Static force fields are fields, such as a simple
electric,
magnetic
Magnetism is the class of physical attributes that are mediated by a magnetic field, which refers to the capacity to induce attractive and repulsive phenomena in other entities. Electric currents and the magnetic moments of elementary particle ...
or
gravitational field
In physics, a gravitational field is a model used to explain the influences that a massive body extends into the space around itself, producing a force on another massive body. Thus, a gravitational field is used to explain gravitational phenome ...
s, that exist without excitations. The
most common approximation method that physicists use for
scattering calculations can be interpreted as static forces arising from the interactions between two bodies mediated by
virtual particle
A virtual particle is a theoretical transient particle that exhibits some of the characteristics of an ordinary particle, while having its existence limited by the uncertainty principle. The concept of virtual particles arises in the perturbat ...
s, particles that exist for only a short time determined by the
uncertainty principle
In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physic ...
. The virtual particles, also known as
force carrier In quantum field theory, a force carrier, also known as messenger particle or intermediate particle, is a type of particle that gives rise to forces between other particles. These particles serve as the quanta of a particular kind of physical field ...
s, are
bosons, with different bosons associated with each force.
The virtual-particle description of static forces is capable of identifying the spatial form of the forces, such as the inverse-square behavior in
Newton's law of universal gravitation and in
Coulomb's law
Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is conventiona ...
. It is also able to predict whether the forces are attractive or repulsive for like bodies.
The
path integral formulation is the natural language for describing force carriers. This article uses the path integral formulation to describe the force carriers for
spin
Spin or spinning most often refers to:
* Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning
* Spin, the rotation of an object around a central axis
* Spin (propaganda), an intentionally b ...
0, 1, and 2 fields.
Pions,
photon
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they always ...
s, and
gravitons fall into these respective categories.
There are limits to the validity of the virtual particle picture. The virtual-particle formulation is derived from a method known as
perturbation theory
In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle ...
which is an approximation assuming interactions are not too strong, and was intended for scattering problems, not bound states such as atoms. For the strong force binding
quark
A quark () is a type of elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. All commonly o ...
s into
nucleons at low energies, perturbation theory has never been shown to yield results in accord with experiments, thus, the validity of the "force-mediating particle" picture is questionable. Similarly, for
bound states the method fails. In these cases, the physical interpretation must be re-examined. As an example, the calculations of atomic structure in atomic physics or of molecular structure in quantum chemistry could not easily be repeated, if at all, using the "force-mediating particle" picture.
Use of the "force-mediating particle" picture (FMPP) is unnecessary in
nonrelativistic quantum mechanics, and Coulomb's law is used as given in atomic physics and quantum chemistry to calculate both bound and scattering states. A non-perturbative
relativistic quantum theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
, in which Lorentz invariance is preserved, is achievable by evaluating Coulomb's law as a 4-space interaction using the 3-space position vector of a reference electron obeying Dirac's equation and the quantum trajectory of a second electron which depends only on the scaled time. The quantum trajectory of each electron in an ensemble is inferred from the Dirac current for each electron by setting it equal to a velocity field times a quantum density, calculating a position field from the time integral of the velocity field, and finally calculating a quantum trajectory from the expectation value of the position field. The quantum trajectories are of course spin dependent, and the theory can be validated by checking that
Pauli's exclusion principle
In quantum mechanics, the Pauli exclusion principle states that two or more identical particles with half-integer spins (i.e. fermions) cannot occupy the same quantum state within a quantum system simultaneously. This principle was formulate ...
is obeyed for a collection of
fermion
In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks an ...
s.
Classical forces
The force exerted by one mass on another and the force exerted by one charge on another are strikingly similar. Both fall off as the square of the distance between the bodies. Both are proportional to the product of properties of the bodies, mass in the case of gravitation and charge in the case of electrostatics.
They also have a striking difference. Two masses attract each other, while two like charges repel each other.
In both cases, the bodies appear to act on each other over a distance. The concept of
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
was invented to mediate the interaction among bodies thus eliminating the need for
action at a distance. The gravitational force is mediated by the
gravitational field
In physics, a gravitational field is a model used to explain the influences that a massive body extends into the space around itself, producing a force on another massive body. Thus, a gravitational field is used to explain gravitational phenome ...
and the Coulomb force is mediated by the
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 ...
.
