Quantum chemistry

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Quantum_chemistry 

Quantum chemistry, also called molecular quantum mechanics, is a branch of physical chemistry focused on the application of quantum mechanics to chemical systems, particularly towards the quantum-mechanical calculation of electronic contributions to physical and chemical properties of molecules, materials, and solutions at the atomic level. These calculations include systematically applied approximations intended to make calculations computationally feasible while still capturing as much information about important contributions to the computed wave functions as well as to observable properties such as structures, spectra, and thermodynamic properties. Quantum chemistry is also concerned with the computation of quantum effects on molecular dynamics and chemical kinetics.

Chemists rely heavily on spectroscopy through which information regarding the quantization of energy on a molecular scale can be obtained. Common methods are infra-red (IR) spectroscopy, nuclear magnetic resonance (NMR) spectroscopy, and scanning probe microscopy. Quantum chemistry may be applied to the prediction and verification of spectroscopic data as well as other experimental data.

Many quantum chemistry studies are focused on the electronic ground state and excited states of individual atoms and molecules as well as the study of reaction pathways and transition states that occur during chemical reactions. Spectroscopic properties may also be predicted. Typically, such studies assume the electronic wave function is adiabatically parameterized by the nuclear positions (i.e., the Born–Oppenheimer approximation). A wide variety of approaches are used, including semi-empirical methods, density functional theory, Hartree–Fock calculations, quantum Monte Carlo methods, and coupled cluster methods.

Understanding electronic structure and molecular dynamics through the development of computational solutions to the Schrödinger equation is a central goal of quantum chemistry. Progress in the field depends on overcoming several challenges, including the need to increase the accuracy of the results for small molecular systems, and to also increase the size of large molecules that can be realistically subjected to computation, which is limited by scaling considerations — the computation time increases as a power of the number of atoms.

History

Some view the birth of quantum chemistry as starting with the discovery of the Schrödinger equation and its application to the hydrogen atom. However, a 1927 article of Walter Heitler (1904–1981) and Fritz London is often recognized as the first milestone in the history of quantum chemistry. This was the first application of quantum mechanics to the diatomic hydrogen molecule, and thus to the phenomenon of the chemical bond. However, prior to this a critical conceptual framework was provided by Gilbert N. Lewis in his 1916 paper The Atom and the Molecule, wherein Lewis developed the first working model of valence electrons. Important contributions were also made by Yoshikatsu Sugiura and S.C. Wang. A series of articles by Linus Pauling, written throughout the 1930s, integrated the work of Heitler, London, Sugiura, Wang, Lewis, and John C. Slater on the concept of valence and its quantum-mechanical basis into a new theoretical framework. Many chemists were introduced to the field of quantum chemistry by Pauling's 1939 text The Nature of the Chemical Bond and the Structure of Molecules and Crystals: An Introduction to Modern Structural Chemistry, wherein he summarized this work (referred to widely now as valence bond theory) and explained quantum mechanics in a way which could be followed by chemists. The text soon became a standard text at many universities. In 1937, Hans Hellmann appears to have been the first to publish a book on quantum chemistry, in the Russian and German languages.

In the years to follow, this theoretical basis slowly began to be applied to chemical structure, reactivity, and bonding. In addition to the investigators mentioned above, important progress and critical contributions were made in the early years of this field by Irving Langmuir, Robert S. Mulliken, Max Born, J. Robert Oppenheimer, Hans Hellmann, Maria Goeppert Mayer, Erich Hückel, Douglas Hartree, John Lennard-Jones, and Vladimir Fock.

Electronic structure

The electronic structure of an atom or molecule is the quantum state of its electrons. The first step in solving a quantum chemical problem is usually solving the Schrödinger equation (or Dirac equation in relativistic quantum chemistry) with the electronic molecular Hamiltonian, usually making use of the Born–Oppenheimer (B–O) approximation. This is called determining the electronic structure of the molecule. An exact solution for the non-relativistic Schrödinger equation can only be obtained for the hydrogen atom (though exact solutions for the bound state energies of the hydrogen molecular ion within the B-O approximation have been identified in terms of the generalized Lambert W function). Since all other atomic and molecular systems involve the motions of three or more "particles", their Schrödinger equations cannot be solved analytically and so approximate and/or computational solutions must be sought. The process of seeking computational solutions to these problems is part of the field known as computational chemistry.

Valence bond theory

Main article: Valence bond theory

As mentioned above, Heitler and London's method was extended by Slater and Pauling to become the valence-bond (VB) method. In this method, attention is primarily devoted to the pairwise interactions between atoms, and this method therefore correlates closely with classical chemists' drawings of bonds. It focuses on how the atomic orbitals of an atom combine to give individual chemical bonds when a molecule is formed, incorporating the two key concepts of orbital hybridization and resonance.

Molecular orbital theory

An anti-bonding molecular orbital of Butadiene

Main article: Molecular orbital theory

An alternative approach to valence bond theory was developed in 1929 by Friedrich Hund and Robert S. Mulliken, in which electrons are described by mathematical functions delocalized over an entire molecule. The Hund–Mulliken approach or molecular orbital (MO) method is less intuitive to chemists, but has turned out capable of predicting spectroscopic properties better than the VB method. This approach is the conceptual basis of the Hartree–Fock method and further post-Hartree–Fock methods.

Density functional theory

Main article: Density functional theory

The Thomas–Fermi model was developed independently by Thomas and Fermi in 1927. This was the first attempt to describe many-electron systems on the basis of electronic density instead of wave functions, although it was not very successful in the treatment of entire molecules. The method did provide the basis for what is now known as density functional theory (DFT). Modern day DFT uses the Kohn–Sham method, where the density functional is split into four terms; the Kohn–Sham kinetic energy, an external potential, exchange and correlation energies. A large part of the focus on developing DFT is on improving the exchange and correlation terms. Though this method is less developed than post Hartree–Fock methods, its significantly lower computational requirements (scaling typically no worse than n³ with respect to n basis functions, for the pure functionals) allow it to tackle larger polyatomic molecules and even macromolecules. This computational affordability and often comparable accuracy to MP2 and CCSD(T) (post-Hartree–Fock methods) has made it one of the most popular methods in computational chemistry.

