frac{5}{16 \pi}}\left(3 \cos ^{2} \theta-1\right) \end{aligned}$

Legendre Polynomial

To be added in future update

Laguerre Polynomial

To be added in future update

Spherical Bessel Functions

$\begin{array}{l} j_{0}(\rho)=\frac{\sin \rho}{\rho} \\ j_{1}(\rho)=\frac{\sin \rho}{\rho^{2}}-\frac{\cos \rho}{\rho} \\ j_{2}(\rho)=\left(\frac{3}{\rho^{3}}-\frac{1}{\rho}\right) \sin \rho-\frac{3 \cos \rho}{\rho^{2}} \end{array}$

Spherical Neumann Functions

$\begin{array}{l} \eta_{0}(\rho)=-\frac{\cos \rho}{\rho} \\ \eta_{1}(\rho)=-\frac{\cos \rho}{\rho^{2}}-\frac{\sin \rho}{\rho} \\ \eta_{2}(\rho)=-\left(\frac{3}{\rho^{3}}-\frac{1}{\rho}\right) \cos \rho-\frac{3 \sin \rho}{\rho^{2}} \end{array}$