Minimal coupling
Minimal coupling, [ Greiner 1] minimal substitution or principle of minimal electromagnetic interaction [ rollnik2 1] describes a principle of quantum mechanics to introduce the electromagnetic interaction in the equations of free particles. The principle defines the replacement to be performed in the Hamiltonian of a free particle to reach its interaction with an electromagnetic field. The justification of this principle stems from the fact that a coupling of free particles according to this principle leads to interaction fields to gauge invariance of the relevant equations. [ Schleich 1]
- 3.1 gauge freedom in terms of the gauge field theory
The principle
In nonrelativistic quantum mechanics the dynamics of a particle is determined by the Schrödinger equation
Described. Here, the wave function of the particle and the Hamiltonian. For a free particle of mass of the Hamiltonian is given as, on the other hand for a particle in a potential than with the momentum operator.
For the coupling of a charged particle with the electromagnetic field following replacements are performed in the Schrödinger equation:
The momentum operator is
Replaced. This corresponds to the replacement of the pulse by the canonical kinetic momentum. [ Schwabl 1] Here, the strength of the coupling of the particle to the pitch, the electric charge of the particle and the vector potential of the electromagnetic field.
In addition, the time derivative at the left side of the Schrödinger equation by
Replace it and the scalar potential of the electromagnetic field is.
In relativistic quantum mechanics, the analogue of the Schrödinger equation is the Dirac equation, both substitutions can be combined into a single. Within the framework of the theory of relativity Tensorkalküls the scalar and vector potential of the electromagnetic field are combined into a four-potential:
The momentum operator is in relativistic quantum mechanics also a four-vector, the four-momentum:
The principle of minimal coupling is now demanding the replacement
In the coordinate representation, the minimal coupling with the required due to gauge invariance covariant derivative coincides, [ rollnik2 1] although both terms are derived in various ways. The term of the minimal coupling and the replacement rule associated springs from the desire to couple the Schrödinger equation or the Dirac equation for a free particle in an electromagnetic field. On the other hand arises from the replacement rule that all partial derivatives are to be replaced by the covariant derivative, the desire for a gauge invariant equation of motion. It turns out that both rewriting rules are identical. In section gauge freedom in terms of the gauge theory is outlined, as the demand for gauge invariance calls for the coupling of the free equation to an interaction field, thus promoting the covariant derivative to days. It is observed that the derived there covariant derivative corresponds exactly to the minimal coupling. In section covariant derivative is outlined why both rewriting rules must be identical.
Classical Mechanics
In the Hamiltonian is the mechanical movement of a charged particle of the charge and mass of the electromagnetic field with the Hamiltonian
. described, which can be derived starting from the Lorentz force [ greiner 2 ] where the electric field and the magnetic field, as in electrodynamics are common, and described by the potentials:
At this Hamiltonian reaches you by the Hamiltonian of a free particle ( free particle is vanishing potential, the total energy, the kinetic energy)
The substitutions
Lead exactly to the Hamiltonian of a classical charged particle in an electromagnetic field. [ Blöchl 1] These replacements comply with the above for quantum mechanics replacements. The first replacement is the same as in the quantum mechanical version. The second replacement is also in just the second replacement for quantum mechanics, as in the time-dependent Schrödinger equation of the energy operator is straight.
One motivation of the minimal coupling is that they to gauge invariance in the sense of classical electrodynamics in the equations of motion arising from the Hamiltonian equations leads. The Hamiltonian function itself, however, is not gauge invariant in this sense. [ Greiner 3]
Gauge freedom in the sense of classical electrodynamics
This is called gauge freedom, if the conditions of potentials and freely choose, without changing the equations of motion of the particle. In other words, the resultant force on the particle must not be changed by the Umeichen potentials. The force on charged particles due to electromagnetic fields, the Lorentz force.
One can now perform such calibrations and the potentials that do not change the Lorentz force, so it must be true. It turns out that the following calibrations allow the equations of motion invariant:
With an arbitrary scalar function.
So If you choose the Weyl gauge, [ Notes 1 ], a calibration in which the scalar potential always vanishes
Only the first replacement must be carried out in the Hamiltonian of a free particle for coupling to the electromagnetic field.
