Minimal coupling

Minimal coupling , minimal substitution or the principle of minimal electromagnetic interaction describes a principle of quantum mechanics for introducing electromagnetic interaction into the equations of free particles . The principle specifies the replacement to be carried out in the Hamilton operator of a free particle in order to achieve its interaction with an electromagnetic field . The justification of this principle arises from the fact that a coupling of free particles to interaction fields according to this principle leads to calibration invariance of the relevant equations.

The principle

In non-relativistic quantum mechanics , the dynamics of a particle is determined by the Schrödinger equation

${\displaystyle i\hbar {\frac {\partial \psi ({\vec {x}},t)}{\partial t}}={\hat {H}}\psi ({\vec {x}},t)}$

described. Here is the wave function of the particle and the Hamilton operator . For a free particle of mass the Hamilton operator is as ${\displaystyle \psi }$ ${\displaystyle {\hat {H}}}$ ${\displaystyle m}$

${\displaystyle {\hat {H}}={\frac {{\hat {\vec {p}}}^{2}}{2m}}}$

given, for a particle in a potential, however, as ${\displaystyle V}$

${\displaystyle {\hat {H}}={\frac {{\hat {\vec {p}}}^{2}}{2m}}+V({\vec {x}})}$

with the momentum operator . ${\displaystyle {\hat {\vec {p}}}}$

To couple a charged particle to the electromagnetic field, the following substitutions are carried out in the Schrödinger equation:

The momentum operator is given by ${\displaystyle {\hat {\vec {p}}}}$

${\displaystyle {\hat {\vec {p}}}\longrightarrow {\hat {\vec {p}}}-q{\vec {A}}({\vec {x}},t)}$

replaced. This corresponds to the replacement of the kinetic momentum by the canonical momentum . The strength of the coupling of the particle to the field is the electrical charge of the particle and the vector potential of the electromagnetic field. ${\displaystyle q}$ ${\displaystyle {\vec {A}}}$

In addition, the time derivative is on the left side of the Schrödinger equation

${\displaystyle i\hbar {\frac {\partial }{\partial t}}\longrightarrow i\hbar {\frac {\partial }{\partial t}}-q\phi ({\vec {x}},t)}$

replaced, where is the scalar potential of the electromagnetic field. ${\displaystyle \phi }$

In relativistic quantum mechanics, whose analogue of the Schrödinger equation is the Dirac equation , both substitutions can be combined into one. Within the framework of the tensor calculus of relativity theory, the scalar potential and vector potential of the electromagnetic field are combined to form a four-potential :

${\displaystyle A^{\mu }=(\phi /c,{\vec {A}})^{T},\,\mu =0,1,2,3}$.

In relativistic quantum mechanics , the momentum operator is also a four-vector , the four- momentum :

${\displaystyle p^{\mu }=({\hat {E}}/c,{\hat {\vec {p}}})^{T},\,\mu =0,1,2,3}$, is the energy operator. ${\displaystyle {\hat {E}}}$

The principle of minimal coupling now requires replacement

${\displaystyle p^{\mu }\longrightarrow p^{\mu }-qA^{\mu }}$.

In the position representation , the minimum coupling agrees with the covariant derivation required due to gauge invariance , although both terms are derived in different ways. The term of the minimum coupling and the associated substitution rule arises from the desire to couple the Schrödinger equation or Dirac equation of a free particle to an electromagnetic field. In contrast, the replacement rule that all partial derivatives should be replaced by the covariant derivative arises from the need for a gauge invariant equation of motion. It turns out that both replacement rules are identical. In the section freedom from calibration in terms of calibration theory , it is outlined how the requirement for calibration invariance demands the coupling of the free equation to an interaction field and thus promotes the covariant derivation. One observes that the covariant derivative derived there corresponds exactly to the minimum coupling . The section Covariant Derivation outlines why both replacement rules must be identical.

