Probability current
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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 ph ...
) is a mathematical quantity describing the flow of
probability Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), 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 ...
. 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 Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
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) and ...
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 of a ...
. 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) can ...
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. S ...
. The probability current is
invariant Invariant and invariance may refer to: Computer science * Invariant (computer science), an expression whose value doesn't change during program execution ** Loop invariant, a property of a program loop that is true before (and after) each iteratio ...
under
gauge transformation In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie group ...
. 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.


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 mad ...
\Psi of a particle of mass m in one dimension is defined as j = \frac \left(\Psi^* \frac- \Psi \frac \right) = \frac \Re\left\ = \frac \Im\left\, where \Psi^* 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 mad ...
, \Re denotes the
real part In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
, and \Im denotes the
imaginary part In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
. Note that the probability current is proportional to a
Wronskian In mathematics, the Wronskian (or Wrońskian) is a determinant introduced by and named by . It is used in the study of differential equations, where it can sometimes show linear independence in a set of solutions. Definition The Wronskian of ...
W(\Psi,\Psi^*). In three dimensions, this generalizes to \mathbf j = \frac \left( \Psi^* \mathbf \nabla \Psi - \Psi \mathbf \nabla \Psi^ \right) = \frac \Re\left\ = \frac\Im\left\ \,, where \hbar 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 ...
, m 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 elementar ...
, \Psi 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 mad ...
, and \nabla denotes the
del Del, or nabla, is an operator used in mathematics (particularly in vector calculus) as a vector differential operator, usually represented by the nabla symbol ∇. When applied to a function defined on a one-dimensional domain, it denotes ...
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 gradi ...
operator. This can be simplified in terms of the kinetic momentum operator, \mathbf = -i\hbar\nabla to obtain \mathbf j = \frac \left(\Psi^* \mathbf \Psi - \Psi \mathbf \Psi^*\right)\,. These definitions use the position basis (i.e. for a wavefunction in
position space In physics and geometry, there are two closely related vector spaces, usually three-dimensional but in general of any finite dimension. Position space (also real space or coordinate space) is the set of all ''position vectors'' r in space, and h ...
), but
momentum space In physics and geometry, there are two closely related vector spaces, usually three-dimensional but in general of any finite dimension. Position space (also real space or coordinate space) is the set of all ''position vectors'' r in space, and h ...
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 c ...
. In
SI units The International System of Units, known by the international abbreviation SI in all languages and sometimes Pleonasm#Acronyms and initialisms, pleonastically as the SI system, is the modern form of the metric system and the world's most wid ...
, 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 particle, ...
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 respe ...
''q'' includes a term due to the interaction with the electromagnetic field; \mathbf j = \frac\left \Psi, ^2 \right/math> where is the
magnetic potential Magnetic potential may refer to: * Magnetic vector potential, the vector whose curl is equal to the magnetic B field * Magnetic scalar potential Magnetic scalar potential, ''ψ'', is a quantity in classical electromagnetism analogous to electr ...
(aka "A-field"). The term ''q''A has dimensions of momentum. Note that \mathbf = -i\hbar\nabla used here is the
canonical momentum In mathematics and classical mechanics, canonical coordinates are sets of coordinates on phase space which can be used to describe a physical system at any given point in time. Canonical coordinates are used in the Hamiltonian formulation of cla ...
and is not
gauge invariant In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie group ...
, unlike the kinetic momentum operator \mathbf = -i\hbar\nabla-q\mathbf. In
Gaussian units Gaussian units constitute a metric system of physical units. This system is the most common of the several electromagnetic unit systems based on cgs (centimetre–gram–second) units. It is also called the Gaussian unit system, Gaussian-cgs uni ...
: \mathbf j = \frac\left \Psi, ^2 \right/math> where ''c'' is the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
.


