Continuity equation
Encyclopedia
A continuity equation in physics is a differential equation
that describes the transport of a conserved quantity
. Since mass
, energy
, momentum
, electric charge
and other natural quantities are conserved under their respective appropriate conditions, a variety of physical phenomena may be described using continuity equations.
Continuity equations are the (stronger) local form of conservation law
s. All the examples of continuity equations below express the same idea, which is roughly that: the total amount (of the conserved quantity) inside any region can only change by the amount that passes in or out of the region through the boundary. A conserved quantity cannot increase or decrease, it can only move from place to place.
Any continuity equation can be expressed in an "integral form" (in terms of a flux
integral), which applies to any finite region, or in a "differential form" (in terms of the divergence
operator) which applies at a point.
where
In the case that q is a conserved quantity
that cannot be created or destroyed (such as energy
), the continuity equation is:
because σ = 0.
This general equation may be used to derive any continuity equation, ranging from as simple as the volume continuity equation to as complicated as the Navier–Stokes equations. This equation also generalizes the advection equation.
Note the flux f should represent some flow or transport, which has dimensions [quantity][T]−1[L]−2, where [quantity]/[L]3 is the dimension of φ.
Other equations in physics, such as Gauss's law of the electric field
and Gauss's law for gravity, have a similar mathematical form to the continuity equation, but are not usually called by the term "continuity equation", because f in those cases does not represent the flow of a real physical quantity.
(see below), the continuity equation can be rewritten in an equivalent way, called the "integral form":
where
In a simple example, V could be a building, and q could be the number of people in the building. The surface S would consist of the walls, doors, roof, and foundation of the building. Then the continuity equation states that the number of people in the building increases when people enter the building (an inward flux through the surface), decreases when people exit the building (an outward flux through the surface), increases when someone in the building gives birth (a "source" where σ > 0), and decreases when someone in the building dies (a "sink" where σ < 0).
Note the flux f emerging normal from/to the local surface dS satisfies
The rate of change of q leaving the region is simply the time derivative:
where the minus sign has been inserted since the amount of q is decreasing in the region. Partial derivatives are used since they enter the integrand, which is not only a function of time, but also space due to the density nature of φ - differentiation needs only to be with respect to t. The rate of change of q crossing the boundary and leaving the region can be formulated as follows.
Using the divergence theorem on the right-hand side:
Equating these expressions leads to an identity of volume integrals,
this is only true if the integrands are equal, which directly leads to the differential continuity equation:
Either form may be useful and quoted, both can appear in hydrodynamics and electromagnetism, but for quantum mechanics and energy conservation, only the first may be used. Therefore the first is more general.
V is constant in shape for the calculation, so it is independent of time and the time derivatives can be freely moved out of the first integral on the left side,
where ordinary derivatives replace partial derivatives since the integral becomes a function of time only (the integral is evaluated over the region - so the spatial variables become removed from the final expression and t remains the only variable). Now using the divergence theorem
for the second integral on the left side:
The right side becomes:
Substituting these in obtains the integral form:
, or can be derived as a consequence of two of Maxwell's equations
. It states that the divergence
of the current density
J (in amperes per square meter) is equal to the negative rate of change of the charge density
ρ (in coulombs per cubic metre),
, Ampère's law (with Maxwell's correction)
, states that
Taking the divergence of both sides results in
but the divergence of a curl is zero, so that
Another one of Maxwell's equations, Gauss's law
, states that
substitution into the previous equation yeilds the continuity equation
To actually see how this is done - click here (see also and ).
Suppose an amount of charge q is contained in a region of volume V. This is equal to:
the rate of change of charge leaving the region is simply the time derivative:
where the minus sign has been inserted since the charge is decreasing in the region. Again partial derivatives are used since they enter the integrand - not only a function of time, but space also due to the density function. The rate of change of charge crossing the boundary and leaving the region is also equal to
by the divergence theorem:
Equating these expressions leads to an identity of volume integrals,
hence the integrands must be equal, which directly leads to the continuity equation:
of a four-current
:
where
so that since
then
implies that the current is conserved:
, the continuity equation states that, in any steady state
process, the rate at which mass enters a system is equal to the rate at which mass leaves the system. In fluid dynamics, the continuity equation is analogous to Kirchhoff's current law in electric circuits.
