 Klein–Gordon equation

Quantum mechanics
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Sommerfeld · Wien · WignerThe 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. Any solution to the Dirac equation is automatically a solution to the Klein–Gordon equation, but the converse is not true.
Contents
Statement
The Klein–Gordon equation is
It is most often written in natural units:
The form is determined by requiring that plane wave solutions of the equation:
obey the energy momentum relation of special relativity:
Unlike the Schrödinger equation, there are two values of ω 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. For the timeindependent case, the Klein–Gordon equation becomes
which is the homogeneous screened Poisson equation.
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, although the Higgs boson is theorized to exist as a spinzero boson, according to the Standard Model.
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 nth energy level. In January 1926, Schrödinger submitted for publication instead his equation, a nonrelativistic approximation that predicts the Bohr energy levels of hydrogen without fine structure.
In 1927, 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 nonrelativistic equation for the energy of a free particle is
By quantizing this, we get the nonrelativistic Schrödinger equation for a free particle,
where
is the momentum operator ( 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
for the energy; then, just inserting the quantum mechanical momentum operator, yields the equation
This, however, is a cumbersome expression to work with because the differential operator cannot be evaluated while under the square root sign. In addition, this equation, as it stands, is nonlocal.
Klein and Gordon instead began with the square of the above identity, i.e.
which, when quantized, gives
which simplifies to
Rearranging terms yields
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.
Using the reciprocal of the Minkowski metric diag( − c^{2},1,1,1), we get
in covariant notation. This is often abbreviated as
where
and
This operator is called the d'Alembert operator. Today this form is interpreted as the relativistic field equation for a scalar (i.e. spin0) particle. Furthermore, any solution to the Dirac equation (for a spinonehalf particle) is automatically a solution to the Klein–Gordon equation, though not all solutions of the Klein–Gordon equation are solutions of the Dirac equation. It is noteworthy that the Klein–Gordon equation is very similar to the Proca equation.
Relativistic free particle solution
The Klein–Gordon equation for a free particle can be written as
with the same solution as in the nonrelativistic case:
except with the constraint
Just as with the nonrelativistic particle, we have for energy and momentum:
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:
For massless particles, we may set m = 0 in the above equations. We then recover the relationship between energy and momentum for massless particles:
Action
The Klein–Gordon equation can also be derived from the following action
where is the Klein–Gordon field and is its mass. The complex conjugate of is written If the scalar field is taken to be realvalued, then
From this we can derive the stressenergy tensor of the scalar field. It is
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. The Klein Gordon equation becomes:
in natural units, where A is the vector potential. While it is possible to add many higher order terms, for example,
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:
Gravitational interaction
In general relativity, we include the effect of gravity and the Klein–Gordon equation becomes
or equivalently
where g^{αβ} is the reciprocal of the metric tensor that is the gravitational potential field, g is the determinant of the metric tensor, is the covariant derivative and Γ^{σ}_{μν} is the Christoffel symbol that is the gravitational force field.
See also
 Dirac equation
 Rarita–Schwinger equation
 Quantum field theory
 Scalar field theory
References
 Sakurai, J. J. (1967). Advanced Quantum Mechanics. Addison Wesley. ISBN 0201067102.
 Davydov, A.S. (1976). Quantum Mechanics, 2nd Edition. Pergamon. ISBN 0080204376.
External links
 Weisstein, Eric W., "Klein–Gordon equation" from MathWorld.
 Linear Klein–Gordon Equation at EqWorld: The World of Mathematical Equations.
 Nonlinear Klein–Gordon Equation at EqWorld: The World of Mathematical Equations.
Categories: Fundamental physics concepts
 Partial differential equations
 Equations
 Special relativity
 Waves
 Quantum field theory

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