- Klein–Gordon 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.
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 time-independent case, the Klein–Gordon equation becomes
which is the homogeneous screened Poisson equation.
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 spin-zero 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 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 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.
The non-relativistic equation for the energy of a free particle is
By quantizing this, we get the non-relativistic Schrödinger equation for a free particle,
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
Using the reciprocal of the Minkowski metric diag( − c2,1,1,1), we get
in covariant notation. This is often abbreviated as
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. Furthermore, any solution to the Dirac equation (for a spin-one-half 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 non-relativistic case:
except with the constraint
Just as with the non-relativistic 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:
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 real-valued, then
From this we can derive the stress-energy tensor of the scalar field. It is
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:
In general relativity, we include the effect of gravity and the Klein–Gordon equation becomes
where gαβ 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 Γσμν is the Christoffel symbol that is the gravitational force field.
- 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.
Wikimedia Foundation. 2010.
Look at other dictionaries:
Equation de Klein-Gordon — Équation de Klein Gordon L équation de Klein Gordon (1926), parfois également appelée équation de Klein Gordon Fock, est une version relativiste de l équation de Schrödinger décrivant des particules massives de spin nul, sans ou avec charge… … Wikipédia en Français
Équation de klein-gordon — L équation de Klein Gordon (1926), parfois également appelée équation de Klein Gordon Fock, est une version relativiste de l équation de Schrödinger décrivant des particules massives de spin nul, sans ou avec charge électrique. Sommaire 1 L… … Wikipédia en Français
Équation de Klein-Gordon — L équation de Klein Gordon (1926), parfois également appelée équation de Klein Gordon Fock, est une version relativiste de l équation de Schrödinger décrivant des particules massives de spin nul, sans ou avec charge électrique. Sommaire 1 L… … Wikipédia en Français
Sine-Gordon equation — The sine Gordon equation is a nonlinear hyperbolic partial differential equation in 1+1 dimensions involving the d Alembert operator and the sine of the unknown function. It was originally considered in the nineteenth century in the course of… … Wikipedia
Equation de Schrodinger — Équation de Schrödinger Cet article fait partie de la série Mécanique quantique Postulats de la mécanique quantique Histoire … Wikipédia en Français
Equation de schrodinger — Équation de Schrödinger Cet article fait partie de la série Mécanique quantique Postulats de la mécanique quantique Histoire … Wikipédia en Français
Équation de Schrodinger — Équation de Schrödinger Cet article fait partie de la série Mécanique quantique Postulats de la mécanique quantique Histoire … Wikipédia en Français
Équation de Shroedinger — Équation de Schrödinger Cet article fait partie de la série Mécanique quantique Postulats de la mécanique quantique Histoire … Wikipédia en Français
Équation de schrodinger — Équation de Schrödinger Cet article fait partie de la série Mécanique quantique Postulats de la mécanique quantique Histoire … Wikipédia en Français
Équation de schrödinger — Cet article fait partie de la série Mécanique quantique Postulats de la mécanique quantique Histoire … Wikipédia en Français