Chemical dynamics

A further step can consist of solving the Schrödinger equation with the total molecular Hamiltonian in order to study the motion of molecules. Direct solution of the Schrödinger equation is called quantum dynamics, whereas its solution within the semiclassical approximation is called semiclassical dynamics. Purely classical simulations of molecular motion are referred to as molecular dynamics (MD). Another approach to dynamics is a hybrid framework known as mixed quantum-classical dynamics; yet another hybrid framework uses the Feynman path integral formulation to add quantum corrections to molecular dynamics, which is called path integral molecular dynamics. Statistical approaches, using for example classical and quantum Monte Carlo methods, are also possible and are particularly useful for describing equilibrium distributions of states.

Adiabatic chemical dynamics

Main article: Born–Oppenheimer approximation

In adiabatic dynamics, interatomic interactions are represented by single scalar potentials called potential energy surfaces. This is the Born–Oppenheimer approximation introduced by Born and Oppenheimer in 1927. Pioneering applications of this in chemistry were performed by Rice and Ramsperger in 1927 and Kassel in 1928, and generalized into the RRKM theory in 1952 by Marcus who took the transition state theory developed by Eyring in 1935 into account. These methods enable simple estimates of unimolecular reaction rates from a few characteristics of the potential surface.

Non-adiabatic chemical dynamics

Main article: Vibronic coupling

Non-adiabatic dynamics consists of taking the interaction between several coupled potential energy surfaces (corresponding to different electronic quantum states of the molecule). The coupling terms are called vibronic couplings. The pioneering work in this field was done by Stueckelberg, Landau, and Zener in the 1930s, in their work on what is now known as the Landau–Zener transition. Their formula allows the transition probability between two adiabatic potential curves in the neighborhood of an avoided crossing to be calculated. Spin-forbidden reactions are one type of non-adiabatic reactions where at least one change in spin state occurs when progressing from reactant to product.

Static forces and virtual-particle exchange

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Static_forces_and_virtual-particle_exchange 

Static force fields are fields, such as a simple electric, magnetic or gravitational fields, 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 particles, particles that exist for only a short time determined by the uncertainty principle. The virtual particles, also known as force carriers, 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. 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 0, 1, and 2 fields. Pions, photons, 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 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 quarks 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 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 is obeyed for a collection of fermions.

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 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 and the Coulomb force is mediated by the electromagnetic field.

Gravitational force

The gravitational force on a mass ${\displaystyle m}$ exerted by another mass ${\displaystyle M}$ is $${\displaystyle \mathbf{F}=-G\frac{mM}{r^2}\hat{\mathbf{r}}=m\mathbf{g}\left(\mathbf{r}\right),}$$ where G is the Newtonian constant of gravitation, r is the distance between the masses, and ${\displaystyle \hat{\mathbf{r}}}$ is the unit vector from mass ${\displaystyle M}$ to mass ${\displaystyle m}$.

The force can also be written $${\displaystyle \mathbf{F}=m\mathbf{g}\left(\mathbf{r}\right),}$$ where ${\displaystyle \mathbf{g}\left(\mathbf{r}\right)}$ is the gravitational field described by the field equation $${\displaystyle \mathrm{\nabla }\cdot \mathbf{g}=-4\pi G{\rho }_m,}$$ where ${\displaystyle {\rho }_m}$ is the mass density at each point in space.

Coulomb force

The electrostatic Coulomb force on a charge ${\displaystyle q}$ exerted by a charge ${\displaystyle Q}$ is (SI units) $${\displaystyle \mathbf{F}=\frac{1}{4\pi {\epsilon }_0}\frac{qQ}{r^2}\hat{\mathbf{r}},}$$ where ${\displaystyle {\epsilon }_0}$ is the vacuum permittivity, ${\displaystyle r}$ is the separation of the two charges, and ${\displaystyle \hat{\mathbf{r}}}$ is a unit vector in the direction from charge ${\displaystyle Q}$ to charge ${\displaystyle q}$.

The Coulomb force can also be written in terms of an electrostatic field: $${\displaystyle \mathbf{F}=q\mathbf{E}\left(\mathbf{r}\right),}$$ where $${\displaystyle \mathrm{\nabla }\cdot \mathbf{E}=\frac{{\rho }_q}{{\epsilon }_0};}$$ ${\displaystyle {\rho }_q}$ being the charge density at each point in space.

Virtual-particle exchange

In perturbation theory, forces are generated by the exchange of virtual particles. 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.

Probability amplitude

Using natural units, ${\displaystyle \hslash =c=1}$, the probability amplitude for the creation, propagation, and destruction of a virtual particle is given, in the path integral formulation by $${\displaystyle Z\equiv ⟨0|\exp \left(-i\hat{H}T\right)|0⟩=\exp \left(-iET\right)=\int D\phi \exp \left(i\mathcal{S}[\phi ]\right)=\exp \left(iW\right)}$$ where ${\displaystyle \hat{H}}$ is the Hamiltonian operator, ${\displaystyle T}$ is elapsed time, ${\displaystyle E}$ is the energy change due to the disturbance, ${\displaystyle W=-ET}$ is the change in action due to the disturbance, ${\displaystyle \phi }$ is the field of the virtual particle, the integral is over all paths, and the classical action is given by $${\displaystyle \mathcal{S}[\phi ]=\int {\mathrm{d}}^{4}x\mathcal{L}[\phi (x)]}$$ where ${\displaystyle \mathcal{L}[\phi (x)]}$ is the Lagrangian density.

