Klein–Gordon equation

The Klein–Gordon equation (Klein–Fock–Gordon equation or sometimes Klein–Gordon–Fock equation) is a relativistic version of the Schrödinger equation.

It is the equation of motion of a quantum scalar or pseudoscalar field, a field whose quanta are spinless particles. It cannot be straightforwardly interpreted as a Schrödinger equation for a quantum state, because it is second order in time and because it does not admit a positive definite conserved probability density. Still, with the appropriate interpretation, it does describe the quantum amplitude for finding a point particle in various places, the relativistic wavefunction, but the particle propagates both forwards and backwards in time.

Statement

The Klein–Gordon equation is

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

It is most often written in natural units:

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

The form is determined 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, there are two values of $\omega$ for each k, one positive and one negative. Only by separating out the positive and negative frequency parts does the equation describe a relativistic wavefunction.

History

The equation was named after the physicists Oskar Klein and Walter Gordon, who in 1927 proposed that it describes relativistic electrons. Although it turned out that the Dirac equation describes the spinning electron, the Klein Gordon equation correctly describes the spinless pion. The pion is a composite particle; no spinless elementary particles have yet been found.

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, without taking into account the electron's spin, the Klein–Gordon 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. 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, 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,

${\frac {\mathbf {p} ^{2}}{2m}}\psi =i\hbar {\frac {\partial }{\partial t}}\psi$

where

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

is the momentum operator ($\nabla$ being the del operator).

The Schrödinger equation suffers from not being relativistically covariant, meaning it does not take into account Einstein's special relativity.

It is natural to try to use the identity from special relativity

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

for the energy; then, just inserting the quantum mechanical momentum operator, yields the equation

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

This, however, is a cumbersome expression to work with because of the square root. In addition, this equation, as it stands, is nonlocal.

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

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

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 which are real valued as well as those which have complex values.

Using the reciprocal of the Minkowski metric ${\text{diag}}(-c^{2},1,1,1)$, we get

$-\eta ^{\mu \nu }\partial _{\mu }\partial _{\nu }\psi +{\frac {m^{2}c^{2}}{\hbar ^{2}}}\psi =0$

in covariant notation. This is often abbreviated as

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

where

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

and

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

This operator is called the d'Alembert operator. Today this form is interpreted as the relativistic field equation for a scalar (i.e. spin-0) particle.

Relativistic free particle solution

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

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

with the same solution as in the non-relativistic case:

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

except with the constraint

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

Just as with the non-relativistic particle, we have for energy and momentum:

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

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

Except that now when we solve for k and ω and substitute into the constraint equation, we recover the relationship between energy and momentum for relativistic massive particles:

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

For massless particles, we may set m = 0 in the above equations. We then recover the relationship between energy and momentum for massless particles:

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

Action

The Klein–Gordon equation can also be derived from the following 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 $\psi \!$ is the Klein–Gordon field and $m\!$ is its mass. The complex conjugate of $\psi \!$ is written ${\bar {\psi }}.$ If the scalar field is taken to be real-valued, then ${\bar {\psi }}=\psi .$

From this we can derive the stress-energy tensor of the scalar field. It is

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

Electromagnetic interaction

The 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. The Klein Gordon equation 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 that the field must be complex. In order for a field to be charged, it must have two components which can be rotated into each other, the real and imaginary parts.

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

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

Gravitational interaction

In general relativity, we include the effect of gravity and the Klein–Gordon equation becomes

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

or equivalently

$0=-g^{\mu \nu }\nabla _{\mu }\nabla _{\nu }\psi +{\frac {m^{2}c^{2}}{\hbar ^{2}}}\psi$

$=-g^{\mu \nu }\partial _{\mu }\partial _{\nu }\psi +g^{\mu \nu }\Gamma ^{\sigma }{}_{\mu \nu }\partial _{\sigma }\psi +{\frac {m^{2}c^{2}}{\hbar ^{2}}}\psi$

where $g^{\alpha \beta }$ is the reciprocal of the metric tensor which is the gravitational potential field, $g$ is the determinant of the metric tensor, $\nabla _{\mu }$ is the covariant derivative and $\Gamma ^{\sigma }{}_{\mu \nu }$ is the Christoffel symbol which is the gravitational force field.

See also

• Dirac equation
• Quantum field theory
• Scalar field theory
• Mathematical aspects of the Klein-Gordon equation are discussed on the Dispersive PDE Wiki.

References

• Sakurai, J. J. (1967). Advanced Quantum Mechanics. Addison Wesley. ISBN 0-201-06710-2.
• Davydov, A.S. (1976). Quantum Mechanics, 2nd Edition. Pergamon. ISBN 0-08-020437-6.

External links

• Weisstein, Eric W. "Klein–Gordon equation". MathWorld. {{cite web}}: Unknown parameter |urltitle= ignored (help)
• Linear Klein-Gordon Equation at EqWorld: The World of Mathematical Equations.
• Nonlinear Klein-Gordon Equation at EqWorld: The World of Mathematical Equations.
• generalizing the Klein-Gordon equation to include a generalized space