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In
quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
, 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 in physics. For transport phe ...
) is a mathematical quantity describing the flow of
probability Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an e ...
. Specifically, if one thinks of probability as a
heterogeneous Homogeneity and heterogeneity are concepts relating to the uniformity of a substance, process or image. A homogeneous feature is uniform in composition or character (i.e., color, shape, size, weight, height, distribution, texture, language, i ...
fluid, then the probability current is the rate of flow of this fluid. It is a real
vector Vector most often refers to: * Euclidean vector, a quantity with a magnitude and a direction * Disease vector, an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematics a ...
that changes with space and time. Probability currents are analogous to mass currents in
hydrodynamic In physics, physical chemistry and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids – liquids and gases. It has several subdisciplines, including (the study of air and other gases in moti ...
s and
electric currents An electric current is a flow of charged particles, such as electrons or ions, moving through an electrical conductor or space. It is defined as the net rate of flow of electric charge through a surface. The moving particles are called charge ...
in
electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge via electromagnetic fields. The electromagnetic force is one of the four fundamental forces of nature. It is the dominant force in the interacti ...
. As in those fields, the probability current (i.e. the probability current density) is related to the
probability density function In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a Function (mathematics), function whose value at any given sample (or point) in the sample space (the s ...
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 In the physics of gauge theory, gauge theories, gauge fixing (also called choosing a gauge) denotes a mathematical procedure for coping with redundant Degrees of freedom (physics and chemistry), degrees of freedom in field (physics), field variab ...
. 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 is the random motion of particles suspended in a medium (a liquid or a gas). The traditional mathematical formulation of Brownian motion is that of the Wiener process, which is often called Brownian motion, even in mathematical ...
and the
Fokker–Planck equation In statistical mechanics and information theory, 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 (physi ...
. The relativistic equivalent of the probability current is known as the probability four-current.


Definition (non-relativistic 3-current)


Free spin-0 particle

In non-relativistic quantum mechanics, the probability current of the
wave function In quantum physics, a wave function (or wavefunction) is a mathematical description of the quantum state of an isolated quantum system. The most common symbols for a wave function are the Greek letters and (lower-case and capital psi (letter) ...
of a particle of mass in one dimension is defined as j = \frac \left(\Psi^* \frac- \Psi \frac \right) = \frac \Re\left\ = \frac \Im\left\, where *\hbar is the reduced
Planck constant The Planck constant, or Planck's constant, denoted by h, is a fundamental physical constant of foundational importance in quantum mechanics: a photon's energy is equal to its frequency multiplied by the Planck constant, and the wavelength of a ...
; *\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 numbers, then the complex conjugate of a + bi is a - ...
of the
wave function In quantum physics, a wave function (or wavefunction) is a mathematical description of the quantum state of an isolated quantum system. The most common symbols for a wave function are the Greek letters and (lower-case and capital psi (letter) ...
; *\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 ...
; *\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 of ''n'' differentiable functions is the determinant formed with the functions and their derivatives up to order . It was introduced in 1812 by the Polish mathematician Józef Wroński, and is used in the study 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 \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 f of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p gives the direction and the rate of fastest increase. The g ...
operator Operator may refer to: Mathematics * A symbol indicating a mathematical operation * Logical operator or logical connective in mathematical logic * Operator (mathematics), mapping that acts on elements of a space to produce elements of another sp ...
. 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 \left( \mathbf \Psi \right)^*\right)\,. These definitions use the position basis (i.e. for a wavefunction in position space), 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 Euclidean sp ...
is possible. In fact, one can write the probability current operator as \mathbf(\mathbf) = \frac which do not depend on a particular choice of basis. The probability current is then the expectation of this operator, \mathbf(\mathbf,t) = \langle \Psi(t), \hat(\mathbf), \Psi(t)\rangle.


