# Matrix elements of position operator in infinite well basis

## Homework Statement

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

## The Attempt at a Solution

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

Last edited:

maajdl
Gold Member
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.

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
Gold Member
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

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

1 person
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.