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:

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.

#### CODE:

#include<stdio.h> #include<math.h> 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){ if(n==0){ return h0(x); } else if(n==1){ return h1(x); } else{ return 2*x*hn(x,n-1)-2*(n-1)*hn(x,n-2); } } main(){ //We will create a data-file and store the values of first few Hermite polynomials for -1<x<5 FILE *fp=NULL; //create data-file fp=fopen("hermite1.txt","w"); double x; //write the values of first 5 Hermite polynomials to data-file for(x=-2;x<=2;x=x+0.1){ fprintf(fp,"%lf\t%lf\t%lf\t%lf\t%lf\t%lf\n",x,hn(x,0),hn(x,1),hn(x,2),hn(x,3),hn(x,4)); } }

#### OUTPUT:

The above program will create a data-file called `legendre1.txt `

and store the values of the first 5 Hermite polynomials for . Now, you can just open the file and select the data and plot it using Excel, GnuPlot, Origin, etc.

For GnuPlot, the command is:

### Using Series Representation

### Using Numerical Integration

### References:

http://mathworld.wolfram.com/HermitePolynomial.html

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.