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) is a physical field, varying in space and time, that represents the electric and magnetic influences generated by and acting upon electric charges. The field at any point in space and time can be regarde ...
. In
SI units The International System of Units, internationally known by the abbreviation SI (from French ), is the modern form of the metric system and the world's most widely used system of measurement. It is the only system of measurement with official st ...
, a
charged particle In physics, a charged particle is a particle with an electric charge. For example, some elementary particles, like the electron or quarks are charged. Some composite particles like protons are charged particles. An ion, such as a molecule or atom ...
of mass and
electric charge Electric charge (symbol ''q'', sometimes ''Q'') is a physical property of matter that causes it to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative''. Like charges repel each other and ...
includes a term due to the interaction with the electromagnetic field; \mathbf j = \frac\left \Psi, ^2 \right/math> where is the
magnetic vector potential In classical electromagnetism, magnetic vector potential (often denoted A) is the vector quantity defined so that its curl is equal to the magnetic field, B: \nabla \times \mathbf = \mathbf. Together with the electric potential ''φ'', the ma ...
. The term 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, unlike the kinetic momentum operator \mathbf = -i\hbar\nabla-q\mathbf. In
Gaussian units Gaussian units constitute a metric system of units of measurement. This system is the most common of the several electromagnetic unit systems based on the centimetre–gram–second system of units (CGS). It is also called the Gaussian unit syst ...
: \mathbf j = \frac\left \Psi, ^2 \right/math> where is the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant exactly equal to ). It is exact because, by international agreement, a metre is defined as the length of the path travelled by light in vacuum during a time i ...
.


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

If the particle has
spin Spin or spinning most often refers to: * Spin (physics) or particle spin, a fundamental property of elementary particles * Spin quantum number, a number which defines the value of a particle's spin * Spinning (textiles), the creation of yarn or thr ...
, it has a corresponding
magnetic moment In electromagnetism, the magnetic moment or magnetic dipole moment is the combination of strength and orientation of a magnet or other object or system that exerts a magnetic field. The magnetic dipole moment of an object determines the magnitude ...
, so an extra term needs to be added incorporating the spin interaction with the electromagnetic field. According to Landau-Lifschitz's ''
Course of Theoretical Physics The ''Course of Theoretical Physics'' is a ten-volume series of books covering theoretical physics that was initiated by Lev Landau and written in collaboration with his student Evgeny Lifshitz starting in the late 1930s. It is said that Landau ...
'' the electric current density is in Gaussian units: \mathbf_e = \frac \left \Psi, ^2 \right+ \frac\nabla\times(\Psi^* \mathbf\Psi) And in SI units: \mathbf j_e = \frac\left \Psi, ^2 \right+ \frac\nabla\times(\Psi^* \mathbf\Psi) Hence the probability current (density) is in SI units: \mathbf = \mathbf_e/q = \frac\left \Psi, ^2 \right+ \frac\nabla\times(\Psi^* \mathbf\Psi) where is the
spin Spin or spinning most often refers to: * Spin (physics) or particle spin, a fundamental property of elementary particles * Spin quantum number, a number which defines the value of a particle's spin * Spinning (textiles), the creation of yarn or thr ...
vector of the particle with corresponding
spin magnetic moment Spin is an intrinsic form of angular momentum carried by elementary particles, and thus by composite particles such as hadrons, atomic nuclei, and atoms. Spin is quantized, and accurate models for the interaction with spin require relativistic ...
and
spin quantum number In physics and chemistry, the spin quantum number is a quantum number (designated ) that describes the intrinsic angular momentum (or spin angular momentum, or simply ''spin'') of an electron or other particle. It has the same value for all ...
. It is doubtful if this formula is valid for particles with an interior structure. The
neutron The neutron is a subatomic particle, symbol or , that has no electric charge, and a mass slightly greater than that of a proton. The Discovery of the neutron, neutron was discovered by James Chadwick in 1932, leading to the discovery of nucle ...
has zero charge but non-zero magnetic moment, so \frac would be impossible (except \nabla \times (\Psi^* \mathbf\Psi) would also be zero in this case). For composite particles with a non-zero charge – like the
proton A proton is a stable subatomic particle, symbol , Hydron (chemistry), H+, or 1H+ with a positive electric charge of +1 ''e'' (elementary charge). Its mass is slightly less than the mass of a neutron and approximately times the mass of an e ...
which has spin quantum number s=1/2 and μS= 2.7927· μN or the
deuteron Deuterium (hydrogen-2, symbol H or D, also known as heavy hydrogen) is one of two Stable isotope ratio, stable isotopes of hydrogen; the other is protium, or hydrogen-1, H. The deuterium atomic nucleus, nucleus (deuteron) contains one proton and ...
(H-2 nucleus) which has s=1 and μS=0.8574·μN – it is mathematically possible but doubtful.