Spin-''s'' particle in an electromagnetic field

If the particle has spin, it has a corresponding
magnetic moment In electromagnetism, the magnetic moment is the magnetic strength and orientation of a magnet or other object that produces a magnetic field. Examples of objects that have magnetic moments include loops of electric current (such as electromagnets ...
, so an extra term needs to be added incorporating the spin interaction with the electromagnetic field. In SI units: \mathbf j = \frac\left \Psi, ^2 \right+ \frac\nabla\times(\Psi^* \mathbf\Psi) where S is the spin vector of the particle with corresponding spin magnetic moment μ''S'' and spin quantum number ''s''. In Gaussian units: \mathbf j = \frac\left \Psi, ^2 \right+ \frac\nabla\times(\Psi^* \mathbf\Psi)


Connection with classical mechanics

The wave function can also be written in the
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
exponential Exponential may refer to any of several mathematical topics related to exponentiation, including: *Exponential function, also: **Matrix exponential, the matrix analogue to the above * Exponential decay, decrease at a rate proportional to value *Exp ...
(
polar Polar may refer to: Geography Polar may refer to: * Geographical pole, either of two fixed points on the surface of a rotating body or planet, at 90 degrees from the equator, based on the axis around which a body rotates * Polar climate, the c ...
) form: \Psi = R e^ where ''R'' and ''S'' are real functions of r and ''t''. Written this way, the probability density is \rho = \Psi^* \Psi = R^2 and the probability current is: \begin \mathbf & = \frac\left(\Psi^ \mathbf \Psi - \Psi \mathbf\Psi^ \right) \\ pt & = \frac\left(R e^ \mathbfR e^ - R e^ \mathbfR e^\right) \\ pt & = \frac\left R e^ \left( e^ \mathbfR + \fracR e^ \mathbfS \right) - R e^ \left( e^ \mathbfR - \frac R e^ \mathbf S \right)\right \end The exponentials and ''R''∇''R'' terms cancel: \mathbf = \frac\left frac R^2 \mathbf S + \frac R^2 \mathbf S \right Finally, combining and cancelling the constants, and replacing ''R''2 with ''ρ'', \mathbf = \rho \frac. If we take the familiar formula for the mass flux in hydrodynamics: \mathbf = \rho \mathbf, where \rho is the mass density of the fluid and v is its velocity (also the
group velocity The group velocity of a wave is the velocity with which the overall envelope shape of the wave's amplitudes—known as the ''modulation'' or ''envelope'' of the wave—propagates through space. For example, if a stone is thrown into the middl ...
of the wave), we can associate the velocity with ∇''S''/''m'', which is the same as equating ∇''S'' with the classical momentum p = ''m''v. This interpretation fits with Hamilton–Jacobi theory, in which \mathbf = \nabla S in Cartesian coordinates is given by ∇''S'', where ''S'' is
Hamilton's principal function Buck Meadows (formerly Hamilton's and Hamilton's Station) is a census-designated place in Mariposa County, California, United States. It is located east-northeast of Smith Peak, at an elevation of . The population was 21 at the 2020 census. Buck ...
.


Motivation


Continuity equation for quantum mechanics

The definition of probability current and Schrödinger's equation can be used to derive the
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. S ...
, which has ''exactly'' the same forms as those for
hydrodynamics 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) and ...
and
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 of a ...
:Quantum Mechanics, E. Abers, Pearson Ed., Addison Wesley, Prentice Hall Inc, 2004, \frac + \mathbf \nabla \cdot \mathbf j = 0 where the probability density \rho\, is defined as \rho(\mathbf,t) = , \Psi, ^2 = \Psi^*(\mathbf,t)\Psi(\mathbf,t) . If one were to integrate both sides of the continuity equation with respect to volume, so that \int_V \left( \frac \right) \mathrmV + \int_V \left( \mathbf \nabla \cdot \mathbf j \right) \mathrmV = 0 then the
divergence theorem In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a theorem which relates the ''flux'' of a vector field through a closed surface to the ''divergence'' of the field in the vol ...
implies the continuity equation is equivalent to the
integral equation In mathematics, integral equations are equations in which an unknown Function (mathematics), function appears under an integral sign. In mathematical notation, integral equations may thus be expressed as being of the form: f(x_1,x_2,x_3,...,x_n ; ...
: where the ''V'' is any volume and ''S'' is the boundary of ''V''. This is the
conservation law In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves over time. Exact conservation laws include conservation of energy, conservation of linear momentum, c ...
for probability in quantum mechanics. In particular, if ''Ψ'' is a wavefunction describing a single particle, the integral in the first term of the preceding equation, sans time derivative, is the probability of obtaining a value within ''V'' when the position of the particle is measured. The second term is then the rate at which probability is flowing out of the volume ''V''. Altogether the equation states that the time derivative of the probability of the particle being measured in ''V'' is equal to the rate at which probability flows into ''V''.