Gravitational force
The
gravitational force on a mass
exerted by another mass
is
where is the
gravitational constant
The gravitational constant (also known as the universal gravitational constant, the Newtonian constant of gravitation, or the Cavendish gravitational constant), denoted by the capital letter , is an empirical physical constant involved in ...
, is the distance between the masses, and
is the
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 ...
from mass
to mass
.
The force can also be written
where
is the
gravitational field
In physics, a gravitational field is a model used to explain the influences that a massive body extends into the space around itself, producing a force on another massive body. Thus, a gravitational field is used to explain gravitational phenome ...
described by the field equation
where
is the
mass density
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...
at each point in space.
Coulomb force
The electrostatic
Coulomb force
Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is conventiona ...
on a charge
exerted by a charge
is (
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 ...
)
where
is the
vacuum permittivity
Vacuum permittivity, commonly denoted (pronounced "epsilon nought" or "epsilon zero"), is the value of the absolute dielectric permittivity of classical vacuum. It may also be referred to as the permittivity of free space, the electric consta ...
,
is the separation of the two charges, and
is 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 ...
in the direction from charge
to charge
.
The Coulomb force can also be written in terms of an
electrostatic field
An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field fo ...
:
where
being the
charge density
In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. Volume charge density (symbolized by the Greek letter ρ) is the quantity of charge per unit volume, measured in the SI system in co ...
at each point in space.
Virtual-particle exchange
In perturbation theory, forces are generated by the exchange of
virtual particle
A virtual particle is a theoretical transient particle that exhibits some of the characteristics of an ordinary particle, while having its existence limited by the uncertainty principle. The concept of virtual particles arises in the perturbat ...
s. The mechanics of virtual-particle exchange is best described with the
path integral formulation of quantum mechanics. There are insights that can be obtained, however, without going into the machinery of path integrals, such as why classical gravitational and electrostatic forces fall off as the inverse square of the distance between bodies.
Path-integral formulation of virtual-particle exchange
A virtual particle is created by a disturbance to the
vacuum state, and the virtual particle is destroyed when it is absorbed back into the vacuum state by another disturbance. The disturbances are imagined to be due to bodies that interact with the virtual particle’s field.
The probability amplitude
Using
natural units,
, the probability amplitude for the creation, propagation, and destruction of a virtual particle is given, in the
path integral formulation by
where
is the
Hamiltonian operator,
is elapsed time,
is the energy change due to the disturbance,
is the change in action due to the disturbance,
is the field of the virtual particle, the integral is over all paths, and the classical
action is given by
where
is the
Lagrangian
Lagrangian may refer to:
Mathematics
* Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier
** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
density.
Here, the
spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
metric is given by
The path integral often can be converted to the form
where
is a differential operator with
and
functions of
spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
. The first term in the argument represents the free particle and the second term represents the disturbance to the field from an external source such as a charge or a mass.
The integral can be written (see )
where
is the change in the action due to the disturbances and the
propagator
In quantum mechanics and quantum field theory, the propagator is a function that specifies the probability amplitude for a particle to travel from one place to another in a given period of time, or to travel with a certain energy and momentum. In ...
is the solution of
Energy of interaction
We assume that there are two point disturbances representing two bodies and that the disturbances are motionless and constant in time. The disturbances can be written
where the delta functions are in space, the disturbances are located at
and
, and the coefficients
and
are the strengths of the disturbances.
If we neglect self-interactions of the disturbances then W becomes
which can be written
Here
is the Fourier transform of
Finally, the change in energy due to the static disturbances of the vacuum is
If this quantity is negative, the force is attractive. If it is positive, the force is repulsive.
Examples of static, motionless, interacting currents are the
Yukawa potential, the
Coulomb potential in a vacuum, and the
Coulomb potential in a simple plasma or electron gas.
The expression for the interaction energy can be generalized to the situation in which the point particles are moving, but the motion is slow compared with the speed of light. Examples are the Darwin interaction
in a vacuum and
in a plasma.
Finally, the expression for the interaction energy can be generalized to situations in which the disturbances are not point particles, but are possibly line charges, tubes of charges, or current vortices. Examples include:
two line charges embedded in a plasma or electron gas,
Coulomb potential between two current loops embedded in a magnetic field, and the
magnetic interaction between current loops in a simple plasma or electron gas. As seen from the Coulomb interaction between tubes of charge example, shown below, these more complicated geometries can lead to such exotic phenomena as
fractional quantum numbers.
