Wave function in terms of Basis Functions

In summary, the problem is to express the function g(x)=x(x-a) \cdot e^{ikx} in the form of a Fourier series with coefficients \{a_n\} and eigenfunctions \{\psi_n(x)\} that are complete and determined by Fourier's theorem. The solution involves using Fourier's principle of orthogonality and recognizing that the set of eigenfunctions, \{\psi_n(x)\}, is complete for a particle in a 1D box.
  • #1
Domnu
178
0
Problem
We have the function [tex]g(x)=x(x-a) \cdot e^{ikx}[/tex]. Express [tex]g(x)[/tex] in the form

[tex]\sum_{n=1}^\infty a_n \psi_n (x)[/tex]

where

[tex]\psi_n = \sqrt{\frac{2}{a}} \sin \(\frac{n\pi x}{a}\)[/tex]

Solution
I have absolutely no clue as to how to start... I know a bit about Fourier series, but here, the function [tex]g(x)[/tex] has an infinite period.
 
Physics news on Phys.org
  • #2
Actually wait... we can just let [tex]g(x)[/tex] go from 0 to a, since it's supposed to model a particle in a box from potential walls 0 to a.
 
  • #3
my guess would be that you could do it from 0 to a. So, I believe the problem now is to find the expansion coefficients An. To do this you will have to use Fourier's principle of orthogonality. I'm not 100 percent sure though, but it seems reasonable.
 
  • #4
The set [itex]\{\psi_{n}(x)\}[/itex] is complete, which means you can express any (well behaved, piecewise continuous) function as a linear combination of its elements. The coefficients which act as weights are to be determined by Fourier's theorem.

These are actually eigenfunctions for a particle in a 1D box. Eigenfunctions are complete by definition.
 

1. What is a wave function?

A wave function is a mathematical function that describes the state of a quantum system. It provides information about the probability of finding a particle at a specific position or with a specific momentum.

2. What are basis functions?

Basis functions are a set of mathematical functions that are used to describe the wave function of a system. These functions serve as a basis for constructing more complex wave functions.

3. How are basis functions related to wave functions?

Basis functions are used to represent the wave function of a quantum system. The wave function is expressed as a linear combination of these basis functions, which allows for the calculation of the probability of finding a particle at a specific position or with a specific momentum.

4. What is the significance of using basis functions in quantum mechanics?

Basis functions are essential in quantum mechanics as they provide a mathematical framework for describing the behavior of quantum systems. They allow for the calculation of probabilities and the prediction of the behavior of particles in a quantum system.

5. Can different sets of basis functions be used to describe the same wave function?

Yes, different sets of basis functions can be used to describe the same wave function. This is known as basis set variation and is commonly used in quantum chemistry to improve the accuracy of calculations.

Similar threads

Replies
1
Views
734
  • Advanced Physics Homework Help
Replies
15
Views
2K
  • Advanced Physics Homework Help
Replies
4
Views
881
  • Advanced Physics Homework Help
Replies
1
Views
1K
  • Advanced Physics Homework Help
Replies
7
Views
936
  • Advanced Physics Homework Help
Replies
2
Views
1K
  • Advanced Physics Homework Help
Replies
9
Views
1K
  • Advanced Physics Homework Help
Replies
1
Views
1K
  • Advanced Physics Homework Help
Replies
7
Views
1K
  • Advanced Physics Homework Help
Replies
3
Views
1K
Back
Top