Hermite Polynomial

#Math
Hermite polynomials are used for solution of the QHO

$\displaystyle H_{n}(\xi)=(-1)^{n}e^{\xi^{2}}\left( \frac{ \mathrm{d} }{ \mathrm{d}\xi } \right)^{n}e^{-\xi^{2}}$

• This formula is the Rodrigues formula
• Expands to $\displaystyle n$th order polynomials due to cancellations of $\displaystyle e^{\xi^{2}}$ with $\displaystyle e^{-\xi^{2}}$
• Example:
• $\displaystyle H_{1}(\xi)=(-1)^{1}e^{\xi^{2}}\frac{ \partial }{ \partial \xi }e^{-\xi^{2}}=-1\cdot e^{\xi^{2}}(-2\xi e^{-\xi^{2}})=2\xi$

$\displaystyle \int_{-\infty}^{\infty} H_{n}(x)H_{m}(x) e^{-x^{2}}, \mathrm{d}x=\delta_{nm}(x)2^{n}n!\sqrt{ \pi }$

• Orthogonality condition of Hermite polynomials
• $\displaystyle \delta_{nm}(x)$ is the Kronecker delta

$\displaystyle H_{n}(\xi)=\sum_{j = 0}^{\infty}a_{j}X^{j},a_{j+2}=\frac{2j-k+1}{(j+1)(j+2)}a_{j}$

• Power series/recurrence relation representation of the above equation

$\displaystyle H_{n}(z)=\frac{n!}{2\pi i}\oint e^{-t^{2}+2tz}t^{-n-1}\mathrm{d}t$

• Contour integral definition
• Wolfram article

Example Values

TableForm[Table[Simplify[(-1)^n*E^\[Xi]^2*D[E^(-\[Xi]^2), {\[Xi], n}]], {n, 4}]] (* 1st Version*)
TableForm[Table[Simplify[HermiteH[n,\[Xi]]],{n,4}]] (* Mathematica version*)