Here, the spacetime metric is given by $${\displaystyle {\eta }_{\mu \nu }=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& -1& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& -1\end{array}\right).}$$

The path integral often can be converted to the form $${\displaystyle Z=\int \exp \left[i\int d^{4}x\left(\frac{1}{2}\phi \hat{O}\phi +J\phi \right)\right]D\phi }$$ where ${\displaystyle \hat{O}}$ is a differential operator with ${\displaystyle \phi }$ and ${\displaystyle J}$ functions of spacetime. 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 Common integrals in quantum field theory § Integrals with differential operators in the argument) $${\displaystyle Z\propto \exp \left(iW\left(J\right)\right)}$$ where $${\displaystyle W\left(J\right)=-\frac{1}{2}\iint d^{4}x{d}^{4}yJ\left(x\right)D\left(x-y\right)J\left(y\right)}$$ is the change in the action due to the disturbances and the propagator ${\displaystyle D\left(x-y\right)}$ is the solution of $${\displaystyle \hat{O}D\left(x-y\right)={\delta }^4\left(x-y\right).}$$

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 $${\displaystyle J(x)=\left(J_1+J_2,0,0,0\right)}$$ $${\displaystyle \begin{array}{rl}{J}_1& =a_1{\delta }^3\left(\mathbf{x}-{\mathbf{x}}_1\right)\\ J_2& =a_2{\delta }^3\left(\mathbf{x}-{\mathbf{x}}_2\right)\end{array}}$$ where the delta functions are in space, the disturbances are located at ${\displaystyle {\mathbf{x}}_1}$ and ${\displaystyle {\mathbf{x}}_2}$, and the coefficients ${\displaystyle a_1}$ and ${\displaystyle a_2}$ are the strengths of the disturbances.

If we neglect self-interactions of the disturbances then W becomes $${\displaystyle W\left(J\right)=-\iint d^{4}x{d}^{4}y{J}_1\left(x\right)\frac{1}{2}\left[D\left(x-y\right)+D\left(y-x\right)\right]J_2\left(y\right),}$$

which can be written $${\displaystyle W\left(J\right)=-T{a}_1{a}_2\int \frac{d^{3}k}{(2\pi {)}^3}D\left(k\right){\mid }_{k_0=0}\exp \left(i\mathbf{k}\cdot \left({\mathbf{x}}_1-{\mathbf{x}}_2\right)\right).}$$

Here ${\displaystyle D\left(k\right)}$ is the Fourier transform of $${\displaystyle \frac{1}{2}\left[D\left(x-y\right)+D\left(y-x\right)\right].}$$

Finally, the change in energy due to the static disturbances of the vacuum is $${\displaystyle E=-\frac{W}{T}=a_1{a}_2\int \frac{d^{3}k}{(2\pi {)}^3}D\left(k\right){\mid }_{k_0=0}\exp \left(i\mathbf{k}\cdot \left({\mathbf{x}}_1-{\mathbf{x}}_2\right)\right).}$$

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

Yukawa potential: the force between two nucleons in an atomic nucleus

Consider the spin-0 Lagrangian density^{[2]: 21–29} $${\displaystyle \mathcal{L}[\phi (x)]=\frac{1}{2}\left[{\left(\mathrm{\partial }\phi \right)}^2-m^2{\phi }^2\right].}$$

The equation of motion for this Lagrangian is the Klein–Gordon equation $${\displaystyle {\mathrm{\partial }}^2\phi +m^2\phi =0.}$$

If we add a disturbance the probability amplitude becomes $${\displaystyle Z=\int D\phi \exp \left\{i\int d^4\mathbf{x}\left[\frac{1}{2}\left({\left(\mathrm{\partial }\phi \right)}^2-m^2{\phi }^2\right)+J\phi \right]\right\}.}$$

If we integrate by parts and neglect boundary terms at infinity the probability amplitude becomes $${\displaystyle Z=\int D\phi \exp \left\{i\int d^{4}x\left[-\frac{1}{2}\phi \left({\mathrm{\partial }}^2+m^2\right)\phi +J\phi \right]\right\}.}$$

With the amplitude in this form it can be seen that the propagator is the solution of $${\displaystyle -\left({\mathrm{\partial }}^2+m^2\right)D\left(x-y\right)={\delta }^4\left(x-y\right).}$$

From this it can be seen that $${\displaystyle D\left(k\right){\mid }_{k_0=0}=-\frac{1}{k^2+m^2}.}$$

The energy due to the static disturbances becomes (see Common integrals in quantum field theory § Yukawa Potential: The Coulomb potential with mass) $${\displaystyle E=-\frac{a_1{a}_2}{4\pi r}\exp \left(-mr\right)}$$ with $${\displaystyle r^2={\left({\mathbf{x}}_1-{\mathbf{x}}_2\right)}^2}$$ which is attractive and has a range of $${\displaystyle \frac{1}{m}.}$$

Yukawa 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

Coulomb potential in vacuum

Consider the spin-1 Proca Lagrangian with a disturbance

$${\displaystyle \mathcal{L}[\phi (x)]=-\frac{1}{4}{F}_{\mu \nu }{F}^{\mu \nu }+\frac{1}{2}{m}^2{A}_{\mu }{A}^{\mu }+A_{\mu }{J}^{\mu }}$$ where $${\displaystyle F_{\mu \nu }={\mathrm{\partial }}_{\mu }{A}_{\nu }-{\mathrm{\partial }}_{\nu }{A}_{\mu },}$$ charge is conserved $${\displaystyle {\mathrm{\partial }}_{\mu }{J}^{\mu }=0,}$$ and we choose the Lorenz gauge $${\displaystyle {\mathrm{\partial }}_{\mu }{A}^{\mu }=0.}$$

Moreover, we assume that there is only a time-like component ${\displaystyle J^0}$ 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 $${\displaystyle \begin{array}{rl}-\frac{1}{4}\int d^{4}x{F}_{\mu \nu }{F}^{\mu \nu }& =-\frac{1}{4}\int d^{4}x\left({\mathrm{\partial }}_{\mu }{A}_{\nu }-{\mathrm{\partial }}_{\nu }{A}_{\mu }\right)\left({\mathrm{\partial }}^{\mu }{A}^{\nu }-{\mathrm{\partial }}^{\nu }{A}^{\mu }\right)\\ & =\frac{1}{2}\int d^{4}x{A}_{\nu }\left({\mathrm{\partial }}^2{A}^{\nu }-{\mathrm{\partial }}^{\nu }{\mathrm{\partial }}_{\mu }{A}^{\mu }\right)\\ & =\frac{1}{2}\int d^{4}x{A}^{\mu }\left({\eta }_{\mu \nu }{\mathrm{\partial }}^2\right)A^{\nu },\end{array}}$$ which implies $${\displaystyle {\eta }_{\mu \alpha }\left({\mathrm{\partial }}^2+m^2\right)D^{\alpha \nu }\left(x-y\right)={\delta }_{\mu }^{\nu }{\delta }^4\left(x-y\right)}$$ and $${\displaystyle D_{\mu \nu }\left(k\right){\mid }_{k_0=0}={\eta }_{\mu \nu }\frac{1}{-k^2+m^2}.}$$

This yields $${\displaystyle D\left(k\right){\mid }_{k_0=0}=\frac{1}{{\mathbf{k}}^2+m^2}}$$ for the timelike propagator and $${\displaystyle E=+\frac{a_1{a}_2}{4\pi r}\exp \left(-mr\right)}$$ which has the opposite sign to the Yukawa case.

In the limit of zero photon mass, the Lagrangian reduces to the Lagrangian for electromagnetism $${\displaystyle E=\frac{a_1{a}_2}{4\pi r}.}$$

Therefore the energy reduces to the potential energy for the Coulomb force and the coefficients ${\displaystyle a_1}$ and ${\displaystyle a_2}$ 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 waves is $${\displaystyle {\omega }^2={\omega }_p^2+\gamma \left(\omega \right)\frac{T_{\text{e}}}{m}{\mathbf{k}}^2.}$$ where ${\displaystyle \omega }$ is the angular frequency of the wave, $${\displaystyle {\omega }_p^2=\frac{4\pi n{e}^2}{m}}$$ is the plasma frequency, ${\displaystyle e}$ is the magnitude of the electron charge, ${\displaystyle m}$ is the electron mass, ${\displaystyle T_{\text{e}}}$ is the electron temperature (the Boltzmann constant equal to one), and ${\displaystyle \gamma \left(\omega \right)}$ 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 and ${\displaystyle \gamma \left(\omega \right)}$ is equal to three. At low frequencies, the compression is an isothermal process and ${\displaystyle \gamma \left(\omega \right)}$ is equal to one. Retardation effects have been neglected in obtaining the plasma-wave dispersion relation.

For low frequencies, the dispersion relation becomes $${\displaystyle {\mathbf{k}}^2+{\mathbf{k}}_{\text{D}}^2=0}$$ where $${\displaystyle k_{\text{D}}^2=\frac{4\pi n{e}^2}{T_e}}$$ is the Debye number, which is the inverse of the Debye length. This suggests that the propagator is $${\displaystyle D\left(k\right){\mid }_{k_0=0}=\frac{1}{k^2+k_{\text{D}}^2}.}$$

In fact, if the retardation effects are not neglected, then the dispersion relation is $${\displaystyle -k_0^2+k^2+k_{\text{D}}^2-\frac{m}{T_{\text{e}}}{k}_0^2=0,}$$ 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 $${\displaystyle E=\frac{a_1{a}_2}{4\pi r}\exp \left(-k_{\text{D}}r\right).}$$ 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 to yield $${\displaystyle E=\frac{a_1{a}_2}{4\pi r}\exp \left(-k_{\text{s}}r\right)}$$ where the inverse of the Thomas–Fermi screening length is $${\displaystyle k_{\text{s}}^2=\frac{6\pi n{e}^2}{{\epsilon }_{\text{F}}}}$$ and ${\displaystyle {\epsilon }_{\text{F}}}$ is the Fermi energy ${\epsilon }_{\text{F}}=\frac{{\hslash }^2}{2m}{\left(3{\pi }^{2}n\right)}^{2/3}.$

This expression can be derived from the chemical potential for an electron gas and from Poisson's equation. The chemical potential for an electron gas near equilibrium is constant and given by $${\displaystyle \mu =-e\phi +{\epsilon }_{\text{F}}}$$ where ${\displaystyle \phi }$ is the electric potential. 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.

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 $${\displaystyle J_1\left(x\right)=\frac{a_1}{L_B}\frac{1}{2\pi r}{\delta }^2\left(r\right)}$$ where ${\displaystyle r}$ is the distance in the xy-plane from the line of charge, ${\displaystyle L_B}$ is the width of the material in the z direction. The superscript 2 indicates that the Dirac delta function is in two dimensions. The propagator is $${\displaystyle D\left(k\right){\mid }_{k_0=0}=\frac{1}{{\mathbf{k}}^2+k_{Ds}^2}}$$ where ${\displaystyle k_{Ds}}$ is either the inverse Debye–Hückel screening length or the inverse Thomas–Fermi screening length.