Selected examples
The Yukawa potential: The force between two nucleons in an atomic nucleus
Consider the
spin
Spin or spinning most often refers to:
* Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning
* Spin, the rotation of an object around a central axis
* Spin (propaganda), an intentionally b ...
-0 Lagrangian density
The equation of motion for this Lagrangian is the
Klein–Gordon equation
If we add a disturbance the probability amplitude becomes
If we integrate by parts and neglect boundary terms at infinity the probability amplitude becomes
With the amplitude in this form it can be seen that the propagator is the solution of
From this it can be seen that
The energy due to the static disturbances becomes (see )
with
which is attractive and has a range of
Yukawa Yukawa (written: 湯川) is a Japanese surname, but is also applied to proper nouns.
People
* Diana Yukawa (born 1985), Anglo-Japanese solo violinist. She has had two solo albums with BMG Japan, one of which opened to #1
* Hideki Yukawa (1907–1 ...
proposed that this field describes the force between two
nucleons in an atomic nucleus. It allowed him to predict both the range and the mass of the particle, now known as the
pion, associated with this field.
Electrostatics
The Coulomb potential in a vacuum
Consider the
spin
Spin or spinning most often refers to:
* Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning
* Spin, the rotation of an object around a central axis
* Spin (propaganda), an intentionally b ...
-1
Proca Lagrangian
In physics, specifically field theory and particle physics, the Proca action describes a massive spin-1 field of mass ''m'' in Minkowski spacetime. The corresponding equation is a relativistic wave equation called the Proca equation. The Proca a ...
with a disturbance
where
charge is conserved
and we choose the
Lorenz gauge
In electromagnetism, the Lorenz gauge condition or Lorenz gauge, for Ludvig Lorenz, is a partial gauge fixing of the electromagnetic vector potential by requiring \partial_\mu A^\mu = 0. The name is frequently confused with Hendrik Lorentz, who ha ...
Moreover, we assume that there is only a time-like component
to the disturbance. In ordinary language, this means that there is a charge at the points of disturbance, but there are no electric currents.
If we follow the same procedure as we did with the Yukawa potential we find that
which implies
and
This yields
for the
timelike propagator and
which has the opposite sign to the Yukawa case.
In the limit of zero
photon
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they always ...
mass, the Lagrangian reduces to the Lagrangian for
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 ...
Therefore the energy reduces to the potential energy for the Coulomb force and the coefficients
and
are proportional to the electric charge. Unlike the Yukawa case, like bodies, in this electrostatic case, repel each other.
Coulomb potential in a simple plasma or electron gas
=Plasma waves
=
The
dispersion relation for
plasma wave In plasma physics, waves in plasmas are an interconnected set of particles and fields which propagate in a periodically repeating fashion. A plasma is a quasineutral, electrically conductive fluid. In the simplest case, it is composed of electron ...
s is
where
is the angular frequency of the wave,
is the
plasma frequency,
is the magnitude of the
electron charge,
is the
electron mass
The electron mass (symbol: ''m''e) is the mass of a stationary electron, also known as the invariant mass of the electron. It is one of the fundamental constants of physics. It has a value of about or about , which has an energy-equivalent of a ...
,
is the electron
temperature
Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer.
Thermometers are calibrated in various temperature scales that historically have relied o ...
(
Boltzmann's constant equal to one), and
is a factor that varies with frequency from one to three. At high frequencies, on the order of the plasma frequency, the compression of the electron fluid is an
adiabatic process
In thermodynamics, an adiabatic process (Greek: ''adiábatos'', "impassable") is a type of thermodynamic process that occurs without transferring heat or mass between the thermodynamic system and its environment. Unlike an isothermal process, ...
and
is equal to three. At low frequencies, the compression is an
isothermal process
In thermodynamics, an isothermal process is a type of thermodynamic process in which the temperature ''T'' of a system remains constant: Δ''T'' = 0. This typically occurs when a system is in contact with an outside thermal reservoir, and ...
and
is equal to one.
Retardation effects have been neglected in obtaining the plasma-wave dispersion relation.
