Linear momentum of particle

Where did the integral in your first step come from? Start with the definition of [itex]\langle \hat{P} \rangle[/itex]...

 

The average linear momentum of what state?

Remember, [itex]\psi_k(x)=e^{ikx}[/itex] is just one k-mode of the full wavefunction for a free particle [itex]\psi(x)=\int_{-\infty}^{\infty} A(k) \psi_k(x)dk [/itex].

 

Does [itex]\frac{\infty}{\infty}=1[/itex]? That is essentially what you are claiming in your last step.

As for your first step, is that the definition of average momentum you are using in your course? Usually one defines the expectation value, or average, of an operator [itex]\hat{A}[/itex] in a given state [itex]|\psi\rangle[/itex] as [itex]\langle \hat{A} \rangle \equiv \langle \psi |\hat{A}|\psi \rangle[/itex].

Frankly, I'm still not 100% clear on what the original problem is. Can you post the original problem verbatim (word for word)?

 

Just evaluate the integral [;\int_{-\infty}^{\infty} dx;]. What do you get?
Also, [; <p>= m\frac{d<x>}{dt} ;], so if you can find [;<x>;], you should also be able to find [;<p>;].