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 * Ex ...
( polar) form: \Psi = R e^ where are real functions of and . 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 terms cancel: \mathbf = \frac\left frac R^2 \mathbf S + \frac R^2 \mathbf S \right Finally, combining and cancelling the constants, and replacing with , \mathbf = \rho \frac. Hence, the spatial variation of the phase of a wavefunction is said to characterize the probability flux of the wavefunction. 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 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 (waves), envelope'' of the wave—propagates through space. For example, if a stone is thro ...
of the wave). In the classical limit, we can associate the velocity with \tfrac, which is the same as equating with the classical momentum however, it does not represent a physical velocity or momentum at a point since simultaneous measurement of position and velocity violates
uncertainty principle The uncertainty principle, also known as Heisenberg's indeterminacy principle, is a fundamental concept in quantum mechanics. It states that there is a limit to the precision with which certain pairs of physical properties, such as position a ...
. This interpretation fits with Hamilton–Jacobi theory, in which \mathbf = \nabla S in Cartesian coordinates is given by , where is Hamilton's principal function. The de Broglie-Bohm theory equates the velocity with \tfrac in general (not only in the classical limit) so it is always well defined. It is an interpretation of quantum mechanics.


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 ...
, which has ''exactly'' the same forms as those for
hydrodynamics In physics, physical chemistry and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids – liquids and gases. It has several subdisciplines, including (the study of air and other gases in ...
and
electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge via electromagnetic fields. The electromagnetic force is one of the four fundamental forces of nature. It is the dominant force in the interacti ...
.Quantum Mechanics, E. Abers, Pearson Ed., Addison Wesley, Prentice Hall Inc, 2004, For some
wave function In quantum physics, a wave function (or wavefunction) is a mathematical description of the quantum state of an isolated quantum system. The most common symbols for a wave function are the Greek letters and (lower-case and capital psi (letter) ...
, let: \rho(\mathbf,t) = , \Psi, ^2 = \Psi^*(\mathbf,t)\Psi(\mathbf,t) .be the
probability density In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values ...
(probability per unit volume, denotes
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 numbers, then the complex conjugate of a + bi is a - ...
). Then, \begin \frac\int_\mathcaldV\, \rho &= \int_\mathcaldV\, \left(\frac \psi^*+ \psi \frac \right) \\ &=\int_\mathcaldV\,\left \frac i\hbar\left( -\frac \nabla^2 \psi + V \psi\right)\psi^* + \frac i\hbar\left( -\frac \nabla^2 \psi^* + V \psi^*\right)\psi\right\\ &=\int_\mathcaldV\,\frac \left left(\nabla^2\psi\right) \psi^* - \psi \left(\nabla^2 \psi^*\right)\right\ &=\int_\mathcaldV\, \nabla\cdot\left(\frac(\psi^*\nabla\psi-\psi\nabla\psi^*)\right)\\ &=\int_\mathcald\mathbf\cdot\left(\frac(\psi^*\nabla\psi-\psi\nabla\psi^*)\right) \end where is any volume and is the boundary of . 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 mass-energy, conservation of linear momen ...
for probability in quantum mechanics. The integral form is stated as: \int_V \left( \frac \right) \mathrmV + \int_V \left( \mathbf \nabla \cdot \mathbf j \right) \mathrmV = 0where \mathbf = \frac \left( \Psi^*\hat\Psi - \Psi\hat\Psi^* \right) = -\frac(\psi^*\nabla\psi-\psi\nabla\psi^*) = \frac \hbar m \operatorname (\psi^*\nabla \psi) is the probability current or probability flux (flow per unit area). Here, equating the terms inside the integral gives 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 ...
for probability:\frac \rho\left(\mathbf,t\right) + \nabla \cdot \mathbf = 0, and the integral equation can also be restated using the
divergence theorem In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a theorem relating the '' flux'' of a vector field through a closed surface to the ''divergence'' of the field in the volume ...
as: In particular, if 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 when the position of the particle is measured. The second term is then the rate at which probability is flowing out of the volume . Altogether the equation states that the time derivative of the probability of the particle being measured in is equal to the rate at which probability flows into . By taking the limit of volume integral to include all regions of space, a well-behaved wavefunction that goes to zero at infinities in the surface integral term implies that the time derivative of total probability is zero ie. the normalization condition is conserved. This result is in agreement with the unitary nature of time evolution operators which preserve length of the vector by definition.


