Expectation values for a harmonic oscillator

[KaiserBrandon]

Homework Statement

I need to find <x>, <x²>, <p>, and <p²> for a particle in the first state of a harmonic oscillator.

Homework Equations

The harmonic oscillator in the first state is described by $$\psi$$(x)=A$$\alpha$$^1/2*x*e^{-$$\alpha$$*x2/2}. I'm using the definition <Q>=($$\int$$$$\psi$$1*Q*$$\psi$$)dx where $$\psi$$1 is the complex conjugate of $$\psi$$, and Q is the specific operator.

The Attempt at a Solution

I solved for <x>, and found it was zero. <p> I'll solve for in a similar fashion. However, for <x²> and <p²>, I am unsure of what operators I use. For the <x> operator, it is simply x, so for <x²>, would I use x² as an operator?
(note that I the only superscripts here are the ones above e, I don't know why latex is putting all of my symbols so high

 

[kuruman]

For the <x> operator, it is simply x, so for <x²>, would I use x² as an operator?

Yes, that's what you should use.

 

[dx]

(Put the whole equation in the tex barackets instead of individual symbols)

The operator for <x²> is simply multiplication by x², so <x²> = ∫ψ*(x)x²ψ(x)dx, and <p²> is

$$-\int \psi^{*}(x) \hbar^2\frac{\partial^2}{\partial x^2}\psi(x) dx$$

 

[KaiserBrandon]

ok, thank you so much guys. It's a good thing I have physics forum to at least make my new ventures into the realm of quantum mechanics a bit easier :)