Klein-Gordon equation

The Klein-Gordon equation (also Klein-Fock-Gordon equation or Klein-Gordon-Schrödinger equation ) is the relativistic field equation which determines the kinematics of free scalar fields or particles (i.e. spin  0). It is a homogeneous partial differential equation of the second order that is relativistically covariant , i.e. H. form invariant under Lorentz transformation .

history

Oskar Klein, Copenhagen 1963

After Schrödinger's publication in 1926, many physicists, including Oskar Klein and Walter Gordon , tried to find the relativistic analogue of the Schrödinger equation in order to characterize wave functions that correspond to the states of a free particle in quantum mechanics . Schrödinger himself and Wladimir Fock also came across the Klein-Gordon equation independently , which is why it is sometimes also named after them.

The Klein-Gordon equation yields the correct relationship between energy and momentum , but not the spin of the examined particles. For this reason, for charged spin 1/2 particles such as the electron and the proton in the hydrogen atom, the binding energies derived from the Klein-Gordon equation do not agree with the observed energies; the correct equation of motion for these particles is the Dirac equation . Instead, the Klein-Gordon equation correctly describes spinless particles as a scalar differential equation, e.g. B. Pions .

Derivation

The derivation is based on the energy-momentum relationship

${\displaystyle E^{2}-\mathbf {p} ^{2}c^{2}-m^{2}c^{4}=0}$

between the energy and the momentum of a particle of mass in the special theory of relativity . The first quantization interprets this relation as an equation for operators that act on wave functions. Where and are the operators ${\displaystyle E}$ ${\displaystyle \mathbf {p} }$ ${\displaystyle m}$ ${\displaystyle \phi (t,\mathbf {x} )}$ ${\displaystyle E}$ ${\displaystyle \mathbf {\widehat {p}} }$

${\displaystyle E=\mathrm {i} \hbar {\frac {\partial }{\partial t}}\,,\ \mathbf {\widehat {p}} =-\mathrm {i} \,\hbar \,\mathbf {\nabla } .}$

This gives the Klein-Gordon equation

${\displaystyle \left[{\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}-\mathbf {\nabla } ^{2}+{\frac {m^{2}c^{2}}{\hbar ^{2}}}\right]\phi (t,\mathbf {x} )=0.}$

In these units, with the D'Alembert operator

${\displaystyle \Box :=\partial ^{\mu }\partial _{\mu }={\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}-\mathbf {\nabla } ^{2}={\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}-{\frac {\partial ^{2}}{\partial x^{2}}}-{\frac {\partial ^{2}}{\partial y^{2}}}-{\frac {\partial ^{2}}{\partial z^{2}}}}$

and with the abbreviation for the space-time coordinates, the Klein-Gordon equation reads: ${\displaystyle x=(t,\mathbf {x} )}$

${\displaystyle \left(\Box +{\frac {1}{{\lambda \!\!\!^{-}}_{\text{C}}^{2}}}\right)\phi (x)=0}$

Since the wave operator and the reduced Compton wavelength transform in Minkowski space-time like scalar quantities, the relativistic invariance of the scalar equation is evident in this representation. In relativistic quantum theory, natural units , in which and have the value 1, are used instead of SI units . This gives the Klein-Gordon equation to ${\displaystyle \Box :=\partial ^{\mu }\partial _{\mu }}$ ${\displaystyle {\lambda \!\!\!^{-}}_{\text{C}}={\frac {\hbar }{m\,c}}}$ ${\displaystyle \hbar }$ ${\displaystyle c}$

${\displaystyle \partial _{t}^{2}\psi -\nabla ^{2}\psi +m^{2}\psi =0}$.

solution

The plane wave

${\displaystyle A\cdot \mathrm {e} ^{\mathrm {i} {\bigl (}\mathbf {k} \cdot \mathbf {x} -\omega \,t{\bigr )}}}$

is a solution of the Klein-Gordon equation if the angular frequency is according to ${\displaystyle \omega }$

${\displaystyle \omega (\mathbf {k} )={\sqrt {{\frac {m^{2}c^{4}}{\hbar ^{2}}}+c^{2}\mathbf {k} ^{2}}}}$

or in the Planck units

${\displaystyle \omega (\mathbf {k} )={\sqrt {m^{2}+\mathbf {k} ^{2}}}}$

is related to the wave vector . Similarly, the complex conjugate wave solves ${\displaystyle \mathbf {k} }$

${\displaystyle A\cdot \mathrm {e} ^{-\mathrm {i} {\bigl (}\mathbf {k} \cdot \mathbf {x} -\omega \,t{\bigr )}}}$

the Klein-Gordon equation, since it is real.