The interaction energy is $${\displaystyle E=\left(\frac{a_1{a}_2}{2\pi L_B}\right){\int }_0^{\mathrm{\infty }}\frac{kdk}{k^2+k_{Ds}^2}{\mathcal{J}}_0(k{r}_{12})=\left(\frac{a_1{a}_2}{2\pi L_B}\right)K_0\left(k_{Ds}{r}_{12}\right)}$$ where ${\displaystyle {\mathcal{J}}_n(x)}$ and ${\displaystyle K_0(x)}$ are Bessel functions and ${\displaystyle r_{12}}$ is the distance between the two line charges. In obtaining the interaction energy we made use of the integrals (see Common integrals in quantum field theory § Integration of the cylindrical propagator with mass) $${\displaystyle {\int }_0^{2\pi }\frac{d\phi }{2\pi }\exp \left(ip\cos \left(\phi \right)\right)={\mathcal{J}}_0(p)}$$ and $${\displaystyle {\int }_0^{\mathrm{\infty }}\frac{kdk}{k^2+m^2}{\mathcal{J}}_0(kr)=K_0(mr).}$$

For ${\displaystyle k_{Ds}{r}_{12}\ll 1}$, we have $${\displaystyle K_0\left(k_{Ds}{r}_{12}\right)\to -\ln \left(\frac{k_{Ds}{r}_{12}}{2}\right)+\mathrm{0.5772.}}$$

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 $${\displaystyle J_1\left(x\right)=\frac{a_1}{L_b}\frac{1}{2\pi r}{\delta }^2\left(r-r_{B1}\right)}$$ where ${\displaystyle r}$ is the distance from the guiding center, ${\displaystyle L_B}$ is the width of the material in the direction of the magnetic field $${\displaystyle r_{B1}=\frac{\sqrt{4\pi }{m}_1{v}_1}{a_{1}B}=\sqrt{\frac{2\hslash }{m_1{\omega }_c}}}$$ where the cyclotron frequency is (Gaussian units) $${\displaystyle {\omega }_c=\frac{a_{1}B}{\sqrt{4\pi }{m}_{1}c}}$$ and $${\displaystyle v_1=\sqrt{\frac{2\hslash {\omega }_c}{m_1}}}$$ 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 the quantum treatment of a charged particle in a magnetic field.

In this geometry, the interaction energy can be written $${\displaystyle E=\left(\frac{a_1{a}_2}{2\pi L_B}\right){\int }_0^{\mathrm{\infty }}kdkD\left(k\right){\mid }_{k_0=k_B=0}{\mathcal{J}}_0\left(k{r}_{B1}\right){\mathcal{J}}_0\left(k{r}_{B2}\right){\mathcal{J}}_0\left(k{r}_{12}\right)}$$ where ${\displaystyle r_{12}}$ is the distance between the centers of the current loops and ${\displaystyle {\mathcal{J}}_n(x)}$ is a Bessel function of the first kind. In obtaining the interaction energy we made use of the integral $${\displaystyle {\int }_0^{2\pi }\frac{d\phi }{2\pi }\exp \left(ip\cos (\phi )\right)={\mathcal{J}}_0(p).}$$

Electric field due to a density perturbation

The chemical potential near equilibrium, is given by $${\displaystyle \mu =-e\phi +N\hslash {\omega }_c=N_0\hslash {\omega }_c}$$ where ${\displaystyle -e\phi }$ is the potential energy of an electron in an electric potential and ${\displaystyle N_0}$ and ${\displaystyle N}$ 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 $${\displaystyle \delta n=\frac{e\phi }{\hslash {\omega }_c{A}_{\text{M}}{L}_B}}$$ where ${\displaystyle A_{\text{M}}}$ is the area of the material in the plane perpendicular to the magnetic field.

Poisson's equation yields $${\displaystyle \left(k^2+k_B^2\right)\phi =0}$$ where $${\displaystyle k_B^2=\frac{4\pi e^2}{\hslash {\omega }_c{A}_{\text{M}}{L}_B}.}$$

The propagator is then $${\displaystyle D\left(k\right){\mid }_{k_0=k_B=0}=\frac{1}{k^2+k_B^2}}$$ and the interaction energy becomes $${\displaystyle E=\left(\frac{a_1{a}_2}{2\pi L_B}\right){\int }_0^{\mathrm{\infty }}\frac{kdk}{k^2+k_B^2}{\mathcal{J}}_0\left(k{r}_{B1}\right){\mathcal{J}}_0\left(k{r}_{B2}\right){\mathcal{J}}_0\left(k{r}_{12}\right)=\left(\frac{2e^2}{L_B}\right){\int }_0^{\mathrm{\infty }}\frac{kdk}{k^2+k_B^2{r}_B^2}{\mathcal{J}}_0^2\left(k\right){\mathcal{J}}_0\left(k\frac{r_{12}}{r_B}\right)}$$ where in the second equality (Gaussian units) we assume that the vortices had the same energy and the electron charge.

In analogy with plasmons, the force carrier is the quantum version of the upper hybrid oscillation which is a longitudinal plasma wave that propagates perpendicular to the magnetic field.

Currents with angular momentum

Delta function currents

Figure 1. Interaction energy vs. r for angular momentum states of value one. The curves are identical to these for any values of ${\displaystyle \ell ={\ell }^{\prime }}$. Lengths are in units are in ${\displaystyle r_{\ell }}$, and the energy is in units of $\frac{e^2}{L_B}$. Here ${\displaystyle r=r_{12}}$. Note that there are local minima for large values of ${\displaystyle k_B}$.

Figure 2. Interaction energy vs. r for angular momentum states of value one and five.

Figure 3. Interaction energy vs. r for various values of theta. The lowest energy is for $\theta =\frac{\pi }{4}$ or ${\displaystyle \frac{\ell }{{\ell }^{\prime }}=1}$. The highest energy plotted is for $\theta =0.90\frac{\pi }{4}$. Lengths are in units of ${\displaystyle r_{\ell {\ell }^{\prime }}}$.

Figure 4. Ground state energies for even and odd values of angular momenta. Energy is plotted on the vertical axis and r is plotted on the horizontal. When the total angular momentum is even, the energy minimum occurs when ${\displaystyle \ell ={\ell }^{\prime }}$ or $\frac{\ell }{{\ell }^{\ast }}=\frac{1}{2}$. When the total angular momentum is odd, there are no integer values of angular momenta that will lie in the energy minimum. Therefore, there are two states that lie on either side of the minimum. Because ${\displaystyle \ell \ne {\ell }^{\prime }}$, the total energy is higher than the case when ${\displaystyle \ell ={\ell }^{\prime }}$ for a given value of ${\displaystyle {\ell }^{\ast }}$.