Classic mechanics

In Hamiltonian mechanics , the movement of a charged particle of charge and mass in the electromagnetic field is given by the Hamilton function ${\displaystyle q}$ ${\displaystyle m}$

${\displaystyle H={\frac {1}{2m}}\left({\vec {p}}-q{\vec {A}}\right)^{2}+q\phi }$

which can be derived from the Lorentz force . The electric field and the magnetic field , as usual in electrodynamics , are described by the potentials and : ${\displaystyle {\vec {E}}({\vec {x}},t)}$ ${\displaystyle {\vec {B}}({\vec {x}},t)}$ ${\displaystyle \phi ({\vec {x}},t)}$ ${\displaystyle {\vec {A}}({\vec {x}},t)}$

${\displaystyle {\vec {E}}=-\nabla \phi -{\frac {\partial {\vec {A}}}{\partial t}}}$,

${\displaystyle {\vec {B}}=\nabla \times {\vec {A}}}$.

This Hamilton function can also be obtained from the Hamilton function of a free particle (free particle means vanishing potential , is the total energy, the kinetic energy ) ${\displaystyle V=0}$ ${\displaystyle E}$ ${\displaystyle T}$

${\displaystyle H=E=T+V={\frac {{\vec {p}}^{2}}{2m}}}$.

The replacements

${\displaystyle {\vec {p}}\longrightarrow {\vec {p}}-q{\vec {A}}}$,

${\displaystyle E\longrightarrow E-q\phi }$,

lead exactly to the Hamilton function of a classical charged particle in the electromagnetic field. These substitutions correspond to the substitutions given above for quantum mechanics. The first substitution is the same as in the quantum mechanical version. The second replacement also corresponds to the second replacement for quantum mechanics, since in the time-dependent Schrödinger equation the energy operator is even . ${\displaystyle {\hat {E}}}$ ${\displaystyle i\hbar \partial _{t}}$

One motivation of the minimal coupling is that it leads to gauge invariance in the sense of classical electrodynamics in the equations of motion that result from Hamilton's equations . The Hamilton function itself is not gauge invariant in this sense.

Calibration-free in the sense of classical electrodynamics

One speaks of freedom from calibration if the potentials and can be freely chosen without changing the equations of motion of the particle. In other words: The resulting force on the particle must not be changed by changing the potential. The force on charged particles due to electromagnetic fields is the Lorentz force . ${\displaystyle \phi }$ ${\displaystyle {\vec {A}}}$ ${\displaystyle {\vec {F}}_{L}}$

${\displaystyle {\vec {F}}_{L}=q{\vec {E}}+q{\vec {v}}\times {\vec {B}}}$

One can now carry out such calibrations or potentials that do not change the Lorentz force, so it must apply. It turns out that the following calibrations leave the equations of motion invariant: ${\displaystyle \phi \longrightarrow \phi '}$ ${\displaystyle {\vec {A}}\longrightarrow {\vec {A}}'}$ ${\displaystyle {\vec {F}}_{L}={\vec {F}}_{L}'}$

${\displaystyle {\vec {A}}'={\vec {A}}+\nabla f}$

${\displaystyle \phi '=\phi -{\frac {\mathrm {d} f}{\mathrm {d} t}}}$

with any scalar function . ${\displaystyle f({\vec {x}},t)}$

If one chooses the Weyl calibration , i.e. a calibration in which the scalar potential always disappears,

${\displaystyle f({\vec {x}})=\int \phi (({\vec {x}}),t)\mathrm {d} t}$,

so only the first replacement to the Hamilton function of a free particle for coupling to the electromagnetic field has to be carried out.

Schrödinger equation without spin

The Schrödinger equation of a free particle without spin is as follows

${\displaystyle i\hbar {\frac {\partial \psi ({\vec {x}},t)}{\partial t}}={\frac {{\hat {\vec {p}}}^{2}}{2m}}\psi ({\vec {x}},t)}$.

The Hamilton operator of the free particle is therefore . Applying the principle of minimal coupling leads to the Hamilton operator of a charged particle without a spinterm in the magnetic field or in the electromagnetic field, with the addition of the Weyl calibration ${\displaystyle {\hat {H}}={\frac {{\hat {\vec {p}}}^{2}}{2m}}}$

${\displaystyle {\hat {H}}={\frac {1}{2m}}\left({\hat {\vec {p}}}-q{\vec {A}}\right)^{2}}$.