For low frequencies, the dispersion relation becomes
where
is the Debye number, which is the inverse of the
Debye length. This suggests that the propagator is
In fact, if the retardation effects are not neglected, then the dispersion relation is
which does indeed yield the guessed propagator. This propagator is the same as the massive Coulomb propagator with the mass equal to the inverse Debye length. The interaction energy is therefore
The Coulomb potential is screened on length scales of a Debye length.
=Plasmons
=
In a quantum
electron gas, plasma waves are known as
plasmons. Debye screening is replaced with
Thomas–Fermi screening Thomas–Fermi screening is a theoretical approach to calculate the effects of electric field screening by electrons in a solid.N. W. Ashcroft and N. D. Mermin, ''Solid State Physics'' (Thomson Learning, Toronto, 1976) It is a special case of the mo ...
to yield
[ pp. 296-299.]
where the inverse of the Thomas–Fermi screening length is
and
is the
Fermi energy
This expression can be derived from the
chemical potential
In thermodynamics, the chemical potential of a species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potential of a species ...
for an electron gas and from
Poisson's equation
Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with th ...
. The chemical potential for an electron gas near equilibrium is constant and given by
where
is the
electric potential
The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in ...
. Linearizing the Fermi energy to first order in the density fluctuation and combining with Poisson's equation yields the screening length. The force carrier is the quantum version of the
plasma wave In plasma physics, waves in plasmas are an interconnected set of particles and fields which propagate in a periodically repeating fashion. A plasma is a quasineutral, electrically conductive fluid. In the simplest case, it is composed of electron ...
.
=Two line charges embedded in a plasma or electron gas
=
We consider a line of charge with axis in the ''z'' direction embedded in an electron gas
where
is the distance in the ''xy''-plane from the line of charge,
is the width of the material in the z direction. The superscript 2 indicates that the
Dirac delta function
In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...
is in two dimensions. The propagator is
where
is either the inverse
Debye–Hückel screening length or the inverse
Thomas–Fermi screening Thomas–Fermi screening is a theoretical approach to calculate the effects of electric field screening by electrons in a solid.N. W. Ashcroft and N. D. Mermin, ''Solid State Physics'' (Thomson Learning, Toronto, 1976) It is a special case of the mo ...
length.
The interaction energy is
where
and
are
Bessel function
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrary ...
s and
is the distance between the two line charges. In obtaining the interaction energy we made use of the integrals (see )
and
For
, we have
Coulomb potential between two current loops embedded in a magnetic field
=Interaction energy for vortices
=
We consider a charge density in tube with axis along a magnetic field embedded in an electron gas
where
is the distance from the
guiding center,
is the width of the material in the direction of the magnetic field
where the
cyclotron frequency
Cyclotron resonance describes the interaction of external forces with charged particles experiencing a magnetic field, thus already moving on a circular path. It is named after the cyclotron, a cyclic particle accelerator that utilizes an oscillati ...
is (
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 unit ...
)
and
is the speed of the particle about the magnetic field, and B is the magnitude of the magnetic field. The speed formula comes from setting the classical kinetic energy equal to the spacing between
Landau levels In quantum mechanics, Landau quantization refers to the quantization of the cyclotron orbits of charged particles in a uniform magnetic field. As a result, the charged particles can only occupy orbits with discrete, equidistant energy values, call ...
in the quantum treatment of a charged particle in a magnetic field.
In this geometry, the interaction energy can be written
where
is the distance between the centers of the current loops and
is a
Bessel function
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrary ...
of the first kind. In obtaining the interaction energy we made use of the integral
=Electric field due to a density perturbation
=
The
chemical potential
In thermodynamics, the chemical potential of a species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potential of a species ...
near equilibrium, is given by
where
is the
potential energy
In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors.
Common types of potential energy include the gravitational potentia ...
of an electron in an
electric potential
The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in ...
and
and
are the number of particles in the electron gas in the absence of and in the presence of an electrostatic potential, respectively.
The density fluctuation is then
where
is the area of the material in the plane perpendicular to the magnetic field.
Poisson's equation
Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with th ...
yields
where
The propagator is then
and the interaction energy becomes
where in the second equality (
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 unit ...
) we assume that the vortices had the same energy and the electron charge.
