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## Homework Statement

Compute ##\left \langle x^2 \right\rangle## for the states ##\psi _0## and ##\psi _1## by explicit integration.

## Homework Equations

##\xi\equiv \sqrt{\frac{m \omega}{\hbar}}x##

##α \equiv (\frac{m \omega}{\pi \hbar})^{1/4}##

##\psi _0 = α e^{\frac{\xi ^2}{2}}##

## The Attempt at a Solution

I'm not getting the right answer here:

$$\left \langle x^2 \right\rangle= α^2\int ^{\infty} _{-\infty} \psi ^* x^2 \psi dx = α^2\frac{\hbar}{2m\omega}\int ^{\infty} _{-\infty} \psi ^* [(a_+)^2 + (a_+ a_- )+(a_- a_ )+(a_- )^2]\psi dx$$

This follows from the fact that ##x^2## can be defined by ##x=\sqrt{\frac{\hbar}{2m\omega}}(a_+ +a_- )##

Because of the orthogonality of ##\psi _n## with ##\psi _m## we can cancel the first and last terms out, leaving ##(a_+ a_-)+(a_- a_ )##

If I'm not mistaken, this should mean the integral goes to zero because ##a_{\pm}\psi _0 = 0##?

I know the correct answer is ##\frac{\hbar}{2m\omega}##

If somebody could point me in the right direction, I would appreciate it.