Why are wavefunctions represented as eigenvectors?

[Superposed_Cat]

Hi I was learning about eigenvectors, inner products, Dirac notation etc. But I don't get why wave functions are represented as eigenvectors?

 

[Simon Bridge]

...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.

 

[Superposed_Cat]

could you elaborate with an example please?

 

[Vanadium 50]

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.

 

[Simon Bridge]

I'm with Vanadium 50.
There are plenty of examples in standard texts in linear algebra - you should have seen them already.

 

[Superposed_Cat]

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.

 

[Simon Bridge]

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.

 

[Superposed_Cat]

Thanks, I didn't know that vectors=functions in a way.

 

[Simon Bridge]

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.

 

[Superposed_Cat]

Thanks again, is this the basis for the linear algebra formulation of QM?

 

[Simon Bridge]

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.

 

[Khashishi]

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.