The differential form of the continuity equation is:
where
If ρ is a constant, as in the case of incompressible flow
, the mass continuity equation simplifies to a volume continuity equation:
which means that the divergence
of velocity field is zero everywhere. Physically, this is equivalent to saying that the local volume dilation rate is zero.
Further, the Navier-Stokes equations
form a vector continuity equation describing the conservation of linear momentum.
To see this done click below.
Suppose an amount of mass M is contained in a region of volume V. This is equal to:
the rate of change of mass leaving the region is simply the time derivative:
where the minus sign has been inserted since the mass is decreasing in the region. Again partial derivatives are used since they enter the integrand - not only a function of time, but space also due to the density function. The rate of change of mass crossing the boundary and leaving the region is also equal to
by the divergence theorem:
Equating these expressions leads to an identity of volume integrals,
hence the integrands must be equal, which directly leads to the continuity equation:
.
Letting
the continuity equation is:
where k is the thermal conductivity
(not Boltzmann constant), this can also be written as:
, the conservation of probability also yields a continuity equation. The terms in the equation require these definitions, and are slightly less obvious than the other forms of volume densities, currents, current densities etc., so they are outlined here:
With these definitions the continuity equation reads:
Either form is usually quoted. Intuitively; considering the above quantities this represents the flow of probability. The chance of finding the particle at some r t flows like a fluid
, the particle itself does not flow deterministically in the same vector field
.
and its complex conjugate
(i → –i) throughout are respectively:
where U is the potential function
. The partial derivative
of ρ with respect to t is:
Multiplying the Schrödinger equation by Ψ* then solving for
, and similarly multiplying the complex conjugated Schrödinger equation by Ψ then solving for 
;
substituting into the time derivative of ρ:
The Laplacian
operators (∇2) in the above result suggest that the right hand side is the divergence of j, and the reversed order of terms imply this is the negative of j, altogether:
so the continuity equation is:
The intgeral form follows as for the general equation.
where the last equality follows from the product rule and the fact that the shape of V is fixed for the calculation and therefore independent of time - i.e. the time derivative can be moved through the integral. To simplify this further consider again the time dependent Schrödinger equation
and its complex conjugate, in terms of the time derivatives of Ψ and Ψ* respectively:
Substituting into the preceding equation:
.
From the product rule for the divergence operator
substituting:
On the right side, the argument of the divergence operator is j,
using the divergence theorem again gives the integral form:
To obtain the differential form:
The differential form follows from the fact that the preceding equation holds for all V, and as the integrand is a continuous function of space, it must vanish everywhere:
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
that describes the transport of a conserved quantity
Conserved quantity
In mathematics, a conserved quantity of a dynamical system is a function H of the dependent variables that is a constant along each trajectory of the system. A conserved quantity can be a useful tool for qualitative analysis...
. Since mass
Mass
Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...
, energy
Energy
In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...
, momentum
Momentum
In classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...
, electric charge
Electric charge
Electric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two...
and other natural quantities are conserved under their respective appropriate conditions, a variety of physical phenomena may be described using continuity equations.
Continuity equations are the (stronger) local form of conservation law
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....
s. All the examples of continuity equations below express the same idea, which is roughly that: the total amount (of the conserved quantity) inside any region can only change by the amount that passes in or out of the region through the boundary. A conserved quantity cannot increase or decrease, it can only move from place to place.
Any continuity equation can be expressed in an "integral form" (in terms of a flux
Flux
In the various subfields of physics, there exist two common usages of the term flux, both with rigorous mathematical frameworks.* In the study of transport phenomena , flux is defined as flow per unit area, where flow is the movement of some quantity per time...
integral), which applies to any finite region, or in a "differential form" (in terms of the divergence
Divergence
In vector calculus, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around...
operator) which applies at a point.
Differential form
The differential form for a general continuity equation is |
where
- ∇∙ is divergenceDivergenceIn vector calculus, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around...
, - t is timeTimeTime is a part of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify rates of change such as the motions of objects....
, - q is some quantity in transport (such as mass, charge, energy, momentum or probability),
is the amount of q per unit volume (for example, if q is mass, φ is the mass density),- f is a vector functionVector-valued functionA vector-valued function also referred to as a vector function is a mathematical function of one or more variables whose range is a set of multidimensional vectors or infinite-dimensional vectors. The input of a vector-valued function could be a scalar or a vector...
describing the fluxFluxIn the various subfields of physics, there exist two common usages of the term flux, both with rigorous mathematical frameworks.* In the study of transport phenomena , flux is defined as flow per unit area, where flow is the movement of some quantity per time...
