# Electron density

The **electron density** or , in physics, is a charge carrier density that indicates the location-dependent number of electrons per volume ( density function ). From a mathematical point of view, it is a scalar field of three-dimensional spatial space .

It is a measured variable ( unit ) that is often used in the description of molecules and solids ( density functional theory ) in order to avoid complicated, high-dimensional wave functions or quantum mechanical state vectors . It is also used in plasma physics , in X-ray structure analysis (as a Fourier transform of the structure factor ) and in semiconductor physics .

By definition, the integral of the electron density, which extends over the entire area of space , must be equal to the number of electrons:

The typical electron density for conduction electrons is in metallic solids , in the F-layer of the ionosphere only .

## Expected value of the electron density operator

In general, in quantum mechanics, measured quantities are identified with Hermitian operators whose eigenvectors represent the states in which the system assumes a sharp measured value with regard to the measured quantity, and whose eigenvalues correspond to the associated measured values themselves.

The electron density is identified as the expected value of the electron density operator:

This operator must meet the following properties:

- Integrability of the expected value (more strictly: the integral over the entire volume must correspond to the number of particles)
- Positive semi-finiteness : expectation value must be greater than or equal to 0 everywhere

By identifying the electron density as the marginal distribution of the probability density ( absolute square of the wave function):

In words: one holds any electron in place and adds up the probabilities of all possible arrangements of the other electrons.

After presenting the expected value in the usual form:

the associated operator can be identified as:

and one recognizes that it is not an operator in the strict sense, since it does not convert a square-integrable function into a square-integrable function and therefore does not satisfy the definition of an operator in the space of square-integrable functions .

There is thus no particle density operator , but there is a linear functional ( distribution ) whose integral kernel is commonly referred to as the particle density operator .

In the sense of the topology induced by the 2-norm, it is a non-continuous linear functional on the locally absolutely Lebesgue integrable functions.

Here in particular the absolute Lebesgue integrable functions of the form are valid for and with the extension of a distribution theory known Delta distributions using Delta consequences to represent.

Within the Hartree-Fock approximation , the electron density is obtained from the sum of the orbital densities: