Two particle probability density

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The wave function \Psi(x,x') for two identical fermions leads to the particle density at position x being calculated as n(x)=2∫|\Psi(x,x')|^2dx'. The term |\Psi(x,x')|^2 represents the probability density of locating one particle at x and another at x'. The integral ∫|\Psi(x,x')|^2dx' indeed corresponds to the probability density of finding a particle at position x. Thus, particle density and probability density are effectively equivalent in this context. This highlights the relationship between wave functions and particle distributions in quantum mechanics.
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If I have a wave function \Psi(x,x') for two identical fermions, then I have learned that the particle density at x is
<br /> n(x)=2\int|\Psi(x,x&#039;)|^2dx&#039;<br />

|\Psi(x,x&#039;)|^2 is the probability density of finding a particle at x and a particle at x&#039;. Does this mean that \int|\Psi(x,x&#039;)|^2dx&#039; is the probability density of finding a particle at x ?
 
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Yes, the particle density and probability density are the same thing.
 
Time reversal invariant Hamiltonians must satisfy ##[H,\Theta]=0## where ##\Theta## is time reversal operator. However, in some texts (for example see Many-body Quantum Theory in Condensed Matter Physics an introduction, HENRIK BRUUS and KARSTEN FLENSBERG, Corrected version: 14 January 2016, section 7.1.4) the time reversal invariant condition is introduced as ##H=H^*##. How these two conditions are identical?

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