Quantum probabilities calculation

[facenian]

Homework Statement

In a one-dimensional problem, consider a system of two particles with which is associated
the wave function $\psi(x_1,x_2)$
a) Give the probability of finding at least one of the partichles between a and b
b)Give the probability of finding one and only one particle between a and b

Homework Equations

The Attempt at a Solution

a)
$$\int_{-\infty}^\infty dx_2 \int_a^b dx_1 |\psi(x_1,x_2)|^2 + \int_a^b dx_2\left(\int_{-\infty}^a dx_1|\psi(x_1,x_2)|^2 +\int_b^\infty dx_1|\psi(x_1,x_2)|^2 } \right)$$
b)
$$\int_a^b dx_1\left(\int_{-\infty}^a dx_2 |\psi(x_1,x_2)|^2 +\int_b^\infty dx_2 |\psi(x_1,x_2)|^2 } \right)+\int_a^b dx_2\left(\int_{-\infty}^a dx_1 |\psi(x_1,x_2)|^2 +\int_b^\infty dx_1 |\psi(x_1,x_2)|^2 } \right)$$
I need to know whether these answers are correct or incorrect and some indication in case they are incorrect

 

[Jhero]

Your solution for part a) is correct. The first term in your integral represents the probability of finding both particles between a and b, while the second term represents the probability of finding one particle between a and b and the other particle outside of that range.

For part b), your solution is incorrect. The integral should represent the probability of finding one and only one particle between a and b, which means that the other particle must be outside of that range. Your integral should therefore be:

\int_a^b dx_1\int_{-\infty}^{a} dx_2|\psi(x_1,x_2)|^2 + \int_a^b dx_1 \int_{b}^{\infty} dx_2|\psi(x_1,x_2)|^2

This integral represents the probability of finding one particle between a and b while the other particle is outside of that range.