Klein–Gordon equation

Quantum mechanics
${\hat {H}}\|\psi (t)\rangle =i\hbar {\frac {\partial }{\partial t}}\|\psi (t)\rangle$ Schrödinger equation
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The Klein–Gordon equation (Klein–Fock–Gordon equation or sometimes Klein–Gordon–Fock equation) is a relativistic wave equation, related to the Schrödinger equation. It is second order in space and time and manifestly Lorentz covariant. It is a quantized version of the relativistic energy–momentum relation. Its solutions include a quantum scalar or pseudoscalar field, a field whose quanta are spinless particles. Its theoretical relevance is similar to that of the Dirac equation.^[1] Electromagnetic interactions can be incorporated, forming the topic of scalar electrodynamics, but because common spinless particles like the pi mesons are unstable and also experience the strong interaction (with unknown interaction term in the Hamiltonian^[2]), the practical utility is limited.

The equation can be put into the form of a Schrödinger equation. In this form it is two coupled differential equations, each of first order in time.^[3] The solutions have two components, reflecting the charge degree of freedom in relativity.^[3]^[4] It admits a conserved quantity, but this is not positive definite. The wave function cannot therefore be interpreted as a probability amplitude. The conserved quantity is instead interpreted as electric charge and the norm squared of the wave function is interpreted as a charge density. The equation describes all spinless particles with positive, negative as well as zero charge. Within the Feynman–Stueckelberg interpretation, particles and antiparticles are treated mathematically as if they propagate forward and backward in time respectively, the advanced propagator (as opposed to the retarded propagator) is employed for antiparticles. Physically, all particles move forward in time.

Any solution of the free Dirac equation is componentwise a solution of the free Klein–Gordon equation.

The equation does not form the basis for a consistent quantum relativistic one-particle theory. There is no known such theory for particles of any spin. For full reconciliation of quantum mechanics with special relativity quantum field theory is needed, in which the Klein–Gordon equation reemerges as the equation obeyed by the components of all free quantum fields.^{[nb 1]} In quantum field theory, the solutions of the free (noninteracting) versions of the original equations still play a role. They are needed to build the Hilbert space (Fock space) and to express quantum field by using complete sets (spanning sets of Hilbert space) of wave functions.

Statement

The Klein–Gordon equation with mass parameter $m$ is

{\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}\psi -\nabla ^{2}\psi +{\frac {m^{2}c^{2}}{\hbar ^{2}}}\psi =0.

Solutions of the equation are complex-valued functions $\psi (t,{\mathbf {x}})$ of the time variable $t$ and space variables $\bold{x}$ ; the Laplacian $\nabla ^{2}$ acts on the space variables only.

The equation is often abbreviated as

(\Box +\mu ^{2})\psi =0,

where $μ = mc / ħ$ and $□$ is the d'Alembert operator, defined by

\Box =-\eta ^{\mu \nu }\partial _{\mu }\partial _{\nu }={\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}-\nabla ^{2}.

(We are using the (−, +, +, +) metric signature.)

The Klein–Gordon equation is often written in natural units:

-\partial _{t}^{2}\psi +\nabla ^{2}\psi =m^{2}\psi

The form of the Klein–Gordon equation is derived by requiring that plane wave solutions of the equation:

\psi =e^{-i\omega t+ik\cdot x}=e^{ik_{\mu }x^{\mu }}

obey the energy momentum relation of special relativity:

-p_{\mu }p^{\mu }=E^{2}-P^{2}=\omega ^{2}-k^{2}=-k_{\mu }k^{\mu }=m^{2}

Unlike the Schrödinger equation, the Klein–Gordon equation admits two values of $ω$ for each $k$ , one positive and one negative. Only by separating out the positive and negative frequency parts does one obtain an equation describing a relativistic wavefunction. For the time-independent case, the Klein–Gordon equation becomes

\left[\nabla ^{2}-{\frac {m^{2}c^{2}}{\hbar ^{2}}}\right]\psi (\mathbf {r} )=0

which is formally the same as the homogeneous screened Poisson equation.

History

The equation was named after the physicists Oskar Klein and Walter Gordon, who in 1926 proposed that it describes relativistic electrons. Other authors making similar claims in that same year were Vladimir Fock, Johann Kudar, Théophile de Donder and Frans-H. van den Dungen, and Louis de Broglie. Although it turned out that modeling the electron's spin required the Dirac equation, the Klein–Gordon equation correctly describes the spinless relativisitic composite particles, like the pion. On July 4, 2012 CERN announced the discovery of the Higgs boson. Since the Higgs boson is a spin-zero particle, it is the first observed ostensibly elementary particle to be described by the Klein–Gordon equation. Further experimentation and analysis is required to discern whether the Higgs boson observed is that of the Standard Model, or a more exotic, possibly composite, form.

The Klein–Gordon equation was first considered as a quantum wave equation by Schrödinger in his search for an equation describing de Broglie waves. The equation is found in his notebooks from late 1925, and he appears to have prepared a manuscript applying it to the hydrogen atom. Yet, because it fails to take into account the electron's spin, the equation predicts the hydrogen atom's fine structure incorrectly, including overestimating the overall magnitude of the splitting pattern by a factor of $4 n / 2 n - 1$ for the $n$ -th energy level. The Dirac equation relativistic spectrum is, however, easily recovered if the orbital momentum quantum number $ℓ$ is replaced by total angular momentum quantum number $j$ .^[5] In January 1926, Schrödinger submitted for publication instead his equation, a non-relativistic approximation that predicts the Bohr energy levels of hydrogen without fine structure.

In 1926, soon after the Schrödinger equation was introduced, Vladimir Fock wrote an article about its generalization for the case of magnetic fields, where forces were dependent on velocity, and independently derived this equation. Both Klein and Fock used Kaluza and Klein's method. Fock also determined the gauge theory for the wave equation. The Klein–Gordon equation for a free particle has a simple plane wave solution.

Derivation

The non-relativistic equation for the energy of a free particle is

{\frac {\mathbf {p} ^{2}}{2m}}=E.

By quantizing this, we get the non-relativistic Schrödinger equation for a free particle,

{\frac {\mathbf {\hat {p}} ^{2}}{2m}}\psi ={\hat {E}}\psi

where

\mathbf {\hat {p}} =-i\hbar \mathbf {\nabla }

is the momentum operator ( $\nabla$ being the del operator), and

{\hat {E}}=i\hbar {\dfrac {\partial }{\partial t}}

is the energy operator.

The Schrödinger equation suffers from not being relativistically invariant, meaning that it is inconsistent with special relativity.

It is natural to try to use the identity from special relativity describing the energy:

{\sqrt {\mathbf {p} ^{2}c^{2}+m^{2}c^{4}}}=E

Then, just inserting the quantum mechanical operators for momentum and energy yields the equation

{\sqrt {(-i\hbar \mathbf {\nabla } )^{2}c^{2}+m^{2}c^{4}}}\psi =i\hbar {\frac {\partial }{\partial t}}\psi .

The square root of a differential operator can be defined with the help of Fourier transformations, but due to the asymmetry of space and time derivatives, Dirac found it impossible to include external electromagnetic fields in a relativistically invariant way. So he looked for another equation that can be modified in order to describe the action of electromagnetic forces. In addition, this equation, as it stands, is nonlocal (see also Introduction to nonlocal equations).

Klein and Gordon instead began with the square of the above identity, i.e.

\mathbf {p} ^{2}c^{2}+m^{2}c^{4}=E^{2}

which, when quantized, gives

\left((-i\hbar \mathbf {\nabla } )^{2}c^{2}+m^{2}c^{4}\right)\psi =\left(i\hbar {\frac {\partial }{\partial t}}\right)^{2}\psi

which simplifies to

-\hbar ^{2}c^{2}\mathbf {\nabla } ^{2}\psi +m^{2}c^{4}\psi =-\hbar ^{2}{\frac {\partial ^{2}}{\partial t^{2}}}\psi .

Rearranging terms yields

{\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}\psi -\mathbf {\nabla } ^{2}\psi +{\frac {m^{2}c^{2}}{\hbar ^{2}}}\psi =0.

Since all reference to imaginary numbers has been eliminated from this equation, it can be applied to fields that are real valued as well as those that have complex values.

Rewriting the first two terms using the inverse of the Minkowski metric $diag(- c 2, 1, 1, 1)$ , and writing the Einstein summation convention explicitly we get

-\eta ^{\mu \nu }\partial _{\mu }\partial _{\nu }\psi \equiv \sum _{\mu =0}^{\mu =3}\sum _{\nu =0}^{\nu =3}-\eta ^{\mu \nu }\partial _{\mu }\partial _{\nu }\psi ={\frac {1}{c^{2}}}\partial _{0}^{2}\psi -\sum _{\nu =1}^{\nu =3}\partial _{\nu }\partial _{\nu }\psi ={\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}\psi -\mathbf {\nabla } ^{2}\psi

Thus the Klein–Gordon equation can be written in a covariant notation. This often means an abbreviation in the form of

(\Box +\mu ^{2})\psi =0,

where

\mu ={\frac {mc}{\hbar }}

and

\Box ={\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}-\nabla ^{2}.

This operator is called the d'Alembert operator.

Today this form is interpreted as the relativistic field equation for spin-0 particles.^[3] Furthermore, any component of any solution to the free Dirac equation (for a spin-one-half particle) is automatically a solution to the free Klein–Gordon equation. This generalizes to particles of any spin due extension to the Bargmann–Wigner equations. Furthermore, in quantum field theory, every component of every quantum field must satisfy the free Klein–Gordon equation,^[6] making the equation a generic expression of quantum fields.

Klein–Gordon equation in a potential

The Klein–Gordon equation can be generalized to describe a field in some potential $V (ψ)$ as:^[7]

\Box \psi +{\frac {\partial {}V}{\partial \psi }}=0

Conserved current

The conserved current associated to the U(1) symmetry of a complex field $\phi (x)\in \mathbb {C}$ satisfying the Klein–Gordon equation reads

\partial _{\mu }J^{\mu }(x)=0,\qquad J^{\mu }(x)\equiv \phi ^{*}(x)\partial ^{\mu }\phi (x)-\phi (x)\partial ^{\mu }\phi ^{*}(x).

The form of the conserved current can be derived systematically by applying Noether's theorem to the U(1) symmetry. We will not do so here, but simply give a proof that this conserved current is correct.

Proof using algebraic manipulations from the KG equation

From the Klein Gordon equation for a complex field $\phi (x)$ of mass $m$ written in covariant notation

(\square +m^{2})\phi (x)=0,

and its complex conjugate

(\square +m^{2})\phi ^{*}(x)=0,

we have, multiplying by the left respectively by $\phi ^{*}(x)$ and $\phi (x)$ (and omitting for brevity the explicit $x$ dependence),

\phi ^{*}(\square +m^{2})\phi =0,

\phi (\square +m^{2})\phi ^{*}=0.

Subtracting the former from the latter we obtain

\phi ^{*}\square \phi -\phi \square \phi ^{*}=0

from which we obtain the conservation law for the Klein Gordon field:

\partial _{\mu }J^{\mu }(x)=0,\qquad J^{\mu }(x)\equiv \phi ^{*}(x)\partial ^{\mu }\phi (x)-\phi (x)\partial ^{\mu }\phi ^{*}(x).

Relativistic free particle solution

The Klein–Gordon equation for a free particle can be written as

\mathbf {\nabla } ^{2}\psi -{\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}\psi ={\frac {m^{2}c^{2}}{\hbar ^{2}}}\psi

We look for plane wave solutions of the form

\psi (\mathbf {r} ,t)=e^{i(\mathbf {k} \cdot \mathbf {r} -\omega t)}

for some constant angular frequency $ω \in ℝ$ and wave number $k \in ℝ 3$ . Substitution gives the dispersion relation:

-|\mathbf {k} |^{2}+{\frac {\omega ^{2}}{c^{2}}}={\frac {m^{2}c^{2}}{\hbar ^{2}}}.

Energy and momentum are seen to be proportional to $ω$ and $k$ :

\langle \mathbf {p} \rangle =\left\langle \psi \left|-i\hbar \mathbf {\nabla } \right|\psi \right\rangle =\hbar \mathbf {k} ,

\langle E\rangle =\left\langle \psi \left|i\hbar {\frac {\partial }{\partial t}}\right|\psi \right\rangle =\hbar \omega .

So the dispersion relation is just the classic relativistic equation:

\langle E\rangle ^{2}=m^{2}c^{4}+\langle \mathbf {p} \rangle ^{2}c^{2}.

For massless particles, we may set $m = 0$ , recovering the relationship between energy and momentum for massless particles:

\langle E\rangle =\langle |\mathbf {p} |\rangle c.

Action

The Klein–Gordon equation can also be derived via a variational method by considering the action:

{\mathcal {S}}=\int \left(-{\frac {\hbar ^{2}}{m}}\eta ^{\mu \nu }\partial _{\mu }{\bar {\psi }}\partial _{\nu }\psi -mc^{2}{\bar {\psi }}\psi \right)\mathrm {d} ^{4}x

where $ψ$ is the Klein–Gordon field and $m$ is its mass. The complex conjugate of $ψ$ is written $ψ$ . If the scalar field is taken to be real-valued, then $ψ = ψ$ .

Applying the formula for the Hilbert stress–energy tensor to the Lagrangian density (the quantity inside the integral), we can derive the stress–energy tensor of the scalar field. It is

T^{\mu \nu }={\frac {\hbar ^{2}}{m}}\left(\eta ^{\mu \alpha }\eta ^{\nu \beta }+\eta ^{\mu \beta }\eta ^{\nu \alpha }-\eta ^{\mu \nu }\eta ^{\alpha \beta }\right)\partial _{\alpha }{\bar {\psi }}\partial _{\beta }\psi -\eta ^{\mu \nu }mc^{2}{\bar {\psi }}\psi .

By integration of the time–time component $T 00$ over all space, one may show that both the positive and negative frequency plane wave solutions can be physically associated with particles with positive energy. This is not the case for the Dirac equation and its energy–momentum tensor.^[3]

Electromagnetic interaction

There is a simple way to make any field interact with electromagnetism in a gauge invariant way: replace the derivative operators with the gauge covariant derivative operators. This is because to maintain symmetry of the physical equations for the wavefunction $\phi$ under a local U(1) gauge transformation $\phi \rightarrow \phi '=exp(i\theta )\phi$ where $\theta (t,{\textbf {x}})$ is a locally variable phase angle, which transformation redirects the wavefunction in the complex phase space defined by $exp(i\theta )=cos\theta +isin\theta$ , it is required that ordinary derivatives $\partial _{\mu }$ be replaced by gauge-covariant derivatives $D_{\mu }=\partial _{\mu }-ieA_{\mu }$ while the gauge fields transform as $eA_{\mu }\rightarrow eA'_{\mu }=eA_{\mu }+\partial _{\mu }\theta$ . The Klein Gordon equation therefore becomes:

D_{\mu }D^{\mu }\phi =-(\partial _{t}-ieA_{0})^{2}\phi +(\partial _{i}-ieA_{i})^{2}\phi =m^{2}\phi

in natural units, where $A$ is the vector potential. While it is possible to add many higher order terms, for example,

D_{\mu }D^{\mu }\phi +AF^{\mu \nu }D_{\mu }\phi D_{\nu }(D_{\alpha }D^{\alpha }\phi )=0

these terms are not renormalizable in 3+1 dimensions.

The field equation for a charged scalar field multiplies by $i$ , which means the field must be complex. In order for a field to be charged, it must have two components that can rotate into each other, the real and imaginary parts.