In analogy with
plasmons, the
force carrier In quantum field theory, a force carrier, also known as messenger particle or intermediate particle, is a type of particle that gives rise to forces between other particles. These particles serve as the quanta of a particular kind of physical field ...
is the quantum version of the
upper hybrid oscillation
In plasma physics, an upper hybrid oscillation is a mode of oscillation of a magnetized plasma. It consists of a longitudinal motion of the electrons perpendicular to the magnetic field with the dispersion relation
:
\omega^2 = \omega_^2 + \ome ...
which is a longitudinal
plasma wave In plasma physics, waves in plasmas are an interconnected set of particles and fields which propagate in a periodically repeating fashion. A plasma is a quasineutral, electrically conductive fluid. In the simplest case, it is composed of electron ...
that propagates perpendicular to the magnetic field.
=Currents with angular momentum
=
Delta function currents
Unlike classical currents, quantum current loops can have various values of the
Larmor radius The gyroradius (also known as radius of gyration, Larmor radius or cyclotron radius) is the radius of the circular motion of a charged particle in the presence of a uniform magnetic field. In SI units, the non-relativistic gyroradius is given by
:r_ ...
for a given energy.
Landau level In quantum mechanics, Landau quantization refers to the quantization of the cyclotron orbits of charged particles in a uniform magnetic field. As a result, the charged particles can only occupy orbits with discrete, equidistant energy values, call ...
s, the energy states of a charged particle in the presence of a magnetic field, are multiply
degenerate. The current loops correspond to
angular momentum
In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
states of the charged particle that may have the same energy. Specifically, the charge density is peaked around radii of
where
is the angular momentum
quantum number
In quantum physics and chemistry, quantum numbers describe values of conserved quantities in the dynamics of a quantum system. Quantum numbers correspond to eigenvalues of operators that commute with the Hamiltonian—quantities that can be kno ...
. When
we recover the classical situation in which the electron orbits the magnetic field at the
Larmor radius The gyroradius (also known as radius of gyration, Larmor radius or cyclotron radius) is the radius of the circular motion of a charged particle in the presence of a uniform magnetic field. In SI units, the non-relativistic gyroradius is given by
:r_ ...
. If currents of two angular momentum
and
interact, and we assume the charge densities are delta functions at radius
, then the interaction energy is
The interaction energy for
is given in Figure 1 for various values of
. The energy for two different values is given in Figure 2.
Quasiparticles
For large values of angular momentum, the energy can have local minima at distances other than zero and infinity. It can be numerically verified that the minima occur at
This suggests that the pair of particles that are bound and separated by a distance
act as a single
quasiparticle with angular momentum
.
If we scale the lengths as
, then the interaction energy becomes
where
The value of the
at which the energy is minimum,
, is independent of the ratio
. However the value of the energy at the minimum depends on the ratio. The lowest energy minimum occurs when
When the ratio differs from 1, then the energy minimum is higher (Figure 3). Therefore, for even values of total momentum, the lowest energy occurs when (Figure 4)
or
where the total angular momentum is written as
When the total angular momentum is odd, the minima cannot occur for
The lowest energy states for odd total angular momentum occur when
or
and
which also appear as series for the filling factor in the
fractional quantum Hall effect
The fractional quantum Hall effect (FQHE) is a physical phenomenon in which the Hall conductance of 2-dimensional (2D) electrons shows precisely quantized plateaus at fractional values of e^2/h. It is a property of a collective state in which elec ...
.
Charge density spread over a wave function
The charge density is not actually concentrated in a delta function. The charge is spread over a wave function. In that case the electron density is
The interaction energy becomes
where
is a
confluent hypergeometric function
In mathematics, a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential equation where two of the three regular singularities merge into an irregular ...
or
Kummer function. In obtaining the interaction energy we have used the integral (see )
As with delta function charges, the value of
in which the energy is a local minimum only depends on the total angular momentum, not on the angular momenta of the individual currents. Also, as with the delta function charges, the energy at the minimum increases as the ratio of angular momenta varies from one. Therefore, the series
and
appear as well in the case of charges spread by the wave function.
The
Laughlin wavefunction In condensed matter physics, the Laughlin wavefunction pp. 210-213 is an ansatz, proposed by Robert Laughlin for the ground state of a two-dimensional electron gas placed in a uniform background magnetic field in the presence of a uniform jellium ...
is an
ansatz for the quasiparticle wavefunction. If the expectation value of the interaction energy is taken over a
Laughlin wavefunction In condensed matter physics, the Laughlin wavefunction pp. 210-213 is an ansatz, proposed by Robert Laughlin for the ground state of a two-dimensional electron gas placed in a uniform background magnetic field in the presence of a uniform jellium ...