(flow) per unit area and unit time of q, - σ is the generation of q per unit volume per unit time. Terms that generate (σ > 0) or remove (σ < 0) q are referred to as a "sources" and "sinks" respectively.
In the case that q is a conserved quantity
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....
that cannot be created or destroyed (such as energy
Energy
In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...
), the continuity equation is:
because σ = 0.
This general equation may be used to derive any continuity equation, ranging from as simple as the volume continuity equation to as complicated as the Navier–Stokes equations. This equation also generalizes the advection equation.
Note the flux f should represent some flow or transport, which has dimensions [quantity][T]−1[L]−2, where [quantity]/[L]3 is the dimension of φ.
Other equations in physics, such as Gauss's law of the electric field
Gauss's law
In physics, Gauss's law, also known as Gauss's flux theorem, is a law relating the distribution of electric charge to the resulting electric field. Gauss's law states that:...
and Gauss's law for gravity, have a similar mathematical form to the continuity equation, but are not usually called by the term "continuity equation", because f in those cases does not represent the flow of a real physical quantity.
Integral form
By the divergence theoremDivergence theorem
In vector calculus, the divergence theorem, also known as Gauss' theorem , Ostrogradsky's theorem , or Gauss–Ostrogradsky theorem is a result that relates the flow of a vector field through a surface to the behavior of the vector field inside the surface.More precisely, the divergence theorem...
(see below), the continuity equation can be rewritten in an equivalent way, called the "integral form":
 |
where
- S is a closed surface that encloses a volume V. S is arbitrary but fixed (unchanging in time) for the calculation,
denotes a surface integralSurface integralIn mathematics, a surface integral is a definite integral taken over a surface ; it can be thought of as the double integral analog of the line integral...
, where
is the outward-pointing unit normal to the surface S,
- N.B:
are equivalent notations for the surface integralSurface integralIn mathematics, a surface integral is a definite integral taken over a surface ; it can be thought of as the double integral analog of the line integral...
, because
,
denotes a volume integralVolume integralIn mathematics — in particular, in multivariable calculus — a volume integral refers to an integral over a 3-dimensional domain....
over V. The notation
is used here since V must be closed (
is sufficient but emphasizes a closed volume less),
is the total amount of φ in the volume V;
is the total generation (negative in the case of removal) per unit time by the sources and sinks in the volume V,
In a simple example, V could be a building, and q could be the number of people in the building. The surface S would consist of the walls, doors, roof, and foundation of the building. Then the continuity equation states that the number of people in the building increases when people enter the building (an inward flux through the surface), decreases when people exit the building (an outward flux through the surface), increases when someone in the building gives birth (a "source" where σ > 0), and decreases when someone in the building dies (a "sink" where σ < 0).
Note the flux f emerging normal from/to the local surface dS satisfies
Derivation and equivalence
The differential form can be derived from first principles as follows.Derivation of the differential form
Suppose first an amount of quantity q is contained in a region of volume V. This is equal to the amount in already in V, plus the generated amount s (total - not per unit time or volume):The rate of change of q leaving the region is simply the time derivative:
where the minus sign has been inserted since the amount of q is decreasing in the region. Partial derivatives are used since they enter the integrand, which is not only a function of time, but also space due to the density nature of φ - differentiation needs only to be with respect to t. The rate of change of q crossing the boundary and leaving the region can be formulated as follows.
- When a velocity field u can be applied to the movement of the quantity, such as a velocity field of dynamic mass or charge, we have:
- However there are cases where it makes no sense to do so, such as the flow of energy or probability. In this case, a flux or current field f has to represent the flow:
- In the cases u can be applied, then f can always be applied by the relation:
Using the divergence theorem on the right-hand side:
Equating these expressions leads to an identity of volume integrals,
this is only true if the integrands are equal, which directly leads to the differential continuity equation:
Either form may be useful and quoted, both can appear in hydrodynamics and electromagnetism, but for quantum mechanics and energy conservation, only the first may be used. Therefore the first is more general.
