Indistinguishable Particles

In Quantum Mechanics, a system of $N$ particles with internal spin degrees of freedom $\sigma_i$ is described by a wave function which in the position representation reads

$$\Psi({\bf r}_1,\sigma_1; {\bf r}_2,\sigma_2;...;{\bf r}_N,\sigma_N).$$ (1.1)

Here, $\vert\Psi(...)\vert^2$ is the probability density for finding particle 1 at ${\bf r}_1$ with spin quantum number(s) $\sigma_1$, particle 2 at ${\bf r}_2$ with spin quantum number(s) $\sigma_2$,... etc. Note that for spin $1/2$, one would choose for $\sigma_i$ one of the spin projections, e.g. $\sigma_i=\sigma_i^{(z)}=\pm \frac{1}{2}$.

Remark: Usually, many-particle wave functions and the issue of indistinguishability are discussed in the position representation.