Hermite Polynomial – C PROGRAM

In this post I’m gonna show you how to calculate Hermite polynomials using three different techniques: using recurrence relations, series representations, and numerical integration.
The programs will calculate and plot the first few Hermite polynomials.

Using Recurrence Relation

We will be using the following recurrence relation:
H_{n+1}(x)=2x H_n (x) - 2n H_{n-1}(x)
We would need two more relations, that is the relations for 0th and 1st order Legendre polynomials:
We will create a program that calculates the values of the Legendre polynomial at various x values and for different l and store these values in a txt file. Then just plot it using Gnuplot.
We will create two functions called ‘h0’ and ‘h1’, that contain the definition of respectively.
Then we will create a function ‘hn’ that will use the first two functions and recursion to find the value of Legendre polynomial for different x,n.
NOTE: I am using a slightly modified form of the recurrence relation. To get the form I am using, just replace n by n-1.



double h0(double x){
	return 1;

double h1(double x){
	return 2*x;
//The following is a general function that returns the value of the Hermite Polynomial for any given x and n=0,1,2,3,...
double hn(double x,int n){
		return h0(x);
	else if(n==1){
		return h1(x);
		return 2*x*hn(x,n-1)-2*(n-1)*hn(x,n-2); 	
	//We will create a data-file and store the values of first few Hermite polynomials for -1<x<5
	//create data-file
	double x;
	//write the values of first 5 Hermite  polynomials to data-file


The above program will create a data-file called legendre1.txt and store the values of the first 5 Hermite polynomials for -2\leq x \leq 2 . Now, you can just open the file and select the data and plot it using Excel, GnuPlot, Origin, etc.
For GnuPlot, the command is:

First few Hermite polynomials using recurrence relation

Using Series Representation

Using Numerical Integration



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One thought on “Hermite Polynomial – C PROGRAM

  1. can you please tell me how can we plot the coherent state wavefunctions or probabilities using the above given function? I am not getting a proper Gaussian curve for the probability density of harmonic oscillator coherent states.

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