Equivalence to the integral form
Starting from the differential form which is for unit volume, multiplying throughout by the infinitesimal volume element dV and integrating over the region gives the total amounts quantities in the volume of the region (per unit time):V is constant in shape for the calculation, so it is independent of time and the time derivatives can be freely moved out of the first integral on the left side,
where ordinary derivatives replace partial derivatives since the integral becomes a function of time only (the integral is evaluated over the region - so the spatial variables become removed from the final expression and t remains the only variable). Now using the divergence theorem
Divergence theorem
In vector calculus, the divergence theorem, also known as Gauss' theorem , Ostrogradsky's theorem , or Gauss–Ostrogradsky theorem is a result that relates the flow of a vector field through a surface to the behavior of the vector field inside the surface.More precisely, the divergence theorem...
for the second integral on the left side:
The right side becomes:
Substituting these in obtains the integral form:
3-currents
In electromagnetic theory, the continuity equation can either be regarded as an empirical law expressing (local) charge conservationCharge conservation
In physics, charge conservation is the principle that electric charge can neither be created nor destroyed. The net quantity of electric charge, the amount of positive charge minus the amount of negative charge in the universe, is always conserved...
, or can be derived as a consequence of two of Maxwell's equations
Maxwell's equations
Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.Maxwell's equations...
. It states that the divergence
Divergence
In vector calculus, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around...
of the current density
Current density
Current density is a measure of the density of flow of a conserved charge. Usually the charge is the electric charge, in which case the associated current density is the electric current per unit area of cross section, but the term current density can also be applied to other conserved...
J (in amperes per square meter) is equal to the negative rate of change of the charge density
Charge density
The linear, surface, or volume charge density is the amount of electric charge in a line, surface, or volume, respectively. It is measured in coulombs per meter , square meter , or cubic meter , respectively, and represented by the lowercase Greek letter Rho . Since there are positive as well as...
ρ (in coulombs per cubic metre),
Derivation from Maxwell's equations
Maxwell's equations are a quick way to derive the continuity of charge. One of Maxwell's equationsMaxwell's equations
Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.Maxwell's equations...
, Ampère's law (with Maxwell's correction)
Ampère's law
In classical electromagnetism, Ampère's circuital law, discovered by André-Marie Ampère in 1826, relates the integrated magnetic field around a closed loop to the electric current passing through the loop...
, states that
Taking the divergence of both sides results in
but the divergence of a curl is zero, so that
Another one of Maxwell's equations, Gauss's law
Gauss's law
In physics, Gauss's law, also known as Gauss's flux theorem, is a law relating the distribution of electric charge to the resulting electric field. Gauss's law states that:...
, states that
substitution into the previous equation yeilds the continuity equation
Derivation from first principles
The derivation is simalar to the above general derivation of the continuity equation, simply make the replacements:To actually see how this is done - click here (see also and ).
Suppose an amount of charge q is contained in a region of volume V. This is equal to:
the rate of change of charge leaving the region is simply the time derivative:
where the minus sign has been inserted since the charge is decreasing in the region. Again partial derivatives are used since they enter the integrand - not only a function of time, but space also due to the density function. The rate of change of charge crossing the boundary and leaving the region is also equal to
by the divergence theorem:
Equating these expressions leads to an identity of volume integrals,
hence the integrands must be equal, which directly leads to the continuity equation:
4-currents
Conservation of a current (not necessarily an electromagnetic current) is expressed compactly as the Lorentz invariant divergenceDivergence
In vector calculus, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around...
of a four-current
Four-current
In special and general relativity, the four-current is the Lorentz covariant four-vector that replaces the electromagnetic current density, or indeed any conventional charge current density...
:
where
- c is the speed of lightSpeed of lightThe speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...
- ρ; the charge densityCharge densityThe linear, surface, or volume charge density is the amount of electric charge in a line, surface, or volume, respectively. It is measured in coulombs per meter , square meter , or cubic meter , respectively, and represented by the lowercase Greek letter Rho . Since there are positive as well as...
- j the conventional 3-current densityCurrent densityCurrent density is a measure of the density of flow of a conserved charge. Usually the charge is the electric charge, in which case the associated current density is the electric current per unit area of cross section, but the term current density can also be applied to other conserved...