, these series are also preserved.
Magnetostatics
Darwin interaction in a vacuum
A charged moving particle can generate a magnetic field that affects the motion of another charged particle. The static version of this effect is called the
Darwin interaction. To calculate this, consider the electrical currents in space generated by a moving charge
with a comparable expression for
.
The Fourier transform of this current is
The current can be decomposed into a transverse and a longitudinal part (see
Helmholtz decomposition).
The hat indicates 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 ...
. The last term disappears because
which results from charge conservation. Here
vanishes because we are considering static forces.
With the current in this form the energy of interaction can be written
The propagator equation for the Proca Lagrangian is
The
spacelike solution is
which yields
which evaluates to (see )
which reduces to
in the limit of small . The interaction energy is the negative of the interaction Lagrangian. For two like particles traveling in the same direction, the interaction is attractive, which is the opposite of the Coulomb interaction.
Darwin interaction in a plasma
In a plasma, the
dispersion relation for an
electromagnetic wave
In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visib ...
is
(
)
which implies
Here
is the
plasma frequency. The interaction energy is therefore
Magnetic interaction between current loops in a simple plasma or electron gas
=The interaction energy
=
Consider a tube of current rotating in a magnetic field embedded in a simple
plasma
Plasma or plasm may refer to:
Science
* Plasma (physics), one of the four fundamental states of matter
* Plasma (mineral), a green translucent silica mineral
* Quark–gluon plasma, a state of matter in quantum chromodynamics
Biology
* Blood pla ...
or electron gas. The current, which lies in the plane perpendicular to the magnetic field, is defined as
where
and
is the unit vector in the direction of the magnetic field. Here
indicates the dimension of the material in the direction of the magnetic field. The transverse current, perpendicular to the
wave vector, drives the
transverse wave
In physics, a transverse wave is a wave whose oscillations are perpendicular to the direction of the wave's advance. This is in contrast to a longitudinal wave which travels in the direction of its oscillations. Water waves are an example of t ...
.
The energy of interaction is
where
is the distance between the centers of the current loops and
is a
Bessel function
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrary ...
of the first kind. In obtaining the interaction energy we made use of the integrals
and
See .
A current in a plasma confined to the plane perpendicular to the magnetic field generates an
extraordinary wave.
This wave generates
Hall current
The Hall effect is the production of a voltage difference (the Hall voltage) across an electrical conductor that is transverse to an electric current in the conductor and to an applied magnetic field perpendicular to the current. It was disco ...
s that interact and modify the electromagnetic field. The
dispersion relation for extraordinary waves is
which gives for the propagator
where
in analogy with the Darwin propagator. Here, the upper hybrid frequency is given by
the
cyclotron frequency
Cyclotron resonance describes the interaction of external forces with charged particles experiencing a magnetic field, thus already moving on a circular path. It is named after the cyclotron, a cyclic particle accelerator that utilizes an oscillati ...
is given by (
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 unit ...
)
and the
plasma frequency (
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 unit ...
)
Here is the electron density, is the magnitude of the electron charge, and is the electron mass.
The interaction energy becomes, for like currents,
=Limit of small distance between current loops
=
In the limit that the distance between current loops is small,
where
and
and and are modified Bessel functions. we have assumed that the two currents have the same charge and speed.
We have made use of the integral (see )
For small the integral becomes
For large the integral becomes
=Relation to the quantum Hall effect
=
The screening
wavenumber
In the physical sciences, the wavenumber (also wave number or repetency) is the ''spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to temp ...
can be written (
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 unit ...
)
where
is the
fine-structure constant and the filling factor is
and is the number of electrons in the material and is the area of the material perpendicular to the magnetic field. This parameter is important in the
quantum Hall effect and the
fractional quantum Hall effect
The fractional quantum Hall effect (FQHE) is a physical phenomenon in which the Hall conductance of 2-dimensional (2D) electrons shows precisely quantized plateaus at fractional values of e^2/h. It is a property of a collective state in which elec ...
. The filling factor is the fraction of occupied
Landau states at the ground state energy.
For cases of interest in the quantum Hall effect,
is small. In that case the interaction energy is