# Use of Operators in Quantum Mechanics

I understand the form of operators in use for quantum mechanics such as the momentum operator: $$\hat{\text{P}}=-ih\frac{d}{dx}$$ My question is in what ways can I use it and what am I getting back? For example: if I simply apply th momentum operator to the wave function $$-ih\frac{d}{dx}\Psi$$ Will I get an Equation that will provide the momentum for a given position? Or is that a useless mathematical thing I just did?

If I use it to get an "expected value" by $$\langle \Psi | \hat{\text{ P }} | \Psi \rangle =\int_{-\infty}^\infty \Psi^* \hat{\text{ P } }\Psi$$ am I getting a number representing the probable momentum of that area integrated for? Or the percentage of total momentum there? Essentially is it a probability (if so of what kind?) or a value for the momentum?

I'm trying to understand these basic things because it has always remained unclear. I'm using it to find the momentum of and electron INSIDE the potential energy barrier as it is tunneling, i.e. the electrons between a surface and a Scanning Tunneling Microscope.