Transmission and reflection through potentials

In regions where a step potential or
potential barrier In quantum mechanics, the rectangular (or, at times, square) potential barrier is a standard one-dimensional problem that demonstrates the phenomena of wave-mechanical tunneling (also called "quantum tunneling") and wave-mechanical reflection. ...
occurs, the probability current is related to the transmission and reflection coefficients, respectively ''T'' and ''R''; they measure the extent the particles reflect from the potential barrier or are transmitted through it. Both satisfy: T + R = 1\,, where ''T'' and ''R'' can be defined by: T= \frac \, , \quad R = \frac \, , where ''j''inc, ''j''ref and ''j''trans are the incident, reflected and transmitted probability currents respectively, and the vertical bars indicate the magnitudes of the current vectors. The relation between ''T'' and ''R'' can be obtained from probability conservation: \mathbf_\mathrm + \mathbf_\mathrm=\mathbf_\mathrm\,. In terms of a
unit vector In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''direction vecto ...
n
normal Normal(s) or The Normal(s) may refer to: Film and television * ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie * ''Norma ...
to the barrier, these are equivalently: T= \left, \frac\\,, \qquad R= \left, \frac \ \,, where the absolute values are required to prevent ''T'' and ''R'' being negative.


Examples


Plane wave

For a
plane wave In physics, a plane wave is a special case of wave or field: a physical quantity whose value, at any moment, is constant through any plane that is perpendicular to a fixed direction in space. For any position \vec x in space and any time t, th ...
propagating in space: \Psi(\mathbf,t) = \, A e^ the probability density is constant everywhere; \rho(\mathbf,t) = , A, ^2 \rightarrow \frac = 0 (that is, plane waves are
stationary state A stationary state is a quantum state with all observables independent of time. It is an eigenvector of the energy operator (instead of a quantum superposition of different energies). It is also called energy eigenvector, energy eigenstate, ener ...
s) but the probability current is nonzero – the square of the absolute amplitude of the wave times the particle's speed; \mathbf\left(\mathbf,t\right) = \left, A\^2 = \rho \frac = \rho \mathbf illustrating that the particle may be in motion even if its spatial probability density has no explicit time dependence.


Particle in a box

For a
particle in a box In quantum mechanics, the particle in a box model (also known as the infinite potential well or the infinite square well) describes a particle free to move in a small space surrounded by impenetrable barriers. The model is mainly used as a hypo ...
, in one spatial dimension and of length ''L'', confined to the region 0 < x < L, the energy eigenstates are \Psi_n = \sqrt \sin \left( \frac x \right) and zero elsewhere. The associated probability currents are j_n = \frac\left( \Psi_n^* \frac - \Psi_n \frac \right) = 0 since \Psi_n = \Psi_n^*


Discrete definition

For a particle in one dimension on \ell^2\left(\mathbb\right), we have the Hamiltonian H = -\Delta + V where -\Delta \equiv 2 I - S - S^\ast is the discrete Laplacian, with S being the right shift operator on \ell^2\left(\mathbb\right). Then the probability current is defined as j \equiv 2 \Im\left\, with v the velocity operator, equal to v \equiv -i ,\, H/math> and X is the position operator on \ell^2\left(\mathbb\right). Since V is usually a multiplication operator on \ell^2\left(\mathbb\right), we get to safely write -i ,\, H= -i ,\, -\Delta= -i\left ,\, -S - S^\right= i S - i S^. As a result, we find: j\left(x\right) \equiv 2 \Im\left\ = 2 \Im\left\ = 2 \Im\left\


References


Further reading

*{{cite book , title=Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles , edition=2nd , first=R. , last=Resnick , first2=R. , last2=Eisberg , publisher=John Wiley & Sons , year=1985 , isbn=0-471-87373-X Quantum mechanics