Schrödinger equation without spin
The Schrödinger equation for a free particle without spin is
The Hamiltonian of the free particle is accordingly. Applying the principle of minimal coupling leads to the Hamiltonian of a charged particle without spin term in the magnetic field and the electromagnetic field, with the addition of the Weyl gauge
Gauge freedom in terms of the gauge field theory
All measurable physical quantities are only on the magnitude square of the wave function dependent. [ Greiner 4] Therefore, the wave function is determined only up to a position-dependent phase factor. The states in quantum mechanics therefore have an arbitrary gauge field. However, if one calculates the free Schrödinger equation with a umgeeichten wave function ( it is so ), the Schrödinger equation is not invariant. [ Amsler 1]
If we write the other hand, the Hamiltonian with the minimal coupling, the Schrödinger equation remains invariant under the calibration phase. This is known as covariance. The requirement of local gauge freedom of the phase makes the existence of the electromagnetic fields, therefore, imperative. Theories in which interaction fields are automatically generated due to invariance under certain transformations (local phase transformation here ) are called gauge field theories. In addition, the Hamiltonian is invariant under now also calibrate the electromagnetic potentials. The replacement of the momentum operator by is also called covariant derivative, since replacing the ordinary derivative ( momentum operator ) by a modified derivative ( covariant derivative ) leads to the invariance of the Schrödinger equation. The relationship to the covariant derivative of general relativity is explained in section # covariant derivative.
If we now consider the Hamiltonian with inserted minimal coupling [ Notes 2] and the same Hamiltonian with only umgeeichtem vector potential, thus leading the Painted Schrödinger equation
On the uncoated Schrödinger equation
Both gauge transformations thus cancel each other, so that the Schrödinger equation is written with covariant derivative invariant under gauge transformation of the potentials and the wave functions.
Schrödinger equation with spin
The principle of minimal coupling leads only in relativistic quantum mechanics (ie, when applied to the Dirac equation ) to the quantitative coupling between charged particles and electromagnetic fields, which has so far been experimentally proven. In the " classical" Schrödinger equation nor the proportion of the interaction between the electron and light, which depends on the spin of the electron is missing.. To introduce these spin- share in the non- relativistic quantum mechanics, the principle of minimal coupling, one can apply a trick [ rollnik2 1] For the Pauli matrices applies to any matrix. Now modifying the free Hamiltonian in the Schrödinger equation with this " hidden" Pauli matrix
If one applies the principle of minimal coupling to these modified free Hamiltonian, we obtain
Multiplying out in compliance with the order and the use of the above definition of the magnetic field results
This corresponds to the Pauli equation, which describes the dynamics of a non- relativistic Spin-1/2-Teilchens with charge and mass in an electromagnetic field (without scalar potential ).
Dirac equation
The free Dirac equation is using the Dirac matrices
And is Lorentz invariant. Same as the case of the Schrödinger equation, the equation is, however, not gauge invariant under phase transformation. Insert the minimal coupling in the four-vector notation, ie
With, leads to the relativistically covariant form of the Dirac equation with coupled electromagnetic field. [ Schwabl 1]
Is called, since the replacement of the "normal partial derivative " by the " covariant derivative " leads to the covariance with respect to gauge transformations of the relevant equation also covariant derivative.
Dipole approximation
The Hamiltonian for a charged particle to an electromagnetic field, and a potential of
Given. This Hamiltonian describes a classical charged particle in a potential. The quantum-mechanical version (transition from the Hamiltonian to the Hamiltonian ) would a single electron bound to an atom ( hydrogen atom) describe. For simplicity, but in the following section, the dipole approximation is to be shown on the classical Hamiltonian function.
Hamilton function can be divided into two parts. A part describing the system ( in electron potential ) itself and the other is its interaction with the electromagnetic field.
Looking at the situation in an electromagnetic field in the Strahlungseichung ( and therefore ) [ Hertel 1] and takes into account only the coupling in linear order with, we obtain
The vector potential can also be approximated as. As long as the characteristic wavelength of the electromagnetic field is very much greater than the expansion of the atom, the vector potential can be regarded as a spatially almost uniform over the extent of the atom. If we write the canonical momentum as a kinetic pulse, as follows [ Ehlotzky 1]
In the dipole approximation, the electric field is given as. This leads to
The last term can be omitted, because the Hamiltonian is determined up to the total time derivative of any function. Finally, the interaction Hamiltonian for a bound charged particle in the dipole approximation yields to
This result was derived from the principle of minimal coupling and is also in its quantum mechanical counterpart ( classical derivation here ) used in quantum electrodynamics. A commonly used name for this interaction is also " Hamiltonian ", pronounced " e by r Hamiltonian ", as is often used for the spatial coordinate. [ Creep 2] It is still the dipole
Define ( in analogy of an electric dipole ). Thus it is obvious that the field couples in the dipole approximation only to the dipole moment of the hydrogen atom. In general, the above procedure can be performed with atoms of more than one electron.
