Eigenfunctions and Eigenvalues

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Discussion Overview

The discussion revolves around the concepts of eigenfunctions and eigenvalues, particularly their physical meaning and interactions. Participants explore these concepts in the context of mathematics and quantum mechanics, seeking clarity on their applications and implications.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Homework-related

Main Points Raised

  • One participant suggests starting with a two-dimensional perspective, using a Cartesian coordinate system to illustrate linear transformations and their relation to eigenvectors and eigenvalues.
  • Another participant mentions that eigenvalues and vectors provide information about functions in differential equations.
  • A participant explains that eigenfunctions in quantum mechanics are a generalization of eigenvectors, noting that measurements cause the state to collapse to one of these eigenfunctions.
  • Some participants express that textbooks on quantum mechanics can be confusing, indicating a need for simpler explanations or specific questions to clarify understanding.

Areas of Agreement / Disagreement

Participants express varying levels of understanding and approaches to the concepts, with no consensus on a singular explanation or interpretation. Multiple viewpoints and methods of explanation are presented, indicating ongoing exploration and uncertainty.

Contextual Notes

Some limitations include the complexity of quantum mechanics texts and the varying levels of familiarity with mathematical concepts among participants. The discussion does not resolve all assumptions or definitions related to eigenfunctions and eigenvalues.

Who May Find This Useful

This discussion may be useful for individuals seeking to understand eigenfunctions and eigenvalues, particularly in the contexts of mathematics and quantum mechanics, as well as those looking for simpler explanations or clarifications on these topics.

kipper
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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
 
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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.
 
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
 
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
 

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