Electron density

The electron density and is a physics carrier density, which indicates the location-dependent number of electrons per unit volume (density function). Mathematically, it is a scalar field of the three-dimensional spatial space.

It is a measure ( unit m -3), which is often used in the description of molecules and solids (density functional theory ) to avoid complicated high-dimensional wave functions or quantum mechanical state vectors. In addition, it is used in plasma physics, in the X-ray structure analysis ( as the Fourier transform of the structure factor ) and in semiconductor physics.

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

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

Expectation value of the electron density operator

General measurements with Hermitian operators in quantum mechanics identified whose eigenvectors represent the states in which the system assumes a sharp measurement with respect to the measure, and whose eigenvalues ​​correspond to the corresponding measured values ​​themselves.

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

This operator must satisfy the following properties:

• Integrability of the expected value ( strict: integral over the entire volume must correspond to the number of particles )
• Positive semidefiniteness: expectation value must be greater everywhere equal to 0.

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

In words: Man holding any electron firmly in place and summed over the probabilities of all possible arrangements of the other electrons.

After presentation of the expected value in the usual form:

Can the corresponding operator identified as the following:

And you realize that this is no operator in the strict sense, since he transferred no square integrable function in a square- integrable function and therefore does not meet the definition of an operator in the space of square integrable functions.

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

This is a, in the sense of the induced by the 2-norm topology, not continuous linear functional on the locally absolutely Lebesgue integrable functions.

Here in particular the absolutely Lebesgue integrable functions of the form are true and for which, with an extension of the well-known from the distribution theory delta distributions with the help of Delta consequences.

Within the Hartree -Fock approximation, we obtain the electron density of the sum of the orbital densities: