Two particle probability density

[daudaudaudau]

If I have a wave function $\Psi(x,x')$ for two identical fermions, then I have learned that the particle density at $x$ is
$$n(x)=2\int|\Psi(x,x')|^2dx'$$

$|\Psi(x,x')|^2$ is the probability density of finding a particle at $x$ and a particle at $x'$. Does this mean that $\int|\Psi(x,x')|^2dx'$ is the probability density of finding a particle at $x$ ?

 

[alxm]

Yes, the particle density and probability density are the same thing.

 

[Entropia]

Yes, that is correct. The integral of |\Psi(x,x')|^2 over all possible positions of the second particle, dx', gives the probability density of finding the first particle at x. This is because the wave function \Psi(x,x') describes the probability amplitude for finding the two particles at specific positions (x,x'). By squaring this amplitude and integrating over all possible positions of the second particle, we obtain the probability density of finding the first particle at x. This is a fundamental concept in quantum mechanics and is essential in understanding the behavior of identical particles.