# Probability current density

The quantum mechanical **probability** current density *(more precisely: residence ***probability** current density *)* is a current density that is associated with the quantum mechanical residence probability density within the framework of the quantum mechanical continuity equation . It is determined by the wave function in spatial space and, in the absence of magnetic fields, has the form (for more precise details see below):

## background

In physical field theories , conserved quantities appear as integrals over certain densities. Such densities, which belong to the conservation quantities, then suffice *continuity equations* , which are a special form of a balance equation .

In general, equations of continuity contain a density and a current and link them in shape

or in an integral formulation using the Gaussian integral theorem :

The continuity equations gain clear meaning from the integral formulation, since the change in density over time within a volume element is equal to the flow beyond the boundaries of the volume element ( points out of the volume element).

Since only the divergence of the current density occurs in the continuity equation , a term proportional to the rotation of an arbitrary vector-valued (sufficiently smooth) function can always be added to this , since according to Schwarz's theorem it holds.

## Non-relativistic quantum mechanics

In quantum mechanics, as in statistical mechanics, the probability of presence is a conserved quantity. This probability, if you look at the entire space, is equal to one: the individual particle must be found somewhere in space. In quantum mechanics, the probability density is given by the square of the magnitude of the wave function :

Since the wave function of quantum mechanics represents a complete description of the physical state of the system, it is initially unclear what the associated current density of the probability density could look like, since, unlike in continuum mechanics *,* no additional velocity field has been given *a priori* . Rather, the current density must be a function of the wave function.

### Probability current density without an external electromagnetic field

The probability density can be reformulated using the Schrödinger equation :

where is the Hamilton operator . If you use the explicit form of the Hamilton operator, you can see that the potential falls out of the equation. It remains a term that you can still put into the form

can bring. A comparison with the continuity equation gives the following form of the probability current density:

as described at the beginning of the article.

Alternative formulations:

where is the *canonical* momentum operator.

### Probability current density with an external electromagnetic field

The wave function in the external electromagnetic field obeys the Pauli equation . The following substitutions are made in the Schrödinger equation:

- For a correct description of the spin, the wave function of the scalar Schrödinger equation is replaced by the two-component spinor .
- The momentum operator is replaced by , where is the vector potential and the electric charge of the particle. In contrast to the
*canonical*, the term (without the “operator hat”) is also called*kinetic*momentum. - The Hamilton operator receives an additional term for the electrostatic energy with the electrical potential and becomes .

This results in a possible probability current density

- .

This probability flux density is invariant under the gauge transformations due to the replacement of the canonical by the kinetic momentum

with any real function .

It turns out, however, that a term can be added proportionally to this naive probability current density , which is also gauge invariant (and as a rotation term does not violate the continuity equation). In fact, the Dirac equation in the non-relativistic limit case with the Pauli matrices results :

## References and footnotes

- ^ Marek Nowakowski: The Quantum Mechanical Current of the Pauli Equation. American Journal of Physics 67, 916 (1999). Article on arxiv.org

## Web links

- Elektronium model Description of atoms with the help of the probability current density