Since the Klein-Gordon equation is linear and homogeneous, sums and complex multiples of solutions are also solutions. Hence triggers

${\displaystyle \phi (x)=\int \!\!{\frac {\mathrm {d} ^{3}k}{(2\pi )^{3}\,2\,\omega (\mathbf {k} )}}\left[a(\mathbf {k} )\,\mathrm {e} ^{\mathrm {i} {\bigl (}\mathbf {k} \cdot \mathbf {x} -\omega (\mathbf {k} )\,t{\bigr )}}+b^{\dagger }(\mathbf {k} )\,\mathrm {e} ^{-\mathrm {i} {\bigl (}\mathbf {k} \cdot \mathbf {x} -\omega (\mathbf {k} )\,t{\bigr )}}\right]}$

with any Fourier transformable amplitudes and the Klein-Gordon equation. Conversely, every Fourier transformable solution is of this form. The frequency term in the denominator ensures a covariant normalization in quantum field theory . ${\displaystyle a(\mathbf {k} )}$ ${\displaystyle b^{\dagger }(\mathbf {k} )}$

In this representation of the solution, however, it cannot be seen that at the point it only depends on its initial values ​​on and inside the light cone of . ${\displaystyle x}$ ${\displaystyle x}$

In quantum field theory there is an operator. The operator annihilates in particle states with spin , for example negative pions, generates the oppositely charged antiparticles , positive pions. The adjoint operator then destroys positive pions and creates negative pions. ${\displaystyle \phi }$ ${\displaystyle a(\mathbf {k} )}$ ${\displaystyle s=0}$ ${\displaystyle b^{\dagger }(\mathbf {k} )}$ ${\displaystyle \phi ^{\dagger }}$

For a real field , it is invariant under phase transformations and does not contribute to the electromagnetic current. The particles that the real field destroys and generates are uncharged and agree with their antiparticles, for example neutral pions. ${\displaystyle \varphi }$ ${\displaystyle b^{\dagger }(\mathbf {k} )=(a(\mathbf {k} ))^{\dagger }.}$

Lagrangian

A Lagrangian for a real field that leads to the Klein-Gordon equation is ${\displaystyle \varphi }$

${\displaystyle {\mathcal {L}}={\frac {1}{2}}\left[(\partial _{t}\varphi )^{2}-(\partial _{x}\varphi )^{2}-(\partial _{y}\varphi )^{2}-(\partial _{z}\varphi )^{2}-m^{2}\varphi ^{2}\right]\ ={\frac {1}{2}}\left[(\partial _{\mu }\varphi )(\partial ^{\mu }\varphi )-m^{2}\varphi ^{2}\right]}$

and for a complex field ${\displaystyle \phi }$

${\displaystyle {\mathcal {L}}=\partial _{t}\phi ^{\dagger }\,\partial _{t}\phi -\partial _{x}\phi ^{\dagger }\,\partial _{x}\phi -\partial _{y}\phi ^{\dagger }\,\partial _{y}\phi -\partial _{z}\phi ^{\dagger }\,\partial _{z}\phi -m^{2}\phi ^{\dagger }\,\phi \,=(\partial _{\mu }\phi ^{\dagger })(\partial ^{\mu }\phi )-m^{2}\phi ^{\dagger }\phi .}$

With the standardization of the Lagrangian density chosen here, the same propagators result in quantum field theory for the complex field as for the real one.

Continuity equation

The Lagrangian for the complex field is invariant under the continuous family of transformations

${\displaystyle T_{\alpha }:\ \phi \mapsto \mathrm {e} ^{\mathrm {i} \alpha }\phi \,,\ \phi ^{\dagger }\mapsto (\mathrm {e} ^{\mathrm {i} \alpha }\phi )^{\dagger }\ =\mathrm {e} ^{-\mathrm {i} \alpha }\phi ^{\dagger },}$

that multiply the field by a complex phase . ${\displaystyle \mathrm {e} ^{\mathrm {i} \alpha }\,,0\leq \alpha <2\pi }$

According to Noether's theorem , this continuous symmetry includes a conserved stream with components

${\displaystyle j_{\mu }=\mathrm {i} \left(\phi ^{\dagger }\,\partial _{\mu }\phi -(\partial _{\mu }\phi ^{\dagger })\,\phi \right)\,,\ \mu \in \{0,1,2,3\}.}$

The 0 component is the density of the charge received:

${\displaystyle \rho (x)=j_{0}(x)=\mathrm {i} \left(\phi ^{\dagger }\,\partial _{t}\phi -(\partial _{t}\phi ^{\dagger })\,\phi \right)}$

This density is not positively definite and cannot be interpreted as a probability density . Rather will

${\displaystyle Q=\int \mathrm {d} ^{3}x\,j_{0}=\mathrm {i} \int \mathrm {d} ^{3}x\,\left(\phi ^{\dagger }\,\partial _{t}\phi -(\partial _{t}\phi ^{\dagger })\,\phi \right)}$

interpreted as the electric charge and the electromagnetic four-current density to which the scalar potential and the vector potential of electrodynamics couple. ${\displaystyle j_{\mu }}$

See also

• Wave equation

literature

• NN Bogoliubov , DV Shirkov: Introduction to the Theory of Quantized Fields. Wiley-Interscience, New York 1959.
• R. Courant , D. Hilbert : Methods of mathematical physics. Volume 2. 2nd edition. Springer, 1968.

Individual evidence

1. ↑ Eckhard Rebhan: Theoretical Physics: Relativistic Quantum Mechanics, Quantum Field Theory and Elementary Particle Theory . Springer, Berlin Heidelberg 2010, ISBN 978-3-8274-2602-4 , p. 3.116 .