Quantum Mechanics: Expression for electron density

[handsomecat]

I'm a little confused about the expression for the electron density.

For example in Marder's Condensed Matter Physics, he writes this

\rho(\vec{r}) = N \int \Psi^{*} \delta( \vec{r} - \vec{r}_1) \Psi d\vec{r}_1 \ldots d\vec{r}_N \deltawhat's going on?

 

[nbo10]

What are you confused about? If you're just getting started in QM you might want to pick up Griffiths book.

 

[olgranpappy]

I'm a little confused about the expression for the electron density.

For example in Marder's Condensed Matter Physics, he writes this

\rho(\vec{r}) = N \int \Psi^{*} \delta( \vec{r} - \vec{r}_1) \Psi d\vec{r}_1 \ldots d\vec{r}_N \deltawhat's going on?

He's taking the many-electron wavefunction, squaring it, and integrating over all but one of the coordinates.
I.e.,
<br /> \rho(\vec r) \equiv N \int d^3r_2 d^3r_3 d^3r_4\ldots d^3r_N |\Psi(\vec r,\vec r_2,\vec r_3,\vec r_4,\ldots,\vec r_N)|^2<br />
it desn't matter which one is singled out because the wavefunction is either symmetric or antisymmetric in all it's coordinates and thus the squared wavefunction is symmetric in it's coordinates. multiplication by N is because we want rho normalized such that
\int d^3r \rho = N
rather than normalized to 1.

 

[handsomecat]

Ah thank you! So am I right to say that this integral

<br /> \int d^3r_2 d^3r_3 d^3r_4\ldots d^3r_N |\Psi(\vec r,\vec r_2,\vec r_3,\vec r_4,\ldots,\vec r_N)|^2<br />

is just the probability of finding an electron in any volume element?