Fourier Series from Eigenfunctions

[O_chemist]

Homework Statement

"Using the eigenfunctions for the Hamiltonian of an infinite square-well potential defined over[-1,1] in the standard, dimensionless setting, construct Fourier series representation of the following functions..." the functions are e^(-100x^2), e^(-5x^2), e^(-x^2)
It also requests I find the RMS and say how many basis functions are required for a certain error but that shouldn't be too hard once I get going.

Homework Equations

The Attempt at a Solution

My question is I am a little confused as what the initial part is asking. I am confused because the eigenfunctions for the Hamiltonian are Sine and Cosine for classical mechanics but Hermite polynomials for quantum.(maybe this is wrong but I thought it was) So which ones am I supposed to use and how exactly do I construct the Fourier series from them? I know how to construct a Fourier Series in the most basic sense where you solve for the coefficients ao, an, and bn. I do know that when trying to approximate a function with the form e^(-x^2) form you end up using the error function, however, I don't think this is what the question is getting at. If someone could just help get me started with doing the Fourier Series, or maybe better, how to think about connecting this idea of using the Eigenfunctions to generating a Fourier Series. Thank you in advance.
Regards,
Caleb
PS as indicated by my name my background is mostly in Chemistry but I and doing more physics so I do have some gaps in my physics/math background which is probably why this question is confusing for me.

 

[mark726]

Dear Caleb,

Thank you for your question. It is understandable that you are confused about which eigenfunctions to use for constructing the Fourier series in this problem. In this case, we will be using the eigenfunctions of the Hamiltonian operator for the infinite square-well potential, which are a set of Hermite polynomials. These are different from the sine and cosine functions that you are familiar with for classical mechanics.

To construct the Fourier series, we need to express the given functions as a linear combination of the eigenfunctions. In this case, the eigenfunctions are Hermite polynomials, which are orthogonal to each other. This means that we can use the orthogonality property to determine the coefficients of the linear combination.

For example, for the function e^(-100x^2), we can write it as a linear combination of the Hermite polynomials as follows:

e^(-100x^2) = a0*H0(x) + a1*H1(x) + a2*H2(x) + ...

where a0, a1, a2, ... are the coefficients that we need to determine. Using the orthogonality property, we can find the coefficients by taking the inner product of both sides of the equation with each of the Hermite polynomials. This will give us a system of equations that we can solve to find the coefficients.

Once we have the Fourier series representation of the given functions, we can calculate the RMS by taking the square root of the sum of the squares of the coefficients. The number of basis functions required to achieve a certain error can be determined by gradually increasing the number of terms in the Fourier series until the error falls below the desired value.

I hope this helps to clarify your confusion and get you started with the problem. Best of luck with your work.