PHY.K02UF Molecular and Solid State Physics

Hermite polynomials

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,

$$H_{n+1}(x) = 2xH_n(x)- 2nH_{n-1}(x).$$

$H_n(x)$ 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.

$H_n$ 

$x$