 
 PHY.K02UF Molecular and Solid State Physics  
The Hermite polynomials $H_n(x)$ are a set of orthogonal polynomials that have the following properties:
$$\int\limits_{\infty}^{\infty}e^{x^2}H_n^2(x)dx = \sqrt{\pi}2^nn!,$$ $$\int\limits_{\infty}^{\infty}e^{x^2}H_n(x)H_m(x)dx = 0 \qquad n\ne m.$$The first two Hermite polynomials are:
$$H_0 = 1,$$ $$H_1=2x.$$The rest of the Hermite polynomials can be generated with the recursion relation,
$$H_{n+1}(x) = 2xH_n(x) 2nH_{n1}(x).$$$H_n(x)$ crosses zero $n$ times. The code below will calculate the the value of $H_n(x)$. You can change $n$ and $x$ in the code.

The first few Hermite polynomials are plotted below.
$H_n$  
$x$ 