Conserved current for Klein–Gordon fields

The probability (4-)current arises from
Noether's theorem Noether's theorem states that every continuous symmetry of the action of a physical system with conservative forces has a corresponding conservation law. This is the first of two theorems (see Noether's second theorem) published by the mat ...
as applied to the Lagrangian the Klein-Gordon Lagrangian density \mathcal=\partial_\mu\phi^*\,\partial^\mu\phi +V(\phi^*\,\phi) of the complex scalar field \phi:\mathbb^\mapsto\mathbb . This is invariant under the symmetry transformation \phi\mapsto\phi'=\phi\,e^\, . Defining \delta\phi=\phi'-\phi we find the Noether current j^\mu:= \frac \cdot \mathbf_r = \frac\,\frac\bigg, _+\frac\,\frac\bigg, _= i\,\phi\,(\partial^\mu\phi^*)-i\,\phi^*\,(\partial^\mu\phi)which satisfies the continuity equation. Here \mathbf_r is the generator of the symmetry, which is \frac in the case of a single parameter \alpha . The continuity equation \partial_\mu j^\mu = 0 is satisfied. However, note that now, the analog of the probability density is not \phi \phi^* but rather \phi^* \partial_t \phi - \phi \partial_t \phi^* . As this quantity can now be negative, we must interpret it as a charge density, with an associated current density and 4-current.


Transmission and reflection through potentials

In regions where a
step potential In quantum mechanics and scattering theory, the one-dimensional step potential is an idealized system used to model incident, reflected and transmitted matter waves. The problem consists of solving the time-independent Schrödinger equation for a ...
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 and ; they measure the extent the particles reflect from the potential barrier or are transmitted through it. Both satisfy: T + R = 1\,, where and can be defined by: T= \frac \, , \quad R = \frac \, , where are the incident, reflected and transmitted probability currents respectively, and the vertical bars indicate the magnitudes of the current vectors. The relation between and 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 (mathematics and physics), vector (often a vector (geometry), spatial vector) of Norm (mathematics), length 1. A unit vector is often denoted by a lowercase letter with a circumfle ...
normal to the barrier, these are equivalently: T= \left, \frac\\,, \qquad R= \left, \frac \ \,, where the absolute values are required to prevent and being negative.


Examples


Plane wave

For a
plane wave In physics Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of ...
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, ene ...
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 the movement of a free particle in a small space surrounded by impenetrable barriers. The model is mainly used a ...
, in one spatial dimension and of length , 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(\Z), we have the Hamiltonian H = -\Delta + V where -\Delta \equiv 2 I - S - S^\ast is the discrete Laplacian, with being the right shift operator on \ell^2(\Z). Then the probability current is defined as j \equiv 2 \Im\left\, with the velocity operator, equal to v \equiv -i ,\, H/math> and is the position operator on \ell^2\left(\mathbb\right). Since is usually a multiplication operator on \ell^2(\Z), we get to safely write -i ,\, H= -i ,\, -\Delta= -i\left ,\, -S - S^\right= i S - i S^. As a result, we find: \begin j\left(x\right) \equiv 2 \Im\left\ &= 2 \Im\left\\\ &= 2 \Im\left\ \end


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