Matrix Element of Position Operator

[phrygian]

Homework Statement

Calculate the general matrix element of the position operator in the basis of the eigenstates of the infinite square well.

Homework Equations

$$|\psi\rangle =\sqrt{\frac{2}{a}}\sin{\frac{n \pi x}{a}}$$

$$x_{n,m}=\langle\psi_{n}|\hat{x}|\psi_{m}\rangle=\int^a_0\psi^\star_{n} x \psi_mdx$$

The Attempt at a Solution

What I tried doing is substituting the sin eigenfections into the integral and then using the relation that sin(x)sin(y)=(cos(x-y)-cos(x+y))/2 and then integrating from 0 to a. I keep getting a negative answer, am I using the wrong approach? I know this doesn't work when m=n, but shouldn't it work when they aren't equal?

Thanks for the help

 

[vela]

Sounds like you're doing it right. Why do you think a negative answer for the off-diagonal elements is wrong?

 

[phrygian]

I guess I was thinking of it as an expectation value which I now see is not correct.

When use this same method using the momentum operator, -i hbar d/dx, I get an integral with Sin(npix/a)Cos(npix/a), which must be zero when integrated from 0 to a by integral identities, correct? What does this mean then if the momentum operator is represented by a matrix with all zeros?

 

[vela]

No, it won't be all zeros. Remember you're not always integrating over a full period (depending on what m and n equal), so you can't use orthogonality to say the integrals will all be zero.