Calibration freedom in the sense of the calibration field theory

All measurable physical quantities only depend on the square of the wave function . The wave function is therefore only determined up to a location-dependent phase factor . The states in quantum mechanics therefore have a freely selectable calibration field . But if you calculate the free Schrödinger equation with a recalibrated wave function (it is ), then the Schrödinger equation does not remain form-invariant. ${\displaystyle |\psi |^{2}=\psi \psi ^{*}}$ ${\displaystyle e^{i\chi ({\vec {x}},t)}}$ ${\displaystyle \chi ({\vec {x}},t)}$ ${\displaystyle \psi '=\psi e^{i\chi ({\vec {x}},t)}}$ ${\displaystyle |\psi '|^{2}=\psi e^{i\chi ({\vec {x}},t)}\psi ^{*}e^{-i\chi ({\vec {x}},t)}=\psi \psi ^{*}=|\psi |^{2}}$

If, on the other hand, the Hamilton operator is written with the minimal coupling, the Schrödinger equation remains invariant under calibration of the phase. This is called covariance . The requirement that the phase is not locally calibrated makes the existence of electromagnetic fields imperative. Theories in which interaction fields are automatically generated due to invariances under certain transformations (here local phase transformation) are called gauge field theories . In addition, the Hamilton operator is now form-invariant under calibration of the electromagnetic potentials. Replacing the momentum operator with is also called a covariant derivative, since replacing the ordinary derivative (momentum operator) with a modified derivative (covariant derivative) leads to the form invariance of the Schrödinger equation. The relationship to the covariant derivation from general relativity is explained in the section #Covariant derivation . ${\displaystyle {\hat {\vec {p}}}={\frac {\hbar }{i}}\nabla }$ ${\displaystyle {\frac {\hbar }{i}}\nabla -q{\vec {A}}}$

If one considers now the Hamilton operator with inserted minimal coupling and the same Hamilton operator only with recalibrated vector potential , then the deleted Schrödinger equation leads ${\displaystyle {\hat {H}}={\frac {1}{2m}}\left({\hat {\vec {p}}}-q{\vec {A}}\right)^{2}}$ ${\displaystyle {\hat {H}}'}$ ${\displaystyle {\vec {A}}'={\vec {A}}+\nabla \chi }$

${\displaystyle {\hat {H}}'\psi '=i\hbar {\frac {\partial \psi '}{\partial t}}}$

on the unprimed Schrödinger equation

${\displaystyle {\hat {H}}\psi =i\hbar {\frac {\partial \psi }{\partial t}}}$.

Both gauge transformations cancel each other out so that the Schrödinger equation written with covariant derivation is form-invariant with gauge transformation of the potentials and the wave functions.

Schrödinger equation with spin

The principle of minimal coupling only leads to the quantitative coupling between charged particles and electromagnetic field in relativistic quantum mechanics (i.e. when applied to the Dirac equation ), which has so far been proven experimentally. In the “classical” Schrödinger equation, the part of the interaction between electron and light that depends on the electron's spin is still missing . A trick can be used to introduce this spin component into non-relativistic quantum mechanics using the principle of minimal coupling. For the Pauli matrices applies to any matrix : . Now we modify the free Hamilton operator in the Schrödinger equation with this "hidden" Pauli matrix ${\displaystyle \mathbf {\sigma } }$ ${\displaystyle \mathbf {a} }$ ${\displaystyle (\mathbf {\sigma } \mathbf {a} )^{2}=\mathbf {a} ^{2}}$

${\displaystyle {\frac {1}{2m}}{\hat {\vec {p}}}^{2}={\frac {1}{2m}}(\mathbf {\sigma } {\hat {\vec {p}}})^{2}}$.

If one now applies the principle of minimal coupling to this modified free Hamilton operator, one obtains

${\displaystyle {\hat {H}}={\frac {\hbar ^{2}}{2m}}\left(\mathbf {\sigma } \left(\nabla -{\frac {iq}{\hbar }}{\vec {A}}\right)\right)^{2}}$.

Multiplying out, observing the sequence and using the definition of the magnetic field given above, results ${\displaystyle {\vec {B}}}$

${\displaystyle {\hat {H}}=-{\frac {\hbar ^{2}}{2m}}\left(\nabla -{\frac {iq}{\hbar }}{\vec {A}}\right)^{2}-{\frac {q\hbar }{2m}}\mathbf {\sigma } \cdot {\vec {B}}}$.

