Probability current
The quantum mechanical probability current density (more precisely, probability current density ) is a current density in the quantum mechanical continuity equation associated with the quantum mechanical probability density. It is determined by the wave function in the spatial domain and in the absence of magnetic fields has the form ( for definitions, see the section on gauge invariance ):
Explanation
Continuity equations
In physical field theories conserved quantities appear as integrals over certain densities. Such densities are among the conserved quantities, then satisfy continuity equations, which are a special form of a balance equation. The simplest example of such continuity equation is the mass density of a liquid that nowhere in the room has sources or sinks. The total mass is then a conserved quantity. When fluid flows and therefore in a certain fixed volume changed the ground, this may only be done by mass flows over the edge of the volume on or off. This mass flow over the edge surface can be describes as the integral of the mass flow density, which is a vector quantity, and the strength and direction of the mass flow at a position to be displayed:
Wherein the volume element and the area of the surface element, the edge of the volume, the bulk density and the mass flow density. Assuming the limit to a infinitesimally small volume over, we obtain the differential form of the continuity equation:
The current density is in the case of a liquid by the product of the bulk density at the respective location and the velocity of the liquid there, that is, by optionally.
Application in quantum mechanics
Similar to the above example in field theory in quantum mechanics, as well as in statistical mechanics, the probability that a conserved quantity, ie it is the probability of the described system ( particles ) to be found somewhere in the room. Naturally, this probability is, if one considers the entire space, equal to one: the single particle must be somewhere to be found in the room. In quantum mechanics, the probability density is given by the square of the amplitude of the wave function:
Since the wave function of quantum mechanics is a complete description of the physical state of the system is initially unclear how the associated current density of the probability density might look like, because you have given otherwise than in the continuum mechanics a priori no additional velocity field. Rather, the current density must be a function of the wave function.
To determine the shape of the probability of the current density, is now considered the differential form of equation of continuity of this size. It is found that, with the help of the Schrödinger equation can reformulate the first term which describes the time derivative of the probability density:
Where the Hamiltonian is. Substituting the explicit form of the Hamiltonian on, one can see that the potential drops out of the equation. It is a term, the one still in the form
Can bring. Let us now compare with the continuity equation, we obtain the following form of the probability current density:
As described at the beginning of the article.
Alternative formulations:
Where the canonical momentum operator is.
Gauge invariance
To ensure the gauge invariance of the theory, one must still add a term proportional to the vector potential. Is usually employed, and thus, when no magnetic field occurs. But this is also not really necessary.
Taking into account the additional term with the vector potential, eg in magnetic fields, one obtains the easily remembered formula
The expression in the parentheses is the so-called kinetic momentum: it is gauge invariant, in contrast to the canonical momentum. It corresponds to the mechanism of the formula for the speed. The kinetic momentum is measurable, and thus gauge invariant in any case.
Here, the real part of the complex number. The operator and the ( negative! ) electric charge have the usual meaning.
Of course, you must also replace the Hamiltonian the canonical momentum by the kinetic and under circumstances, if the calibration depends explicitly on time, add additional potential terms.