# Expected value of operator or expected value of observable?

A question about terminology. I have seen both $$\langle p\rangle$$ and $$\langle\hat{p}\rangle$$ to calculate the expected value of momentum (same thing with position, energy etc.). The first one would suggest that we take the expected value of an observable while the second would suggest that we take the expected value of an operator. Which one is correct?

In terms of the notation: $$\langle\hat{p}\rangle$$ is the correct expression, since the mathematical operations are done with the operators, which are mathematical objects, rather than with the observables, which are the quantities measured experimentally.
More specifically, $$\langle\hat{p}\rangle$$ corresponds to $$$$\langle\hat{p}\rangle = \langle\psi|\hat{p}|\psi\rangle = \int dx \psi^*(x)\hat{p}\psi(x),$$$$ if the system is described by a wave function $$\psi$$, and to $$$$\langle\hat{p}\rangle = \mathbf{Tr}[\hat{\rho}\hat{p}],$$$$ if the system is described by densisty matrix $$\hat{\rho}$$.
• @user5744148 I added a couple of equation to my answer, to underscore that $\langle\hat{p}\rangle$ has very specific mathematical meaning. Mar 26, 2020 at 10:15