This corresponds to the Pauli equation , which describes the dynamics of a non-relativistic spin 1/2 particle with charge and mass in an electromagnetic field (without scalar potential). ${\displaystyle q}$ ${\displaystyle m}$

Dirac equation

The free Dirac equation reads using the Dirac matrices ${\displaystyle \gamma ^{\mu }}$

${\displaystyle \left(i\gamma ^{\mu }\partial _{\mu }-{\frac {mc}{\hbar }}\right)\psi (x)=0}$

and is Lorentz invariant . Just as in the case of the Schrödinger equation, the equation is not gauge invariant under phase transformation. Insertion of the minimal coupling in the four-vector notation, i.e.

${\displaystyle \partial _{\mu }\rightarrow \partial _{\mu }+{\frac {iq}{\hbar c}}A_{\mu }}$

with , leads to the relativistic covariant form of the Dirac equation with coupled electromagnetic field. ${\displaystyle A_{\mu }=(A_{0},A_{1},A_{2},A_{3})=(\phi ,-A_{x},-A_{y},-A_{z})}$

${\displaystyle \left(i\gamma ^{\mu }\left(\partial _{\mu }+{\frac {iq}{\hbar c}}A_{\mu }\right)-{\frac {mc}{\hbar }}\right)\psi (x)=0}$

${\displaystyle D_{\mu }:=\partial _{\mu }+{\frac {iq}{\hbar c}}A_{\mu }}$

is also called covariant derivative , because replacing the “normal partial derivative” with the “covariant derivative” leads to covariance with regard to gauge transformations of the equation in question.

Dipole approximation

The Hamilton function for a charged particle in an electromagnetic field and a potential is through ${\displaystyle V}$

${\displaystyle H={\frac {1}{2m}}\left({\vec {p}}-q{\vec {A}}({\vec {x}},t)\right)^{2}+q\phi ({\vec {x}},t)+V({\vec {x}})}$

given. This Hamilton function describes a classical charged particle in a potential. The quantum mechanical version (transition from the Hamilton function to the Hamilton operator) would describe a single electron bound to an atom (hydrogen atom). For the sake of simplicity, the dipole approximation to the classical Hamilton function will be shown in the following section.

The Hamilton function can be divided into two parts. One part describes the system (electron in potential) itself and the other describes its interaction with the electromagnetic field.

${\displaystyle H_{0}={\frac {{\vec {p}}^{2}}{2m}}+V({\vec {x}})}$,

${\displaystyle H_{WW}=-{\frac {q}{2m}}({\vec {p}}\cdot {\vec {A}}({\vec {x}},t)+{\vec {A}}({\vec {x}},t)\cdot {\vec {p}})+{\frac {q^{2}}{2m}}{\vec {A}}^{2}({\vec {x}},t)+q\phi ({\vec {x}},t)}$.

Looking now at the situation in an electromagnetic field in the Strahlungseichung ( , and therefore ), and only takes into account the coupling in linear order with , one obtains ${\displaystyle \phi =0}$ ${\displaystyle \nabla {\vec {A}}=0}$ ${\displaystyle [{\vec {A}},{\hat {\vec {p}}}]=0}$ ${\displaystyle {\vec {A}}}$

${\displaystyle H_{WW}=-{\frac {q}{m}}{\vec {p}}\cdot {\vec {A}}({\vec {x}},t)}$.

The vector potential can also be approximated as. As long as the characteristic wavelength of the electromagnetic field is much larger than the size of the atom, the vector potential can be viewed as spatially almost homogeneous over the size of the atom. If one writes the canonical momentum as a kinetic momentum , it follows ${\displaystyle {\vec {A}}({\vec {x}},t)\approx {\vec {A}}(t)}$ ${\displaystyle \lambda =2\pi k^{-1}}$ ${\displaystyle {\vec {p}}=m{\frac {\mathrm {d} {\vec {x}}}{\mathrm {d} t}}}$

${\displaystyle H_{WW}=-q{\frac {\mathrm {d} {\vec {x}}}{\mathrm {d} t}}\cdot {\vec {A}}(t)=-q{\frac {\mathrm {d} }{\mathrm {d} t}}\left[{\vec {x}}{\vec {A}}(t)\right]+q{\vec {x}}{\frac {\mathrm {d} {\vec {A}}(t)}{\mathrm {d} t}}}$.

