Why are wavefunctions represented as eigenvectors?

Click For Summary

Discussion Overview

The discussion centers around the representation of wavefunctions as eigenvectors in quantum mechanics, exploring the mathematical framework and implications of this representation. Participants delve into concepts such as state-vectors, eigenvalues, and the relationship between functions and vectors, with a focus on linear algebra's role in quantum mechanics.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants suggest that wavefunctions are represented as eigenvectors because it simplifies mathematical operations, similar to resolving a velocity vector into components.
  • One participant emphasizes that wavefunctions are state-vectors and that all wavefunctions transform as vectors, equating "function" and "vector".
  • A request for examples of eigenvectors in the context of quantum mechanics is made, indicating a desire for practical applications.
  • Concerns are raised about the understanding of eigenvectors among participants, with one questioning another's familiarity with linear algebra concepts.
  • Another participant notes that functions can be considered vectors, prompting further clarification on the relationship between vectors and functions.
  • Discussion includes the idea that different representations of vectors can be useful, especially in complex systems where traditional function notation may be cumbersome.
  • One participant asserts that the linear algebra formulation of quantum mechanics arises from the linear nature of Schrödinger's equation, allowing for superposition of solutions.
  • Another participant argues that the distinction between wavefunction and state-vector formulations is primarily a matter of notation.

Areas of Agreement / Disagreement

Participants express varying levels of understanding regarding the concepts of eigenvectors and their application in quantum mechanics. While some agree on the utility of representing wavefunctions as eigenvectors for mathematical convenience, others highlight a lack of consensus on foundational knowledge and the implications of these representations.

Contextual Notes

Some participants indicate limitations in their understanding of linear algebra and its application to quantum mechanics, suggesting that further exploration of these foundational concepts may be necessary for a complete grasp of the topic.

Superposed_Cat
Messages
388
Reaction score
5
Hi I was learning about eigenvectors, inner products, Dirac notation etc. But I don't get why wave functions are represented as eigenvectors?
 
Physics news on Phys.org
...why wave functions are represented as eigenvectors?
Because it makes the math easier.

Technically - the wave-function is a state-vector.
We like to resolve them into eigenvectors (also called eigenfunctions) for the same reason we like to resolve a velocity vector into components: it makes the math easier.

All wave-functions are vectors because they transform as vectors - so "function" and "vector" mean the same thing.
 
could you elaborate with an example please?
 
Do you even know what an eigenvector is? The reason I ask is that four hours ago you asked "Hey all, does anyone a great place to learn linear algebra online? Thanks for any help. " It took me more than four hours to learn linear algebra.
 
I'm with Vanadium 50.
There are plenty of examples in standard texts in linear algebra - you should have seen them already.
 
I only learned eigenvectors, eigenvalues, dirac notation and inner products. I already knew how to work with matrices.I just haven't seen how to use them in qm.

I'm motivated. Science is my life. I don't spend my time playing games like my other friends.
 
Last edited by a moderator:
Do you know what a vector space is?
Have you seen that functions are vectors in their own right?
Or are you thinking that a vector is a column (or row) of numbers?

See also:
https://ece.uwaterloo.ca/~ece204/howtos/functions/

All the rest is just different ways of writing vectors down.
 
Last edited:
  • Like
Likes   Reactions: 1 person
Thanks, I didn't know that vectors=functions in a way.
 
Yep - after grokking that, it remains only to figure which representation of vectors is the one you want to deal with. Some of the cooler ways can look a lot like magic.

This gets very useful when you have to deal with much more complicated systems where writing down the functions normally can occupy a whole page - it can also obscure the relationships that you are interested in.
 
  • #10
Thanks again, is this the basis for the linear algebra formulation of QM?
 
  • #11
Um - it is a demonstration that there is no useful distinction to be made between the wavefunction and state-vector formulation. The formulation is just notation.
 
  • #12
We have a linear algebra formulation of QM because Schrödinger's equation is linear. That is, if A and B are solutions, then c1*A + c2*B is a solution.
 

Similar threads

Graduate What is a spatial wavefunction in QFT?
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Inner products and wavefunctions
  • · Replies 9 ·
Replies
9
Views
2K
Graduate Wavefunction in the context of quantum physics
  • · Replies 16 ·
Replies
16
Views
2K
Undergrad Why observables are represented as operators in QM?
  • · Replies 4 ·
Replies
4
Views
3K
Undergrad Inner and Outer Product of the Wavefunctions
  • · Replies 8 ·
Replies
8
Views
4K
Undergrad Density Operators of Pure States
  • · Replies 21 ·
Replies
21
Views
3K
Graduate Measuring the spin of a moving Dirac spinor particle
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Why do you need infinite size matrix which commute....
  • · Replies 2 ·
Replies
2
Views
2K
Graduate The eigenvalue power method for quantum problems
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Is the Wavefunction a Contravariant Component?
  • · Replies 47 ·
2
Replies
47
Views
3K