Matrix elements of position operator in infinite well basis

[carllacan]

Homework Statement

Find the eigenfunctions of a particle in a infinite well and express the position operator in the basis of said functions.

Homework Equations

The Attempt at a Solution

Tell me if I'm right so far (the |E> are the eigenkets)
$X_{ij}= \langle E_i \vert \hat{X} \vert E_j \rangle = \int dx \int dx' \langle E_i \vert x \rangle \langle x \vert \hat{X} \vert x'\rangle \langle x'\vert E_j \rangle$
$\int dx \int dx' \Psi_i^*(x) x\delta_{x, x'} \Psi_j(x') = \int dx \Psi_i^*(x) x \Psi_j(x)$

 

[maajdl]

Modify your notation for the eigenfunctions.
Using E to denote an eigenfunction is very confusing.
You can read on wikipedia about the eigenfunctions for an infinite well:
http://en.wikipedia.org/wiki/Particle_in_a_box
If I guessed your notations correctly, you are starting in the right direction.

 

[carllacan]

The eigenvalues Ei are only used to denote their respective eigenkets, the wavefunctions are denoted by the index of the eigenvalue. I think I've seen this in many books.

The last integral is correct then? I was not sure how to handle the delta.

 

[maajdl]

Yes it is correct.
By definition, I would say.
It looks more like a change of notation rather than any derivation of something!

$$X_{ij}= \langle \Psi_i \vert \hat{X} \vert \Psi_j \rangle$$

Which is indeed the integral you have written!

The state is independent of its representation.
Therefore I see no need to distinguish between

$$\langle \Psi_i \vert \hat{X} \vert \Psi_j \rangle$$

and

$$\langle E_i \vert \hat{X} \vert E_j \rangle$$

 

[carllacan]

Yes, you are right, but I first needed to state it in terms of the wavefunctions, and I wasn't quite sure if I had done it right.

Thanks.