Multipolar coupling and power Zinau -Woolley transformation
General can bring in the equivalent representation of the multi - polar coupling Hamiltonian of the minimum - coupling Hamiltonian with the unitary power Zinau -Woolley transformation. Here, the electromagnetic field is coupled by the vector potential of the polarization and magnetization. This form of the interaction Hamiltonian light-matter interactions of dielectrics can be described.
General Theory of Relativity
In general relativity, the principle of minimal coupling term refers to a slightly different principle. The Einstein field equations in vacuum can choose from a Lagrangian density of the form
Can be derived with the metric, the Krümmungsskalar and a constant. The coupling to other fields (eg electromagnetic field ) is now to be achieved by the addition of an appropriate interaction Lagrangian density. The decomposition of the Lagrangian density is in principle of minimal gravitational coupling called. [ Anderson 1]
Covariant derivative
A principle of general relativity is the covariance, which states that equations that are valid and therefore Lorentz invariant in special relativity theory, by replacing the partial derivatives by the covariant derivative to general coordinate independent equations ( generally covariant ) are. Mathematically, this corresponds to covariant derivative of the Levi- Civita connection. This is the link to the Tangentialvektorbündel a semi - Riemannian manifold. One hand, the covariant derivative to covariant ( invariant under change of coordinates ) equations, otherwise the covariant derivative defines the parallel transport of tensors in curved spaces.
In the gauge field theory ( for example, all theories regarding the fundamental interactions in the Standard Model of particle physics ) are the wave functions of the particles certain symmetries. These symmetries manifest in the invariance of the Lagrangian theory to the action of a group (in the case of the Schrodinger equation). The wave functions are defined on a manifold. Both structures and are in modern differential geometry into a unitary structure P ( M, G), the principal bundle together. A main fiber bundle is a manifold is attached to which, for each point of a copy of the pattern group. These copies are called fibers, and the representation of group elements from different fibers are located in disjoint vector spaces. Since and are in different rooms, after the definition of a connection on the principal bundle a derivative can be formed. The replacement of the partial derivative by the minimal coupling is just the covariant derivative ( related to coordinates ) in this case. Just as in the case of the Christoffel symbols ART determine the curvature of space (and the Christoffel symbols depend on the metric ab), as determined in the case of gauge field theories the four-potential the curvature. The curvature tensor is obtained in both cases from the commutator of the covariant derivative.
Origin of name
The Lagrangian density of the electromagnetic field with minimal coupling is:
Here is the first part of the kinetic term with the field strength tensor and the second term is the coupling of the field to the " charged current " - the charged matter, according to the principle of minimal coupling. The connection with the procedure described in the introduction of the mini tional coupling is explained in the following.
The name is due to the minimal coupling, since it is the simplest combination of charge current density and electromagnetic field, which meets the following conditions: [ Heron Wolf 1 ]
- Receives Lorentz invariance of the free equation
- Gauge invariance
- Coupled electromagnetic field to charged matter
In addition, performs exactly this minimal coupling procedure on a gauge-invariant action.
The above representation of the minimum coupling of Lagrangian corresponds exactly to the presented in the initiation procedure for a point-like -charged particles. This purpose we consider the four-current density of a point-like particle:
Here, the usual symbols from the special theory of relativity were used: is the charge density, the Dirac delta function in three dimensions, the velocity of the charged particle, the Lorentz factor, the four-velocity and the proper time. It follows: . If, which is now in the effect of the interaction Lagrangian density, we obtain:
If we write now the scalar product of two four-vectors, so we have:
To obtain the total Lagrangian nor the kinetic part must be added on a particle of mass:
The canonical momentum arises from to
The kinetic pulse is accordingly. This result is exactly the replacement is performed at the launch of the minimal coupling in the Lagrangian and Hamiltonian of a free particle. As described in the introduction, the canonical pulse corresponding to the kinetic momentum of a free particle is replaced by the kinetic momentum of the particles in the electromagnetic field.
Thus, the interaction Lagrangian density leads exactly to the result, which is assumed by the specified procedure in the introduction, and explains the name of the coupling procedure.