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iℏ∂∂t|ψ(t)⟩=H^|ψ(t)⟩{displaystyle ihbar {frac {partial }{partial t}}|psi (t)rangle ={hat {H}}|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 expressed as 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.

Any solution of the free Dirac equation is, component-wise, a solution of the free Klein–Gordon equation.

The equation does not form the basis of 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{displaystyle m} is

1c2∂2∂t2ψ−∇2ψ+m2c2ℏ2ψ=0.{displaystyle {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 ψ(t,x){displaystyle psi (t,mathbf {x} )} of the time variable t{displaystyle t} and space variables x{displaystyle mathbf {x} }; the Laplacian ∇2{displaystyle nabla ^{2}} acts on the space variables only.

The equation is often abbreviated as

(◻+μ2)ψ=0,{displaystyle (Box +mu ^{2})psi =0,}

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

◻=−ημν∂μ∂ν=1c2∂2∂t2−∇2.{displaystyle 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:

−∂t2ψ+∇2ψ=m2ψ{displaystyle -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:

ψ=e−iωt+ik⋅x=eikμxμ{displaystyle psi =e^{-iomega t+ikcdot x}=e^{ik_{mu }x^{mu }}}

obey the energy momentum relation of special relativity:

−pμpμ=E2−P2=ω2−k2=−kμkμ=m2{displaystyle -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

[∇2−m2c2ℏ2]ψ(r)=0{displaystyle 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, European Organization for Nuclear Research 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 4n/2n − 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

p22m=E.{displaystyle {frac {mathbf {p} ^{2}}{2m}}=E.}

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

p^22mψ=E^ψ{displaystyle {frac {mathbf {hat {p}} ^{2}}{2m}}psi ={hat {E}}psi }

where

p^=−iℏ∇{displaystyle mathbf {hat {p}} =-ihbar mathbf {nabla } }

is the momentum operator (∇ being the del operator), and

E^=iℏ∂∂t{displaystyle {hat {E}}=ihbar {frac {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:

p2c2+m2c4=E{displaystyle {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

(−iℏ∇)2c2+m2c4ψ=iℏ∂∂tψ.{displaystyle {sqrt {(-ihbar mathbf {nabla } )^{2}c^{2}+m^{2}c^{4}}},psi =ihbar {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.

p2c2+m2c4=E2{displaystyle mathbf {p} ^{2}c^{2}+m^{2}c^{4}=E^{2}}

which, when quantized, gives

((−iℏ∇)2c2+m2c4)ψ=(iℏ∂∂t)2ψ{displaystyle left((-ihbar mathbf {nabla } )^{2}c^{2}+m^{2}c^{4}right)psi =left(ihbar {frac {partial }{partial t}}right)^{2}psi }

which simplifies to

−ℏ2c2∇2ψ+m2c4ψ=−ℏ2∂2∂t2ψ.{displaystyle -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

1c2∂2∂t2ψ−∇2ψ+m2c2ℏ2ψ=0.{displaystyle {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(−c², 1, 1, 1), and writing the Einstein summation convention explicitly we get

−ημν∂μ∂νψ≡∑μ=0μ=3∑ν=0ν=3−ημν∂μ∂νψ=1c2∂02ψ−∑ν=1ν=3∂ν∂νψ=1c2∂2∂t2ψ−∇2ψ.{displaystyle -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

(◻+μ2)ψ=0,{displaystyle (Box +mu ^{2})psi =0,}

where

μ=mcℏ{displaystyle mu ={frac {mc}{hbar }}}

and

◻=1c2∂2∂t2−∇2.{displaystyle 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]

◻ψ+∂V∂ψ=0{displaystyle Box psi +{frac {partial V}{partial psi }}=0}

Conserved current

The conserved current associated to the U(1) symmetry of a complex field φ(x)∈C{displaystyle varphi (x)in mathbb {C} } satisfying the Klein–Gordon equation reads

∂μJμ(x)=0,Jμ(x)≡φ∗(x)∂μφ(x)−φ(x)∂μφ∗(x).{displaystyle partial _{mu }J^{mu }(x)=0,qquad J^{mu }(x)equiv varphi ^{*}(x)partial ^{mu }varphi (x)-varphi (x)partial ^{mu }varphi ^{*}(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.

