Calculation of probabilities in QM

[facenian]

I posted this question in the homework page but I got no answers, so will try here.
Given the wave function of a (spinless) particle I need to expres in terms of $\psi(\vec{r})$ the probability for simultaneous measurements of X y P_z to yield :

$$x_1 \leq x \leq x_2$$
$$p_z \geq 0$$

I got the result:
$$\int_{-\infty}^{\infty}dz\int_{-\infty}^{\infty}dy\int_{x_1}^{x_2}dx \int_{-\infty}^{\infty}dp_x\int_{-\infty}^{\infty}dp_y\int_0^{\infty}dp_z <\vec{p}|\vec{r}>\psi(\vec{r})<\psi|\vec{p}>$$

To get this I evaluated the expression $<\psi|P_2P_1|\psi>$ where P_1 and P_2 are the proyectors:
$$P_1=\int_{-\infty}^{\infty}dz\int_{-\infty}^{\infty}dy\int_{x_1}^{x_2}dx|x,y,z><x,y,z|$$
$$P_2=\int_{-\infty}^{\infty}dp_x\int_{-\infty}^{\infty}dp_y\int_0^{\infty}dp_z|p_x,p_y,p_z><p_x,p_y,p_z|$$

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?

 

[Avodyne]

1) Yes.
2) Yes.

 

[facenian]

1) Yes.
2) Yes.

Thank you Avodyne, could you be so kind as to explain a little how to get an easier or more concrete answer?

 

[Avodyne]

Well, notice that if it wasn't for the pz question, you could do everything with the position-space wave function. So you really only need to Fourier transform on z. Your version has a lot of extra Fourier-transforming and untransforming in x and y.

 

[javierR]

You can go further by writing <p|r> and <Psi|p> in a function form like you did with <x|Psi>. Generally, you want to put things in a single representation, like the position space, so you might ask yourself how far you can go to do that.

If you have a free particle, e.g., then Psi(r)=Psi_x Psi_y Psi_z, and similarly for the momentum space wavefunction. Can you see how doing the above steps would make things much more simplified?

 

[facenian]

Thank you both. I think what javier says is simple to write explictly the expresions for <p|r> and <psi|p>
in space representation but that would only complicate even more the final expression(sorry if I misunderstood you)
So will try Avodyne's approach and I will very much appretiate any comments on this :
Since we need to express the final result in terms of $\psi(\vec{r})$ only, we will change momentarily from X,Y,Z to the C.S.C.O X,Y,P_z. The desired probabilty is $<\psi|P_n|\psi>$

where
$$P_n=\int_{-\infty}^\infty dy\int_{x_1}^{x_2}dx\int_0^\infty dp_z |x,y,p_z><x,y,p_z|$$
$$<\psi|P_n|\psi>=\int_{-\infty}^\infty dy \int_{x_1}^{x_2}dx \int_0^\infty dp_z |<x,y,p_z|\psi>|^2$$
where
$$<x,y,p_z|\psi>=\int<x,y,p_z|\vec{r'}><\vec{r'}|\psi>d^3r'=\int<x,y,p_z|x',y',z'><x',y',z'|\psi>d^3r'$$
$$=\int\delta(x-x')\delta(y-y')<p_z|z'>\psi(\vec{r'})d^3r'=\frac{1}{\sqrt{2\pi\hbar}}\int_{-\infty}^\infty dz' \psi(x,y,z')e^{-ip_z z'/\hbar}$$
is this what you meant Avodyne?

 

[Avodyne]

Yep!

 

[facenian]

Thank you Avodyne! I really appreate your help because a have no other means to deal with this matters.