Eigenfunctions and Eigenvalues

[kipper]

Hi,
I am having a lot of difficulty conceptually understanding what eigenfunctions and eigenvalues actually are, their physical meaning, i.e. what they represent, and how they interact.
Would anybody happen to be able to explain them in relatively simple terms?
I didn't know whether to put this question in maths or here, hopefully i chose right.
Cheers
Kipper

 

[humanino]

I guess the best is to begin by thinking about it in terms of 2 dimensions. Say in a real plane, you define a cartesian orthonormal coordinate system. Now say you define a linear transformation of every vector by multiplying the x-coordinate by a number, and the y-coordinate by another number. Both x and y directions correspond to possible eigenvectors with the corresponding eigenvalues.

In general, you could have represented your vector space with another coordinate system, say just if you had rotated your axis you would still work with an orthonormal system. Then your linear transformation is no longer just multiplication of coordinates by their respective eigenvalues. Now the linear transformation is not represented by a diagonal matrix anymore.

I think it is best to work one's way up building increasingly complex examples. In addition, I am sure others will suggest different points of view.

 

[LostConjugate]

Eigenvalues and vectors give you information about the function in a differential equation.

See here:

http://tutorial.math.lamar.edu/Classes/DE/RealEigenvalues.aspx

 

[ansgar]

an eigenfunction of an operator is just a generalization of eigenvector to a matrix.

one of the postulates of QM (copenhagen interpretation) is that the state is repsented by a eigenfunctions to operators representing observables. A measurment causes the state to collapse to ONE of those eigenfunctions.

It is easier if you put a more specific question, all of this can easily be found in any textbook on QM

 

[kipper]

Hi all,
Thanks for the help,
I went through the link that was posted and that seemed to really help aswell.
I've gone through a few books on QM, but they seem to be too confusing initially.
I just need to get aquainted with it properly before i do anything major with it :P
Thanks once again for the help
Kipper

 

[Edgardo]

Hi kipper,

I hope this post helps:
https://www.physicsforums.com/showthread.php?t=61527