Relativistic free particle solution

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

∇2ψ−1c2∂2∂t2ψ=m2c2ℏ2ψ{displaystyle 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

ψ(r,t)=ei(k⋅r−ωt){displaystyle psi (mathbf {r} ,t)=e^{i(mathbf {k} cdot mathbf {r} -omega t)}}

for some constant angular frequency ω ∈ ℝ and wave number k ∈ ℝ³. Substitution gives the dispersion relation:

−|k|2+ω2c2=m2c2ℏ2.{displaystyle -|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:

⟨p⟩=⟨ψ|−iℏ∇|ψ⟩=ℏk,{displaystyle langle mathbf {p} rangle =leftlangle psi left|-ihbar mathbf {nabla } right|psi rightrangle =hbar mathbf {k} ,}

⟨E⟩=⟨ψ|iℏ∂∂t|ψ⟩=ℏω.{displaystyle langle Erangle =leftlangle psi left|ihbar {frac {partial }{partial t}}right|psi rightrangle =hbar omega .}

So the dispersion relation is just the classic relativistic equation:

⟨E⟩2=m2c4+⟨p⟩2c2.{displaystyle langle Erangle ^{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:

⟨E⟩=|⟨p⟩|c.{displaystyle langle Erangle =|langle mathbf {p} rangle |c.}

Action

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

S=∫(−ℏ2mημν∂μψ¯∂νψ−mc2ψ¯ψ)d4x{displaystyle {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 and it is customary to introduce a factor of 1/2 for both terms.

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

Tμν=ℏ2m(ημαηνβ+ημβηνα−ημνηαβ)∂αψ¯∂βψ−ημνmc2ψ¯ψ.{displaystyle 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 T⁰⁰ 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 φ{displaystyle varphi } under a local U(1) gauge transformation φ→φ′=exp⁡(iθ)φ{displaystyle varphi rightarrow varphi '=exp(itheta )varphi } where θ(t,x){displaystyle theta (t,{textbf {x}})} is a locally variable phase angle, which transformation redirects the wavefunction in the complex phase space defined by exp⁡(iθ)=cos⁡θ+isin⁡θ{displaystyle exp(itheta )=cos theta +isin theta }, it is required that ordinary derivatives ∂μ{displaystyle partial _{mu }} be replaced by gauge-covariant derivatives Dμ=∂μ−ieAμ{displaystyle D_{mu }=partial _{mu }-ieA_{mu }} while the gauge fields transform as eAμ→eAμ′=eAμ+∂μθ{displaystyle eA_{mu }rightarrow eA'_{mu }=eA_{mu }+partial _{mu }theta }. The Klein–Gordon equation therefore becomes:

DμDμφ=−(∂t−ieA0)2φ+(∂i−ieAi)2φ=m2φ{displaystyle D_{mu }D^{mu }varphi =-(partial _{t}-ieA_{0})^{2}varphi +(partial _{i}-ieA_{i})^{2}varphi =m^{2}varphi }

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

DμDμφ+AFμνDμφDν(DαDαφ)=0{displaystyle D_{mu }D^{mu }varphi +AF^{mu nu }D_{mu }varphi D_{nu }(D_{alpha }D^{alpha }varphi )=0}

these terms are not renormalizable in 3 + 1 dimensions.

The field equation for a charged scalar field multiplies by i,^{[clarification needed]} 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=∫x(∂μφ∗+ieAμφ∗)(∂νφ−ieAνφ)ημν=∫x|Dφ|2{displaystyle S=int _{x}left(partial _{mu }varphi ^{*}+ieA_{mu }varphi ^{*}right)left(partial _{nu }varphi -ieA_{nu }varphi right)eta ^{mu nu }=int _{x}|Dvarphi |^{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]

0=−gμν∇μ∇νψ+m2c2ℏ2ψ=−gμν∇μ(∂νψ)+m2c2ℏ2ψ=−gμν∂μ∂νψ+gμνΓσμν∂σψ+m2c2ℏ2ψ{displaystyle {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

−1−g∂μ(gμν−g∂νψ)+m2c2ℏ2ψ=0{displaystyle {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, ∇_μ is the covariant derivative and Γ^σ_μν is the Christoffel symbol that is the gravitational force field.

See also

• Relativistic wave equations


• Dirac equation


• Rarita–Schwinger equation


• Quantum field theory


• Scalar field theory


• Sine–Gordon equation

Remarks

Notes

References

• 
  Davydov, A.S. (1976). Quantum Mechanics, 2nd Edition. Pergamon Press. ISBN 0-08-020437-6..mw-parser-output cite.citation{font-style:inherit}.mw-parser-output q{quotes:"""""""'""'"}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}
  


• 
  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.