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.
Answer
The first thing you did is useless--- multiplying by the momentum doesn't do very much. But if you multiply by functions of the momentum, you can do things like project out the part of the state with a certain momentum.
The momentum operator is most important because if you find its eigenvectors and eignevalues, these are the states of definite momentum.
The expected value is the average of many measurements of the momentum--- it is the average value of momentum measurements. It is given by the expression you wrote down, but only when you integrate over all space. You can't restrict the range of integration to find the momentum in a limited region.
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