Expectation value of the square of the observable

[v0id]

Homework Statement

I know how to compute the expectation value of an observable. But how does one compute the expectation value of an observable's square?

Homework Equations

$$\langle Q \rangle = \int_{-\infty}^{\infty} \Psi^* \hat{Q} \Psi \; dx$$
$$\langle Q^2 \rangle = \int_{-\infty}^{\infty} \Psi^* (\hat{Q} \Psi)^2 \; dx \; ?$$

 

[dextercioby]

No, the square is defined by double application of the operator

$$A^2 \psi =A\left(A \psi \right) \ , \forall \psi\in D(A) \ \mbox{and} \ {} A\psi \in D(A)$$

 

[v0id]

I tried double application after I posted and got more elegant answers. Thanks dextercioby.