# Meaning of wave function squared, notational confusion

For a non-degenerate ground state in a system with $$N$$ electrons, we may write the wave function as,

$$\begin{equation} \psi(r_1, r_2,..., r_N) \end{equation}$$

Where the $$r_i$$ represent the position of the $$i^{th}$$ electron. When we say the square of this wave function represents the probability of finding those $$N$$ electrons at $$r_1,r_2,..., r_N$$, it becomes the probability of finding the entire system in that particular configuration of electronic positions.

My question: what about the probability of finding an electron at any position $$r$$? The definition (notation) above makes me think we have a particular value of probability which is the same at each of those $$r_1, r_2,..., r_N$$ and $$0$$ elsewhere. That thought seems limited in scope.

• Which electron are you talking about? The wavefunction you have written stands for the entire system, not just one electron. – Yejus Jan 2 at 10:31
• Hi @Yejus, I was reading about the electron density in density functional theory and got curious if we could write the electron density in terms of this collective wavefunction. Weirdly, I asked it in a roundabout way 😅. – Hitanshu Sachania Jan 2 at 16:15

For simplicity, let us take a 2-electorn case with wavefunction $$\psi(r_1, r_2)$$ and ignore the spin. Then the electron density is $$\rho(r) = \int dr_1 |\psi(r_1, r)|^2 + \int dr_2 |\psi(r,r_2)|^2 = 2 \int dr_2 |\psi(r, r_2)|^2$$ The last equality holds because the wavefunction is anti-symmetric $$\psi(r_1,r_2) = - \psi(r_2, r_1)$$. (In the general case, anti-symmetry involves exchanging all coordinates including the spin)
• The example I gave explicitly defines the electron density in terms of the collective wave function $\psi(r_1, r_2)$, so I'm not sure what do you mean with your comment. Are you asking about the electron density of several electrons? If so, then you can in general define the $n$-electron density by integrating over the coordinates of all electrons except $n$ ones. – Quantum-Collapse Jan 2 at 16:30
• Is $\psi(r_1,r)$ the same as $\psi(r_1,r_2)$? Also, can you help me understand what does $\psi(r_1,r)$ mean? I thought a single eectron wave function would be denoted as $\psi(r_1)$ and a collective wave function as $\psi(r_1,r_2)$. – Hitanshu Sachania Jan 2 at 16:52
• $\psi(r_1, r)$ and $\psi(r_1, r_2)$ are the same function evaluated at two different points...they give the same value when $r=r_2$. $\psi(r_1,r)$ is the wavefunction value when the first electron is at $r_1$ and the second electron is at $r$. – Quantum-Collapse Jan 2 at 17:00