How to integrate using trapezoidal method

How would I solve the above integral numerically using trapazoidal rule? I have no idea where to start, I am only able to use basic coding language functions, so nothing fancy please.

[David Goodmanson]

Hello Candice,

could you provide more detail on the integral? Since the wave function psi(x) is a function of x only, it is a constant as far as the integral in epsilon is concerned. So psi can be pulled outside the integral, leaving an integral that does not converge.

[Candice Pillow]

I am sorry, I forgot to mention that epsilon=x/sigma. The H_n is the hermite polynomial. So psi(x) would become of a function of x and epsilon. But looking at it further, I believe it would be possible to still pull out psi of the integral since it is divided by epsilon. Psi would be squared and you'd be left with e^-epsilon for the integration. My task is to evaluate the whole expression with different values of n, after the expression is simplified.

[David Goodmanson]

This comment is not going to provide any immediate solution, but using 'a' in place of epsilon the form of the integral is

[constants]*Integral{0,2*n+1}  exp(-a^2)*psi(a)^2*(1/a^2)  da

To lowest order, the hermite polynomials of odd order are proportional to 'a' at the origin, and the hermite polynomials of even order are a constant at the origin. So for odd order the integrand goes like a constant at the origin and the integral converges. You can get an answer by setting up an array of 'a' values, calculating the integrand values and then using the trapz function. But for even order the integrand goes like a^-2 at the origin and the integral diverges.