Is My Quantum Mechanics Probability Calculation Correct?

[facenian]

Homework Statement

Let \psi(x,y,z)=\psi(\vec{r}) be the normalized wave function of a particle.Express in terms of \psi(\vec{r}) the probability for a simultaneous measurements o X y P_z to yield :
x_1 \leq x \leq x_2
p_z \geq 0

Homework Equations

<\vec{p}|\vec{r}>=\frac{1}{(2\pi\hbar)^{3/2}}e^{-i\vec{p}.\vec{r}/\hbar}
<\vec{p}|\psi>=\frac{1}{(2\pi\hbar)^{3/2}}\int \psi(\vec{r}) e^{-i\vec{p}.\vec{r}/\hbar} dr^3

The Attempt at a Solution

I have reached the following result:
\int_{-\infty}^{\infty}dz\int_{-\infty}^{\infty}dy\int_{x_1}^{x_2}dx \int_{-\infty}^{\infty}dp_x\int_{-\infty}^{\infty}dp_y\int_0^{\infty}dp_z <\vec{p}|\vec{r}>\psi(\vec{r})<\psi|\vec{p}>
I need to know two things: 1) is my result correct? 2) in case it is correct, is there any other more simple or concrete answer?

 

[elduderino]

Sorry, I am not answering your question. But could you explain how you arrived at the expression &lt;\vec{p}|\vec{r}&gt;\psi(\vec{r})&lt;\psi|\vec{p}&gt; <br />

 

[facenian]

Sorry, I am not answering your question. But could you explain how you arrived at the expression &lt;\vec{p}|\vec{r}&gt;\psi(\vec{r})&lt;\psi|\vec{p}&gt; <br />

I evalueted de expression &lt;\psi|P_2P_1|\psi&gt; where P_1 and P_2 are the proyectors:
P_1=\int_{-\infty}^{\infty}dz\int_{-\infty}^{\infty}dy\int_{x_1}^{x_2}dx|x,y,z&gt;&lt;x,y,z|
P_2=\int_{-\infty}^{\infty}dp_x\int_{-\infty}^{\infty}dp_y\int_0^{\infty}dp_z|p_x,p_y,p_z&gt;&lt;p_x,p_y,p_z|
But I'm sure whether what I'm doing is correct