In the dipole approximation , the electric field is given. This is listing ${\displaystyle {\vec {E}}(t)}$ ${\displaystyle {\vec {E}}(t)=-{\frac {\mathrm {d} {\vec {A}}(t)}{\mathrm {d} t}}}$

${\displaystyle H_{WW}=-q{\vec {x}}\cdot {\vec {E}}(t)-q{\frac {\mathrm {d} }{\mathrm {d} t}}\left[{\vec {x}}{\vec {A}}(t)\right]}$.

The last term can be omitted because the Hamilton function is only determined up to the total time derivative of any function. Finally, the interaction Hamiltonian function is obtained for a bound charged particles in the dipole approximation to

${\displaystyle H_{WW}=-q{\vec {x}}\cdot {\vec {E}}(t)}$.

This result was derived from the principle of minimal coupling and is also used in its quantum mechanical equivalent (here classical derivation) of quantum electrodynamics . A name that is often used for this interaction is also “ -Hamiltonian”, pronounced “E mal r Hamiltonian”, since it is often used for the location coordinate . One can still use the dipole operator ${\displaystyle E\cdot r}$ ${\displaystyle x}$ ${\displaystyle r}$

${\displaystyle {\vec {d}}=q{\vec {x}}}$

define (by analogy with an electric dipole ). So it is obvious that the field in the dipole approximation only couples to the dipole moment of the hydrogen atom. In general, the above procedure can also be carried out for atoms with more than one electron.

Multipolar coupling and Power-Zienau-Woolley transformation

In general, the minimal coupling Hamilton operator can be brought into the equivalent representation of the multipolar coupling Hamilton operator with the unitary Power-Zinau-Woolley transformation . Here the electromagnetic field is coupled to the polarization and magnetization via the vector potential . This form of the interaction Hamiltonian can describe light-matter interactions of dielectrics .

general theory of relativity

In the general theory of relativity , the term principle of minimal coupling denotes a slightly modified principle. The Einstein field equations in vacuum may be made of a Lagrangian density of the mold

${\displaystyle L_{Vac}=a{\sqrt {g}}R}$

can be derived with the metric , the curvature scalar and a constant . The coupling to other fields (e.g. electromagnetic field) should now be achieved by adding a suitable interaction Lagrangian density . The decomposition of the Lagrangian density in is called the principle of minimal gravitational coupling . ${\displaystyle g}$ ${\displaystyle R}$ ${\displaystyle a}$ ${\displaystyle L_{WW}}$ ${\displaystyle L_{total}=L_{Vac}+L_{WW}}$

Covariant derivative

One principle of general relativity is the covariance principle , which states that equations that are valid in special relativity and are therefore Lorentz invariant , by replacing the partial derivatives with the covariant derivative, become generally coordinate-independent equations (generally covariant). From a mathematical point of view, this covariant derivation corresponds to the Levi-Civita relationship . This is the relationship on the tangential vector bundle of a semi-Riemann manifold . On the one hand, the covariant derivation leads to covariant (form-invariant under change of coordinates) equations, otherwise the covariant derivation defines the parallel transport of tensors in curved spaces. ${\displaystyle {\frac {\partial F^{\alpha }}{\partial x^{\mu }}}}$ ${\displaystyle D_{ART,\mu }F^{\alpha }={\frac {\partial F^{\alpha }}{\partial x^{\mu }}}+\Gamma _{\sigma \mu }^{\alpha }F^{\sigma }}$