Unlike classical currents, quantum current loops can have various values of the Larmor radius for a given energy. Landau levels, the energy states of a charged particle in the presence of a magnetic field, are multiply degenerate. The current loops correspond to angular momentum states of the charged particle that may have the same energy. Specifically, the charge density is peaked around radii of $${\displaystyle r_{\ell }=\sqrt{\ell }{r}_B\ell =0,1,2,\dots }$$ where ${\displaystyle \ell }$ is the angular momentum quantum number. When ${\displaystyle \ell =1}$ we recover the classical situation in which the electron orbits the magnetic field at the Larmor radius. If currents of two angular momentum ${\displaystyle \ell >0}$ and ${\displaystyle {\ell }^{\prime }\ge \ell }$ interact, and we assume the charge densities are delta functions at radius ${\displaystyle r_{\ell }}$, then the interaction energy is $${\displaystyle E=\left(\frac{2e^2}{L_B}\right){\int }_0^{\mathrm{\infty }}\frac{kdk}{k^2+k_B^2{r}_{\ell }^2}{\mathcal{J}}_0\left(k\right){\mathcal{J}}_0\left(\sqrt{\frac{{\ell }^{\prime }}{\ell }}k\right){\mathcal{J}}_0\left(k\frac{r_{12}}{r_{\ell }}\right).}$$

The interaction energy for ${\displaystyle \ell ={\ell }^{\prime }}$ is given in Figure 1 for various values of ${\displaystyle k_B{r}_{\ell }}$. 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 $${\displaystyle r_{12}=r_{\ell {\ell }^{\prime }}=\sqrt{\ell +{\ell }^{\prime }}{r}_B.}$$

This suggests that the pair of particles that are bound and separated by a distance ${\displaystyle r_{\ell {\ell }^{\prime }}}$ act as a single quasiparticle with angular momentum ${\displaystyle \ell +{\ell }^{\prime }}$.

If we scale the lengths as ${\displaystyle r_{\ell {\ell }^{\prime }}}$, then the interaction energy becomes $${\displaystyle E=\frac{2e^2}{L_B}{\int }_0^{\mathrm{\infty }}\frac{kdk}{k^2+k_B^2{r}_{\ell {\ell }^{\prime }}^2}{\mathcal{J}}_0\left(\cos \theta k\right){\mathcal{J}}_0(\sin \theta k){\mathcal{J}}_0\left(k\frac{r_{12}}{r_{\ell {\ell }^{\prime }}}\right)}$$ where $${\displaystyle \tan \theta =\sqrt{\frac{\ell }{{\ell }^{\prime }}}.}$$

The value of the ${\displaystyle r_{12}}$ at which the energy is minimum, ${\displaystyle r_{12}=r_{\ell {\ell }^{\prime }}}$, is independent of the ratio $\tan \theta =\sqrt{\ell /{\ell }^{\prime }}$. However the value of the energy at the minimum depends on the ratio. The lowest energy minimum occurs when $${\displaystyle \frac{\ell }{{\ell }^{\prime }}=1.}$$

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) $${\displaystyle \ell ={\ell }^{\prime }=1}$$ or $${\displaystyle \frac{\ell }{{\ell }^{\ast }}=\frac{1}{2}}$$ where the total angular momentum is written as $${\displaystyle {\ell }^{\ast }=\ell +{\ell }^{\prime }.}$$

When the total angular momentum is odd, the minima cannot occur for ${\displaystyle \ell ={\ell }^{\prime }.}$ The lowest energy states for odd total angular momentum occur when $${\displaystyle \frac{\ell }{{\ell }^{\ast }}=\frac{{\ell }^{\ast }\pm 1}{2{\ell }^{\ast }}}$$ or $${\displaystyle \frac{\ell }{{\ell }^{\ast }}=\frac{1}{3},\frac{2}{5},\frac{3}{7},\text{etc.,}}$$ and $${\displaystyle \frac{\ell }{{\ell }^{\ast }}=\frac{2}{3},\frac{3}{5},\frac{4}{7},\text{etc.,}}$$ which also appear as series for the filling factor in the fractional quantum Hall effect.

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 $${\displaystyle \frac{1}{\pi r_B^2{L}_B}\frac{1}{n!}{\left(\frac{r}{r_B}\right)}^{2l}\exp \left(-\frac{r^2}{r_B^2}\right).}$$

The interaction energy becomes $${\displaystyle E=\left(\frac{2e^2}{L_B}\right){\int }_0^{\mathrm{\infty }}\frac{kdk}{k^2+k_B^2{r}_B^2}M\left(\ell +1,1,-\frac{k^2}{4}\right)M\left({\ell }^{\prime }+1,1,-\frac{k^2}{4}\right){\mathcal{J}}_0\left(k\frac{r_{12}}{r_B}\right)}$$ where ${\displaystyle M}$ is a confluent hypergeometric function or Kummer function. In obtaining the interaction energy we have used the integral (see Common integrals in quantum field theory § Integration over a magnetic wave function)

$${\displaystyle \frac{2}{n!}{\int }_0^{\mathrm{\infty }}dr{r}^{2n+1}{e}^{-r^2}{J}_0(kr)=M\left(n+1,1,-\frac{k^2}{4}\right).}$$

As with delta function charges, the value of ${\displaystyle r_{12}}$ 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 $${\displaystyle \frac{\ell }{{\ell }^{\ast }}=\frac{1}{3},\frac{2}{5},\frac{3}{7},\text{etc.,}}$$ and $${\displaystyle \frac{\ell }{{\ell }^{\ast }}=\frac{2}{3},\frac{3}{5},\frac{4}{7},\text{etc.,}}$$ appear as well in the case of charges spread by the wave function.

