how do you find the expectation value <x> for the 1st 2 states of a harmonic oscillator?
The expectation value of an operator A is <psi|A|psi>. If you're not familiar with Dirac's notation that means that for state 1 you'd integrate psi1 times A times psi1 over all space.
Once you start getting your hands dirty with the integrations pay attention to wether the integrand is even or odd. That will save you from a lot of useless intergration. Also the gamma function may prove to be useful.
Well, they are [itex]\langle \psi_0|x|\psi_0\rangle[/itex] and [itex]\langle \psi_1|x|\psi_1\rangle[/itex] ofcourse.
You could find them either by integration or the application of the ladder operators.
However, a look at the probability distributions [itex]|\psi_0|^2[/itex] and [itex]|\psi_1|^2[/itex] should tell you immediately what the expectation value for the position is.
You can do that, but if you really want to see the math, use the ladder operators.
ladder operators? what's that?
Separate names with a comma.