Quantum Mechanics: Expression for electron density

In summary, the expression for the electron density is given by \rho(\vec{r}) = N \int \Psi^{*} \delta( \vec{r} - \vec{r}_1) \Psi d\vec{r}_1 \ldots d\vec{r}_N \delta , which represents the probability of finding an electron in any volume element. This is obtained by taking the many-electron wavefunction, squaring it, and integrating over all but one of the coordinates. The multiplication by N is for normalization purposes.
  • #1
handsomecat
70
0
I'm a little confused about the expression for the electron density.

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

[tex] \rho(\vec{r}) = N \int \Psi^{*} \delta( \vec{r} - \vec{r}_1) \Psi d\vec{r}_1 \ldots d\vec{r}_N \delta [/tex]what's going on?
 
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  • #2
What are you confused about? If you're just getting started in QM you might want to pick up Griffiths book.
 
  • #3
handsomecat said:
I'm a little confused about the expression for the electron density.

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

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

He's taking the many-electron wavefunction, squaring it, and integrating over all but one of the coordinates.
I.e.,
[tex]
\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
[/tex]
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
[tex]\int d^3r \rho = N[/tex]
rather than normalized to 1.
 
  • #4
Ah thank you! So am I right to say that this integral

[tex]
\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
[/tex]

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

Related to Quantum Mechanics: Expression for electron density

1. What is the expression for electron density in quantum mechanics?

The expression for electron density in quantum mechanics is given by the square of the wave function, which represents the probability of finding an electron at a certain location in space.

2. How is the electron density related to the position of an electron?

The electron density is directly proportional to the square of the wave function, which varies in space. This means that the electron density is a measure of the probability of finding an electron at a particular location in space.

3. What is the significance of the electron density in quantum mechanics?

The electron density is a crucial concept in quantum mechanics as it allows us to understand the distribution of electrons in an atom or molecule. It also helps us to calculate various properties such as bond strengths and molecular structures.

4. How is the electron density calculated in quantum mechanics?

The electron density is calculated by solving the Schrödinger equation, which is a fundamental equation in quantum mechanics that describes the behavior of particles. This equation takes into account factors such as the potential energy of the system and the wave function of the electron.

5. Can the electron density be experimentally measured?

Yes, the electron density can be experimentally measured using techniques such as X-ray diffraction, which allows us to visualize the electron density distribution in a molecule. This information can be used to validate and improve theoretical calculations in quantum mechanics.

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