. - μ; labels the space-timeSpacetimeIn physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
dimensionDimensionIn physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...
so that since
then
implies that the current is conserved:
Interpretation
Current is the movement of charge. The continuity equation says that if charge is moving out of a differential volume (i.e. divergence of current density is positive) then the amount of charge within that volume is going to decrease, so the rate of change of charge density is negative. Therefore the continuity equation amounts to a conservation of charge.Fluid dynamics
In fluid dynamicsFluid dynamics
In physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...
, the continuity equation states that, in any steady state
Steady state
A system in a steady state has numerous properties that are unchanging in time. This implies that for any property p of the system, the partial derivative with respect to time is zero:...
process, the rate at which mass enters a system is equal to the rate at which mass leaves the system. In fluid dynamics, the continuity equation is analogous to Kirchhoff's current law in electric circuits.
The differential form of the continuity equation is:
where
- ρ is fluid densityDensityThe mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...
, - t is time,
- u is the flow velocityFlow velocityIn fluid dynamics the flow velocity, or velocity field, of a fluid is a vector field which is used to mathematically describe the motion of a fluid...
vector fieldVector fieldIn vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...
.
If ρ is a constant, as in the case of incompressible flow
Incompressible flow
In fluid mechanics or more generally continuum mechanics, incompressible flow refers to flow in which the material density is constant within an infinitesimal volume that moves with the velocity of the fluid...
, the mass continuity equation simplifies to a volume continuity equation:
which means that the divergence
Divergence
In vector calculus, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around...
of velocity field is zero everywhere. Physically, this is equivalent to saying that the local volume dilation rate is zero.
Further, the Navier-Stokes equations
Navier-Stokes equations
In physics, the Navier–Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of fluid substances. These equations arise from applying Newton's second law to fluid motion, together with the assumption that the fluid stress is the sum of a diffusing viscous...
form a vector continuity equation describing the conservation of linear momentum.
Derivation from first principles
Again this is simalar to the general derivation above, change the settings to:To see this done click below.
Suppose an amount of mass M is contained in a region of volume V. This is equal to:
the rate of change of mass leaving the region is simply the time derivative:
where the minus sign has been inserted since the mass is decreasing in the region. Again partial derivatives are used since they enter the integrand - not only a function of time, but space also due to the density function. The rate of change of mass crossing the boundary and leaving the region is also equal to
by the divergence theorem:
Equating these expressions leads to an identity of volume integrals,
hence the integrands must be equal, which directly leads to the continuity equation:
Energy
By conservation of energy, which can only be transferred and not created or destroyed leads to a continuity equation, an alternative mathematical statement of energy conservation to the thermodynamic lawsLaws of thermodynamics
The four laws of thermodynamics summarize its most important facts. They define fundamental physical quantities, such as temperature, energy, and entropy, in order to describe thermodynamic systems. They also describe the transfer of energy as heat and work in thermodynamic processes...
.
Letting
- u = local energy densityEnergy densityEnergy density is a term used for the amount of energy stored in a given system or region of space per unit volume. Often only the useful or extractable energy is quantified, which is to say that chemically inaccessible energy such as rest mass energy is ignored...
(energy per unit volume), - q = energy fluxEnergy fluxEnergy flux is the rate of transfer of energy through a surface. The quantity is defined in two different ways, depending on the context:# Rate of energy transfer per unit area...
(transfer of energy per unit cross-sectional area per unit time) as a vector,
the continuity equation is:
Thermodynamics
By Fourier's law for a uniformly conducting medium,where k is the thermal conductivity
Thermal conductivity
In physics, thermal conductivity, k, is the property of a material's ability to conduct heat. It appears primarily in Fourier's Law for heat conduction....
(not Boltzmann constant), this can also be written as:
Quantum mechanics
In quantum mechanicsQuantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
, the conservation of probability also yields a continuity equation. The terms in the equation require these definitions, and are slightly less obvious than the other forms of volume densities, currents, current densities etc., so they are outlined here:
- The wavefunctionWavefunctionNot to be confused with the related concept of the Wave equationA wave function or wavefunction is a probability amplitude in quantum mechanics describing the quantum state of a particle and how it behaves. Typically, its values are complex numbers and, for a single particle, it is a function of...
Ψ for a single particleParticleA particle is, generally, a small localized object to which can be ascribed physical properties. It may also refer to:In chemistry:* Colloidal particle, part of a one-phase system of two or more components where the particles aren't individually visible.In physics:* Subatomic particle, which may be...
in the positionPositionPosition may refer to:* Position , a player role within a team* Position , the orientation of a baby prior to birth* Position , a mathematical identification of relative location...
