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The electron density used in density functional theory for a system of $N$ electrons with wavefunction $\psi$ is defined as

$$\rho(r)=N\int \Psi^*(r,r_2,\dots r_N)\Psi(r,r_2,\dots r_N) d^3r_2\dots d^3r_N$$

The interpretation of this is given as the probability of finding one of the $N$ electrons in the volume element $d^3r$. The following property also holds: $$\int \rho(r)d^3r=N$$ I do not understand this, if $\rho(r)$ is the probability density with aforementioned interpretation, its integral over all space should simply mean: "The probability of finding an electron in all space" and that should be just $1$, not $N$. How can the probability of finding an electron at any point in all space be greater than 1?

Source: A Chemist’s Guide to Density Functional Theory. Second Edition Wolfram Koch, Max C. Holthausen

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    $\begingroup$ Can you provide a source? $\endgroup$ Commented Feb 17, 2022 at 16:50

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The sum of the probabilities of all mutually exclusive events must equal one. However, just because you've found an electron somewhere, it does not mean that you can't find another somewhere else (except for $N=1$).

Reference: A Primer in Density Functional Theory by Fiolhais, C., Nogueira, F., & Marques, M. A. (Eds.), Springer, chapter 1.2. Especially the discussion below equation (1.21).

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