In the gauge field theory (e.g. all theories regarding the fundamental interactions in the standard model of particle physics) the wave functions of the particles are subject to certain symmetries. These symmetries manifest themselves in the invariance of the theory's Lagrange density on the action of a group (in the case of the Schrödinger equation ). The wave functions are defined on a manifold . Both structures and are combined in modern differential geometry to form a uniform structure P (M, G), the main fiber bundle . A main fiber bundle is a manifold to which a copy of the structural group is attached for each point of . These copies are called fibers , and the representation of group elements from different fibers resides in disjoint vector spaces. Since and are located in different rooms, a derivation can only be made after defining a connection on the main fiber bundle. The replacement of the partial derivative by the minimal coupling is precisely the covariant derivative (relationship in coordinates) in this case. Just as in the case of the GTR, the Christoffel symbols determine the curvature of space (and the Christoffelsymbols depend on the metric), so in the case of the gauge field theories, the four potential determines the curvature. The curvature tensor results in both cases from the commutator of the covariant derivative. ${\displaystyle G}$ ${\displaystyle U(1)}$ ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle G}$ ${\displaystyle x}$ ${\displaystyle M}$ ${\displaystyle G(x)}$ ${\displaystyle \phi (x)}$ ${\displaystyle \phi (x)}$ ${\displaystyle \phi (x+\mathrm {d} x)}$

Origin of the designation

The Lagrange density of the electromagnetic field with minimal coupling is:

${\displaystyle {\mathcal {L}}={\mathcal {L}}_{rad}+{\mathcal {L}}_{WW}=-{\frac {1}{16\pi }}F^{\nu \mu }F_{\nu \mu }-{\frac {1}{c}}A_{\mu }j^{\mu }}$.

The first part is the kinetic term with the field strength tensor and the second term the coupling of the field to the “charged current” - the charged matter, according to the principle of minimal coupling. The relationship with the minimal coupling procedure described in the introduction is explained below. ${\displaystyle F^{\mu \nu }}$

The name minimal coupling comes from the fact that it represents the simplest combination of charge current density and electromagnetic field that fulfills the following conditions: ${\displaystyle j^{\mu }}$ ${\displaystyle A^{\mu }}$

• Receives Lorentz invariance of the free equation
• Gauge invariance
• Couples electromagnetic field to charged matter ${\displaystyle A^{\mu }}$ ${\displaystyle j^{\mu }}$

In addition, precisely this minimal coupling procedure leads to a calibration-invariant effect .

The above representation of the minimal coupling in the Lagrange density corresponds exactly to the procedure described in the introduction for a point-like charged particle. To do this, consider the four-current density of a point particle:

${\displaystyle j=(j^{\mu })=(c\rho ,{\vec {j}})=q\delta ^{(3)}({\vec {x}}-{\vec {x}}_{1}(t))(c,{\vec {v}})=qc\delta ^{(3)}({\vec {x}}-{\vec {x}}_{1}(t))\gamma (c,{\vec {v}}){\frac {1}{\gamma c}}=qc\delta ^{(3)}({\vec {x}}-{\vec {x}}_{1}(t))u{\frac {\mathrm {d} \tau }{\mathrm {d} x^{0}}}}$.

Here, the usual symbols are from the special theory of relativity used: is the charge density , the Dirac delta function in three dimensions, the velocity of the charged particle, the Lorentz factor , the quadruple speed and the proper time . It follows: . If we now insert this into the effect of the interaction Lagrange density , we get: ${\displaystyle \rho }$ ${\displaystyle \delta ^{(3)}}$ ${\displaystyle v}$ ${\displaystyle \gamma }$ ${\displaystyle u}$ ${\displaystyle \tau }$ ${\displaystyle j=cq\int \mathrm {d} \tau \delta ^{(4)}(x-x_{1}(\tau ))u(\tau )}$ ${\displaystyle S_{WW}}$ ${\displaystyle {\mathcal {L}}_{WW}}$

${\displaystyle S_{WW}=-{\frac {1}{c^{2}}}\int \mathrm {d} x^{4}A_{\mu }(x)j^{\mu }(x)}$

${\displaystyle =-{\frac {q}{c}}\int \mathrm {d} x^{4}A_{\mu }(x)\int \mathrm {d} \tau \delta ^{(4)}(x-x_{1}(\tau ))u^{\mu }(\tau )}$

${\displaystyle =-{\frac {q}{c}}\int \mathrm {d} \tau A_{\mu }(x_{1}(\tau ))u^{\mu }(\tau )}$

If we write out the scalar product of the two four-vectors, we get:

${\displaystyle S_{WW}=\int \mathrm {d} t\left(-q\phi ({\vec {x}}_{1},t)+q{\vec {v}}_{1}\cdot {\vec {A}}({\vec {x}}_{1},t)\right)=:\int \mathrm {d} tL_{WW}}$.