-timeTimeTime is a part of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify rates of change such as the motions of objects....
space (rather than momentum space) - i.e. functions of position r and time t, Ψ = Ψ(r, t) = (x, y, z, t). - The probability density functionProbability density functionIn probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...
ρ = ρ(r, t) is:
- The probabilityProbabilityProbability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...
that a measurementMeasurementMeasurement is the process or the result of determining the ratio of a physical quantity, such as a length, time, temperature etc., to a unit of measurement, such as the metre, second or degree Celsius...
of the particle's position will yield a value within V at t, denoted by P = Pr ∈ V(t), is:
- The probability currentProbability currentIn quantum mechanics, the probability current is a mathematical quantity describing the flow of probability density. Intuitively; if one pictures the probability density as an inhomogeneous fluid, then the probability current is the rate of flow of this fluid...
(aka probability flux) j:
With these definitions the continuity equation reads:
Either form is usually quoted. Intuitively; considering the above quantities this represents the flow of probability. The chance of finding the particle at some r t flows like a fluid
Fluid
In physics, a fluid is a substance that continually deforms under an applied shear stress. Fluids are a subset of the phases of matter and include liquids, gases, plasmas and, to some extent, plastic solids....
, the particle itself does not flow deterministically in the same vector field
Vector field
In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...
.
Derivation from Schrödinger's equation
For this derivation see for example. The 3-d time dependant Schrödinger equationSchrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....
and its complex conjugate
Complex conjugate
In mathematics, complex conjugates are a pair of complex numbers, both having the same real part, but with imaginary parts of equal magnitude and opposite signs...
(i → –i) throughout are respectively:
where U is the potential function
Potential
*In linguistics, the potential mood*The mathematical study of potentials is known as potential theory; it is the study of harmonic functions on manifolds...
. The partial derivative
Partial derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant...
of ρ with respect to t is:
Multiplying the Schrödinger equation by Ψ* then solving for
, and similarly multiplying the complex conjugated Schrödinger equation by Ψ then solving for ;substituting into the time derivative of ρ:
The Laplacian
Laplace operator
In mathematics the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a function on Euclidean space. It is usually denoted by the symbols ∇·∇, ∇2 or Δ...
operators (∇2) in the above result suggest that the right hand side is the divergence of j, and the reversed order of terms imply this is the negative of j, altogether:
so the continuity equation is:
The intgeral form follows as for the general equation.
Derivation from the wavefunction probability distribution
The time derivative of P iswhere the last equality follows from the product rule and the fact that the shape of V is fixed for the calculation and therefore independent of time - i.e. the time derivative can be moved through the integral. To simplify this further consider again the time dependent Schrödinger equation
Schrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....
and its complex conjugate, in terms of the time derivatives of Ψ and Ψ* respectively:
Substituting into the preceding equation:
.From the product rule for the divergence operator
substituting:
On the right side, the argument of the divergence operator is j,
using the divergence theorem again gives the integral form:
To obtain the differential form:
The differential form follows from the fact that the preceding equation holds for all V, and as the integrand is a continuous function of space, it must vanish everywhere:
See also
- Conservation lawConservation lawIn physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves....
- Euler equationsEuler equationsIn fluid dynamics, the Euler equations are a set of equations governing inviscid flow. They are named after Leonhard Euler. The equations represent conservation of mass , momentum, and energy, corresponding to the Navier–Stokes equations with zero viscosity and heat conduction terms. Historically,...
- Groundwater energy balanceGroundwater energy balanceThe groundwater energy balance is the energy balance of a groundwater body in terms of incoming hydraulic energy associated with groundwater inflow into the body, energy associated with the outflow, energy conversion into heat due to friction of flow, and the resulting change of energy status and...
- Incompressible fluid
- Mass flow rateMass flow rateMass flow rate is the mass of substance which passes through a given surface per unit time. Its unit is mass divided by time, so kilogram per second in SI units, and slug per second or pound per second in US customary units...
- Noether's TheoremNoether's theoremNoether's theorem states that any differentiable symmetry of the action of a physical system has a corresponding conservation law. The theorem was proved by German mathematician Emmy Noether in 1915 and published in 1918...
- Probability density functionProbability density functionIn probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...
- Schrödinger equationSchrödinger equationThe Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....