The Laughlin wavefunction is an ansatz for the quasiparticle wavefunction. If the expectation value of the interaction energy is taken over a Laughlin wavefunction, 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 $${\displaystyle {\mathbf{J}}_1\left(\mathbf{x}\right)=a_1{\mathbf{v}}_1{\delta }^3\left(\mathbf{x}-{\mathbf{x}}_1\right)}$$ with a comparable expression for ${\displaystyle {\mathbf{J}}_2}$.

The Fourier transform of this current is $${\displaystyle {\mathbf{J}}_1\left(\mathbf{k}\right)=a_1{\mathbf{v}}_1\exp \left(i\mathbf{k}\cdot {\mathbf{x}}_1\right).}$$

The current can be decomposed into a transverse and a longitudinal part (see Helmholtz decomposition). $${\displaystyle {\mathbf{J}}_1\left(\mathbf{k}\right)=a_1\left[1-\hat{\mathbf{k}}\hat{\mathbf{k}}\right]\cdot {\mathbf{v}}_1\exp \left(i\mathbf{k}\cdot {\mathbf{x}}_1\right)+a_1\left[\hat{\mathbf{k}}\hat{\mathbf{k}}\right]\cdot {\mathbf{v}}_1\exp \left(i\mathbf{k}\cdot {\mathbf{x}}_1\right).}$$

The hat indicates a unit vector. The last term disappears because $${\displaystyle \mathbf{k}\cdot \mathbf{J}=-k_0{J}^0\to 0,}$$ which results from charge conservation. Here ${\displaystyle k_0}$ vanishes because we are considering static forces.

With the current in this form the energy of interaction can be written $${\displaystyle E=a_1{a}_2\int \frac{d^3\mathbf{k}}{(2\pi {)}^3}D\left(k\right){\mid }_{k_0=0}{\mathbf{v}}_1\cdot \left[1-\hat{\mathbf{k}}\hat{\mathbf{k}}\right]\cdot {\mathbf{v}}_2\exp \left(i\mathbf{k}\cdot \left({\mathbf{x}}_1-{\mathbf{x}}_2\right)\right).}$$

The propagator equation for the Proca Lagrangian is $${\displaystyle {\eta }_{\mu \alpha }\left({\mathrm{\partial }}^2+m^2\right)D^{\alpha \nu }\left(x-y\right)={\delta }_{\mu }^{\nu }{\delta }^4\left(x-y\right).}$$

The spacelike solution is $${\displaystyle D\left(k\right){\mid }_{k_0=0}=-\frac{1}{k^2+m^2},}$$ which yields $${\displaystyle E=-a_1{a}_2\int \frac{d^3\mathbf{k}}{(2\pi {)}^3}\frac{{\mathbf{v}}_1\cdot \left[1-\hat{\mathbf{k}}\hat{\mathbf{k}}\right]\cdot {\mathbf{v}}_2}{k^2+m^2}\exp \left(i\mathbf{k}\cdot \left({\mathbf{x}}_1-{\mathbf{x}}_2\right)\right),}$$ where $k=|\mathbf{k}|$. The integral evaluates to (see Common integrals in quantum field theory § Transverse potential with mass) $${\displaystyle E=-\frac{1}{2}\frac{a_1{a}_2}{4\pi r}{e}^{-mr}\left\{\frac{2}{{\left(mr\right)}^2}\left(e^{mr}-1\right)-\frac{2}{mr}\right\}{\mathbf{v}}_1\cdot \left[1+\hat{\mathbf{r}}\hat{\mathbf{r}}\right]\cdot {\mathbf{v}}_2}$$ which reduces to $${\displaystyle E=-\frac{1}{2}\frac{a_1{a}_2}{4\pi r}{\mathbf{v}}_1\cdot \left[1+\hat{\mathbf{r}}\hat{\mathbf{r}}\right]\cdot {\mathbf{v}}_2}$$ in the limit of small m. 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 plasma

In a plasma, the dispersion relation for an electromagnetic wave is (${\displaystyle c=1}$) $${\displaystyle k_0^2={\omega }_p^2+k^2,}$$ which implies $${\displaystyle D\left(k\right){\mid }_{k_0=0}=-\frac{1}{k^2+{\omega }_p^2}.}$$

Here ${\displaystyle {\omega }_p}$ is the plasma frequency. The interaction energy is therefore $${\displaystyle E=-\frac{1}{2}\frac{a_1{a}_2}{4\pi r}{\mathbf{v}}_1\cdot \left[1+\hat{\mathbf{r}}\hat{\mathbf{r}}\right]\cdot {\mathbf{v}}_2{e}^{-{\omega }_{p}r}\left\{\frac{2}{{\left({\omega }_{p}r\right)}^2}\left(e^{{\omega }_{p}r}-1\right)-\frac{2}{{\omega }_{p}r}\right\}.}$$

Magnetic interaction between current loops in a simple plasma or electron gas

Interaction energy

Consider a tube of current rotating in a magnetic field embedded in a simple plasma or electron gas. The current, which lies in the plane perpendicular to the magnetic field, is defined as $${\displaystyle {\mathbf{J}}_1(\mathbf{x})=a_1{v}_1\frac{1}{2\pi r{L}_B}{\delta }^2\left(r-r_{B1}\right)\left(\hat{\mathbf{b}}\times \hat{\mathbf{r}}\right)}$$ where $${\displaystyle r_{B1}=\frac{\sqrt{4\pi }{m}_1{v}_1}{a_{1}B}}$$ and ${\displaystyle \hat{\mathbf{b}}}$ is the unit vector in the direction of the magnetic field. Here ${\displaystyle L_B}$ 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.