The action for a charged scalar is the covariant version of the uncharged action:

S=\int _{x}\left(\partial _{\mu }\phi ^{*}+ieA_{\mu }\phi ^{*}\right)\left(\partial _{\nu }\phi -ieA_{\nu }\phi \right)\eta ^{\mu \nu }=\int _{x}|D\phi |^{2}

Gravitational interaction

In general relativity, we include the effect of gravity by replacing partial with covariant derivatives and the Klein–Gordon equation becomes (in the mostly pluses signature)^[8]

{\begin{aligned}0&=-g^{\mu \nu }\nabla _{\mu }\nabla _{\nu }\psi +{\dfrac {m^{2}c^{2}}{\hbar ^{2}}}\psi =-g^{\mu \nu }\nabla _{\mu }(\partial _{\nu }\psi )+{\dfrac {m^{2}c^{2}}{\hbar ^{2}}}\psi \\&=-g^{\mu \nu }\partial _{\mu }\partial _{\nu }\psi +g^{\mu \nu }\Gamma ^{\sigma }{}_{\mu \nu }\partial _{\sigma }\psi +{\dfrac {m^{2}c^{2}}{\hbar ^{2}}}\psi \end{aligned}}

or equivalently

{\frac {-1}{\sqrt {-g}}}\partial _{\mu }\left(g^{\mu \nu }{\sqrt {-g}}\partial _{\nu }\psi \right)+{\frac {m^{2}c^{2}}{\hbar ^{2}}}\psi =0

where g $αβ$ is the inverse of the metric tensor that is the gravitational potential field, g is the determinant of the metric tensor, $\nabla μ$ is the covariant derivative and $Γ σ μν$ is the Christoffel symbol that is the gravitational force field.

Remarks

↑ Steven Weinberg makes a point about this. He leaves out the treatment of relativistic wave mechanics altogether in his otherwise complete introduction to modern applications of quantum mechanics, explaining "It seems to me that the way this is usually presented in books on quantum mechanics is profoundly misleading." (From the preface in Lectures on Quantum Mechanics, referring to treatments of the Dirac equation in its original flavor.)

Others, like Walter Greiner does in his series on theoretical physics, give a full account of the historical development and view of relativistic quantum mechanics before they get to the modern interpretation, with the rationale that it is highly desirable or even necessary from a pedagogical point of view to take the long route.

Notes

↑ Gross 1993
↑ Greiner & Müller 1994
1 2 3 4 Greiner 2000, Ch. 1
↑ Feshbach & Villars 1958
↑ See C Itzykson and J-B Zuber, Quantum Field Theory, McGraw-Hill Co., 1985, pp. 73–74. Eq. 2.87 is identical to eq. 2.86 except that it features $j$ instead of $ℓ$ .
↑ Weinberg 2002, Ch. 5
↑ David Tong, Lectures on Quantum Field Theory, Lecture 1, Section 1.1.1
↑ S.A. Fulling, Aspects of Quantum Field Theory in Curved Space–Time, Cambridge University Press, 1996, p. 117

References

Davydov, A.S. (1976). Quantum Mechanics, 2nd Edition. Pergamon Press. ISBN 0-08-020437-6.
Feshbach, H.; Villars, F. (1958). "Elementary relativistic wave mechanics of spin 0 and spin 1/2 particles". Rev. Mod. Phys. 30 (1). Bibcode:1958RvMP...30...24F. doi:10.1103/RevModPhys.30.24.
Greiner, W. (2000). Relativistic Quantum Mechanics. Wave Equations (3rd ed.). Springer Verlag. ISBN 3-5406-74578.
Greiner, W.; Müller, B. (1994). Quantum Mechanics: Symmetries (2nd ed.). Springer. ISBN 978-3540580805.
Gross, F. (1993). Relativistic Quantum Mechanics and Field Theory (1st ed.). Wiley-VCH. ISBN 978-0471591139.
Sakurai, J. J. (1967). Advanced Quantum Mechanics. Addison Wesley. ISBN 0-201-06710-2.
Weinberg, S. (2002). The Quantum Theory of Fields. I. Cambridge University Press. ISBN 0-521-55001-7.

External links

Hazewinkel, Michiel, ed. (2001) [1994], "Klein–Gordon equation", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
Weisstein, Eric W. "Klein–Gordon equation". MathWorld.

Linear Klein–Gordon Equation at EqWorld: The World of Mathematical Equations.
Nonlinear Klein–Gordon Equation at EqWorld: The World of Mathematical Equations.
Introduction to nonlocal equations.

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