In
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, the probability current (sometimes called probability
flux
Flux describes any effect that appears to pass or travel (whether it actually moves or not) through a surface or substance. Flux is a concept in applied mathematics and vector calculus which has many applications to physics. For transport ...
) is a mathematical quantity describing the flow of
probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, ...
. Specifically, if one thinks of probability as a
heterogeneous
Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, siz ...
fluid, then the probability current is the rate of flow of this fluid. It is a
real vector that changes with space and time. Probability currents are analogous to
mass currents in
hydrodynamic
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) ...
s and
electric currents
An electric current is a stream of charged particles, such as electrons or ions, moving through an electrical conductor or space. It is measured as the net rate of flow of electric charge through a surface or into a control volume. The moving pa ...
in
electromagnetism
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions o ...
. As in those fields, the probability current is related to the
probability density function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) ca ...
via a
continuity equation
A continuity equation or transport equation is an equation that describes the transport of some quantity. It is particularly simple and powerful when applied to a conserved quantity, but it can be generalized to apply to any extensive quantity. ...
. The probability current is
invariant under
gauge transformation.
The concept of probability current is also used outside of quantum mechanics, when dealing with probability density functions that change over time, for instance in
Brownian motion
Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas).
This pattern of motion typically consists of random fluctuations in a particle's position insi ...
and the
Fokker–Planck equation
In statistical mechanics, the Fokker–Planck equation is a partial differential equation that describes the time evolution of the probability density function of the velocity of a particle under the influence of drag forces and random forces, ...
.
Definition (non-relativistic 3-current)
Free spin-0 particle
In non-relativistic quantum mechanics, the probability current j of the
wave function
A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements ...
of a particle of mass
in one dimension is defined as
where
denotes the
complex conjugate
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
of the
wave function
A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements ...
,
denotes the
real part, and
denotes the
imaginary part. Note that the probability current is proportional to a
Wronskian .
In three dimensions, this generalizes to
where
is the reduced
Planck constant
The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivale ...
,
is the particle's
mass
Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different ele ...
,
is the
wavefunction
A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements ...
, and
denotes the
del or
gradient
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
operator.
This can be simplified in terms of the
kinetic momentum operator,
to obtain
These definitions use the position basis (i.e. for a wavefunction in
position space), but
momentum space is possible.
Spin-0 particle in an electromagnetic field
The above definition should be modified for a system in an external
electromagnetic field
An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classical ...
. In
SI units
The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric system and the world's most widely used system of measurement. ...
, a
charged particle
In physics, a charged particle is a particle with an electric charge. It may be an ion, such as a molecule or atom with a surplus or deficit of electrons relative to protons. It can also be an electron or a proton, or another elementary pa ...
of mass ''m'' and
electric charge
Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons res ...
''q'' includes a term due to the interaction with the electromagnetic field;