Fourier Series Problem - Representing an Even Function

In summary, the conversation discusses the eigenfunctions of a free particle in an infinite potential well in quantum mechanics and whether they form a basis. The eigenfunctions take the form of a sine function and it is questioned how an even function can be represented using odd eigenfunctions. The response clarifies that not all functions can be represented in this manner and suggests that the eigenfunctions form a basis for the set of solutions of the wave equation.
  • #1
maple
9
0
In quantum mechanics, a free particle in an infinite potential well has the wave function (ie. overlap <x/phi>). Its eigenfunctions take the form:

(2/a)^1/2 * sin(n*pi*x/a), n is ofcourse an integer.

My question is that do all eigenfunctions form a basis? And if so how can you represent an even function with eigenfunctions which are clearly odd- my understanding of the Fourier Series is that its equals a function by representing it as an infinite sum of both sin and cos terms, and if the function is even, the coefficients of the sin terms are zero.

Eg. why can I represent:

cos (pi*x/a) = Infinite Sum (A(subscript)n * sin(n*pi*x/a)).

Any assistance would be much appreciated.
 
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  • #2
You cannot represent any function with sines in the manner you hypothesize. I would guess the answer is that the Eigenfunctions form a basis of the space of the set of solutions of the wave equation. Not all functions are solutions.
 
  • #3

1. What is a Fourier series?

A Fourier series is a mathematical representation of a periodic function using a combination of sine and cosine functions. It allows us to break down a complex function into simpler components.

2. What is an even function?

An even function is a mathematical function that is symmetric about the y-axis. This means that the function outputs the same value for both positive and negative inputs, resulting in a mirror image when graphed.

3. Why is it useful to represent an even function using a Fourier series?

Using a Fourier series allows us to represent an even function using a combination of simpler trigonometric functions, making it easier to analyze and understand the behavior of the function. It also provides a way to approximate the function with a finite number of terms.

4. How do you calculate the coefficients for a Fourier series of an even function?

The coefficients for a Fourier series of an even function can be calculated using the following formula: an = (2/π) ∫-ππ f(x) cos(nx) dx, where n is the order of the coefficient and f(x) is the given even function.

5. Can a Fourier series accurately represent any even function?

Yes, a Fourier series can accurately represent any even function that is piecewise continuous and has a period of 2π. However, the accuracy of the representation may depend on the number of terms used in the series.

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