The energy of interaction is $${\displaystyle E=\left(\frac{a_1{a}_2}{2\pi L_B}\right)v_1{v}_2{\int }_0^{\mathrm{\infty }}kdkD\left(k\right){\mid }_{k_0=k_B=0}{\mathcal{J}}_1\left(k{r}_{B1}\right){\mathcal{J}}_1\left(k{r}_{B2}\right){\mathcal{J}}_0\left(k{r}_{12}\right)}$$ where ${\displaystyle r_{12}}$ is the distance between the centers of the current loops and ${\displaystyle {\mathcal{J}}_n(x)}$ is a Bessel function of the first kind. In obtaining the interaction energy we made use of the integrals $${\displaystyle {\int }_0^{2\pi }\frac{d\phi }{2\pi }\exp \left(ip\cos \left(\phi \right)\right)={\mathcal{J}}_0\left(p\right)}$$ and $${\displaystyle {\int }_0^{2\pi }\frac{d\phi }{2\pi }\cos \left(\phi \right)\exp \left(ip\cos \left(\phi \right)\right)=i{\mathcal{J}}_1\left(p\right).}$$

See Common integrals in quantum field theory § Angular integration in cylindrical coordinates.

A current in a plasma confined to the plane perpendicular to the magnetic field generates an extraordinary wave. This wave generates Hall currents that interact and modify the electromagnetic field. The dispersion relation for extraordinary waves is $${\displaystyle -k_0^2+k^2+{\omega }_p^2\frac{k_0^2-{\omega }_p^2}{k_0^2-{\omega }_H^2}=0,}$$ which gives for the propagator $${\displaystyle D\left(k\right){\mid }_{k_0=k_B=0}=-\left(\frac{1}{k^2+k_X^2}\right)}$$ where $${\displaystyle k_X\equiv \frac{{\omega }_p^2}{{\omega }_H}}$$ in analogy with the Darwin propagator. Here, the upper hybrid frequency is given by $${\displaystyle {\omega }_H^2={\omega }_p^2+{\omega }_c^2,}$$ the cyclotron frequency is given by (Gaussian units) $${\displaystyle {\omega }_c=\frac{eB}{mc},}$$ and the plasma frequency (Gaussian units) $${\displaystyle {\omega }_p^2=\frac{4\pi n{e}^2}{m}.}$$

Here n is the electron density, e is the magnitude of the electron charge, and m is the electron mass.

The interaction energy becomes, for like currents, $${\displaystyle E=-\left(\frac{a^2}{2\pi L_B}\right)v^2{\int }_0^{\mathrm{\infty }}\frac{kdk}{k^2+k_X^2}{\mathcal{J}}_1^2\left(k{r}_B\right){\mathcal{J}}_0\left(k{r}_{12}\right)}$$

Limit of small distance between current loops

In the limit that the distance between current loops is small, $${\displaystyle E=-E_0{I}_1\left(\mu \right)K_1\left(\mu \right)}$$ where $${\displaystyle E_0=\left(\frac{a^2}{2\pi L_B}\right)v^2}$$ and $${\displaystyle \mu =\frac{{\omega }_p^2{r}_B}{{\omega }_H}=k_X{r}_B}$$ and I and K 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 Common integrals in quantum field theory § Integration of the cylindrical propagator with mass) $${\displaystyle {\int }_o^{\mathrm{\infty }}\frac{kdk}{k^2+m^2}{\mathcal{J}}_1^2\left(kr\right)=I_1\left(mr\right)K_1\left(mr\right).}$$

For small mr the integral becomes $${\displaystyle I_1\left(mr\right)K_1\left(mr\right)\to \frac{1}{2}\left[1-\frac{1}{8}{\left(mr\right)}^2\right].}$$

For large mr the integral becomes $${\displaystyle I_1\left(mr\right)K_1\left(mr\right)\to \frac{1}{2}\left(\frac{1}{mr}\right).}$$

Relation to the quantum Hall effect

The screening wavenumber can be written (Gaussian units) $${\displaystyle \mu =\frac{{\omega }_p^2{r}_B}{{\omega }_{H}c}=\left(\frac{2e^2{r}_B}{L_B\hslash c}\right)\frac{\nu }{\sqrt{1+\frac{{\omega }_p^2}{{\omega }_c^2}}}=2\alpha \left(\frac{r_B}{L_B}\right)\left(\frac{1}{\sqrt{1+\frac{{\omega }_p^2}{{\omega }_c^2}}}\right)\nu }$$ where ${\displaystyle \alpha }$ is the fine-structure constant and the filling factor is $${\displaystyle \nu =\frac{2\pi N\hslash c}{eBA}}$$ and N is the number of electrons in the material and A 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 filling factor is the fraction of occupied Landau states at the ground state energy.

For cases of interest in the quantum Hall effect, ${\displaystyle \mu }$ is small. In that case the interaction energy is $${\displaystyle E=-\frac{E_0}{2}\left[1-\frac{1}{8}{\mu }^2\right]}$$ where (Gaussian units) $${\displaystyle E_0=4\pi \frac{e^2}{L_B}\frac{v^2}{c^2}=8\pi \frac{e^2}{L_B}\left(\frac{\hslash {\omega }_c}{m{c}^2}\right)}$$ is the interaction energy for zero filling factor. We have set the classical kinetic energy to the quantum energy $${\displaystyle \frac{1}{2}m{v}^2=\hslash {\omega }_c.}$$

Gravitation

A gravitational disturbance is generated by the stress–energy tensor ${\displaystyle T^{\mu \nu }}$; consequently, the Lagrangian for the gravitational field is spin-2. If the disturbances are at rest, then the only component of the stress–energy tensor that persists is the ${\displaystyle 00}$ component. If we use the same trick of giving the graviton some mass and then taking the mass to zero at the end of the calculation the propagator becomes $${\displaystyle D\left(k\right){\mid }_{k_0=0}=-\frac{4}{3}\frac{1}{k^2+m^2}}$$ and $${\displaystyle E=-\frac{4}{3}\frac{a_1{a}_2}{4\pi r}\exp \left(-mr\right),}$$ which is once again attractive rather than repulsive. The coefficients are proportional to the masses of the disturbances. In the limit of small graviton mass, we recover the inverse-square behavior of Newton's Law.

Unlike the electrostatic case, however, taking the small-mass limit of the boson does not yield the correct result. A more rigorous treatment yields a factor of one in the energy rather than 4/3.