To get the total Lagrangian density , the kinetic part for a particle of mass has to be added: ${\displaystyle L}$ ${\displaystyle m}$

${\displaystyle L={\frac {m}{2}}{\vec {v}}_{1}^{2}-q\phi ({\vec {x}}_{1},t)+q{\vec {v}}_{1}\cdot {\vec {A}}({\vec {x}}_{1},t)}$.

The canonical impulse results from to ${\displaystyle {\vec {p}}=\nabla _{{\vec {v}}_{1}}L}$

${\displaystyle {\vec {p}}=m{\vec {v}}_{1}+q{\vec {A}}}$.

The kinetic momentum is accordingly . This result corresponds exactly to the replacement that is carried out when introducing the minimum coupling into the Lagrange density or Hamilton function of a free particle. As described in the introduction of canonical momentum corresponding to the kinetic momentum of a free particle is replaced by the kinetic momentum of the particles in the electromagnetic field: . ${\displaystyle {\vec {\pi }}=m{\vec {v}}_{1}={\vec {p}}-q{\vec {A}}}$ ${\displaystyle {\vec {p}}={\vec {\pi }}_{frei}}$ ${\displaystyle {\vec {p}}\rightarrow {\vec {\pi }}}$

The interaction Lagrangian density thus leads exactly to the result that is assumed by the procedure given in the introduction and explains the name of the coupling procedure. ${\displaystyle {\mathcal {L}}_{WW}={\frac {1}{c}}A^{\mu }j_{\mu }}$

Remarks

1. ↑ Also called "temporal gauge" in English : Article in the Engl. Wikipedia .
2. ↑ Using the Weyl calibration, otherwise one would have to replace the left side of the Schrödinger equation according to the principle of minimal coupling.

Individual evidence

• Greiner, Quantum Mechanics Part 1. Introduction , Verlag Harri Deutsch (1989), ISBN 3-8171-1064-2

1. ↑ p. 226.
2. ↑ p. 232-p. 234.
3. ↑ p. 232.
4. ↑ ^{a b} p. 228.

• Rollnik, Quantum Theory 2 , Springer-Verlag Berlin Heidelberg New York (2003), ISBN 3-540-43717-7

1. ↑ ^{a b c} p. 235.

• Claude Amsler: Nuclear and Particle Physics . vdf Hochschulverlag AG, March 1, 2007, ISBN 978-3-8252-2885-9 .

1. ↑ p. 170.

• Ingolf H. Hertel, Ingolf V. Hertel, Claus-Peter Schulz: Atoms, Molecules and Optical Physics 1 . Springer, March 5, 2008, ISBN 978-3-540-30613-9 .

1. ↑ pp. 478-482.

• Theoretical Physics II (electrodynamics), script by P. Blöchl

1. ↑ p. 1ff.

• Fritz Ehlotzky: Quantum Mechanics and Its Applications . Springer, September 15, 2004, ISBN 978-3-540-21450-2 .

1. ↑ p. 306.

• Franz Schwabl: Quantum Mechanics for Advanced Students (QM II) . Springer, September 16, 2008, ISBN 978-3-540-85075-5 .

1. ↑ ^{a b} p. 130.

• I. Anderson, The principle of minimal gravitational coupling , Archive for Rational Mechanics and Analysis, Volume 75, Issue 4, Springer (1981), doi : 10.1007 / BF00256383

1. ↑ p. 1ff.

• Wolfgang P. Schleich: Quantum Optics in Phase Space . John Wiley & Sons, February 16, 2011, ISBN 978-3-527-63500-9 .

1. ↑ p. 383.
2. ↑ p. 389.

• Markus Reiher, Alexander Wolf: Relativistic Quantum Chemistry: The Fundamental Theory of Molecular Science . John Wiley & Sons, June 2, 2009, ISBN 978-3-527-62749-3 .

1. ↑ p. 95.