A question about probabilities

[facenian]

In need to know if my answer is correct. I tried to place this question on the homework forum but it seems that this kind of questions are not answered there.
In a one-dimensional problem, consider a system o two particles with wave function $\psi(x_1,x_2)$ the distance X_1 - X_2 between, the two particles is measured. What is the probability of finding a result included between -d and +d
My answer is :

$$\int_{-\infty}^\infty dx_1 \int_{-d}^{+d}de |\psi(x_1,x_1-e)|^2$$

I'd appreciate any comments!

 

[Fredrik]

You should also explain how you got that result.

 

[facenian]

You should also explain how you got that result.

I reason like this: d=x_2-x_1, x_2=d-x_1,the states that give the desired results are $\{|x_1,d-x_1>,\quad -\infty<x_1<\infty\}$
then mean value of the proyector on the eigen-subspace genarated by those kets$P_n=\int_{-\infty}^\infty dx_1 |x_1,d-x_1><x_1,d-x_1|$ should yield the answer
$$<\psi|P_n|\psi>=\int_{-\infty}^\infty dx_1 |<x_1,d-x_1|\psi>|^2 \Delta d$$
for the probability of $x_2-x_1$ being between $d \quad and \quad d +\Delta d$

 

[DaTario]

It seems correct to me although your explanation's first line is a little bit confuse.

best regards

DaTario