Information about Klein Gordon Equation
The Klein-Gordon equation (Klein-Fock-Gordon equation or sometimes Klein-Gordon-Fock equation) is the relativistic version of the Schrödinger equation, which is used to describe spinless particles. It was named after Oskar Klein and Walter Gordon.
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 of the square root. In addition, this equation, as it stands, is nonlocal.
Klein and Gordon instead worked with the more general square of this equation (the Klein-Gordon equation for a free particle), which in covariant notation reads
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. spin-0) particle.
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.
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:
is the Klein-Gordon field and 
is its mass.
In physics, hidden variable theories are espoused by a minority of physicists who argue that the statistical nature of quantum mechanics indicates that quantum
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| Quantum physics |
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| Quantum mechanics |
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Introduction to... Mathematical formulation of... |
| Fundamental concepts |
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Decoherence Interference Uncertainty Exclusion Transformation theory Ehrenfest theorem Measurement |
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Double-slit experiment Davisson-Germer experiment Stern–Gerlach experiment EPR paradox Popper's experiment Schrdinger's cat |
| Equations |
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Schrdinger equation Pauli equation Klein-Gordon equation Dirac equation |
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Quantum field theory Wightman axioms Quantum electrodynamics Quantum chromodynamics Quantum gravity Feynman diagram |
| Interpretations |
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Copenhagen
Ensemble Hidden variables Transactional Many-worlds Consistent histories Quantum logic Consciousness causes collapse |
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Planck Schrdinger Heisenberg Bohr Pauli Dirac Bohm Born de Broglie von Neumann Einstein Feynman Everett Others |
Details
The Schrödinger equation for a free particle is
- 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 of the square root. In addition, this equation, as it stands, is nonlocal.
Klein and Gordon instead worked with the more general square of this equation (the Klein-Gordon equation for a free particle), which in covariant notation reads
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. spin-0) particle.
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.
Relativistic free particle solution
The Klein-Gordon equation for a free particle can be written aswith 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:
Action
The Klein-Gordon equation can be derived from the following action is the Klein-Gordon field and is its mass.
See also
- Dirac equation
- Quantum field theory
- Scalar field
- Mathematical aspects of the Klein-Gordon equation are discussed on the .
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
- 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
special theory of relativity was proposed in 1905 by Albert Einstein in his article "On the Electrodynamics of Moving Bodies". Some three centuries earlier, Galileo's principle of relativity had stated that all uniform motion was relative, and that there was no absolute and
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Schrödinger equation, proposed by the Austrian physicist Erwin Schrödinger in 1926, describes the space- and time-dependence of quantum mechanical systems. It is of central importance in non-relativistic quantum mechanics, playing a role for microscopic particles analogous to
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Oskar Klein (September 15 1894 - February 5 1977) was a Swedish theoretical physicist.
Klein was born in Danderyd outside Stockholm, son of the chief rabbi of Stockholm, Dr. Gottlieb Klein and Antonie (Toni) Levy.
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Klein was born in Danderyd outside Stockholm, son of the chief rabbi of Stockholm, Dr. Gottlieb Klein and Antonie (Toni) Levy.
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Quantum mechanics (QM, or quantum theory) is a physical science dealing with the behaviour of matter and energy on the scale of atoms and subatomic particles / waves.[1]
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This article or section may be confusing or unclear for some readers.
Please [improve the article] or discuss this issue on the talk page. This article has been tagged since April 2007.
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Please [improve the article] or discuss this issue on the talk page. This article has been tagged since April 2007.
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Interference is the addition (superposition) of two or more waves that results in a new wave pattern.
As most commonly used, the term interference usually refers to the interaction of waves which are correlated or coherent with each other, either because they
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As most commonly used, the term interference usually refers to the interaction of waves which are correlated or coherent with each other, either because they
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Heisenberg uncertainty principle, or HUP, gives a lower bound on the product of the standard deviations of position and momentum for a system, implying that it is impossible to have a particle that has an arbitrarily well-defined position and momentum simultaneously.
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The Pauli exclusion principle is a quantum mechanical principle formulated by Wolfgang Pauli in 1925. This principle is significant, because it explains why matter occupies space exclusively for itself and does not allow other material objects to pass through it, while at the same
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The term transformation theory refers to a procedure used by P. A. M. Dirac in his early formulation of quantum theory, from around 1927.
The term is related to the famous wave-particle duality, according to which a particle (a "small" physical object) may display
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The term is related to the famous wave-particle duality, according to which a particle (a "small" physical object) may display
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Measurement from a practical point of view
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π due to reflection at the interface of a denser medium)
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Quantum version of experiment
By the 1920s, various other experiments (such as the photoelectric effect) had demonstrated that light interacts with matter only in discrete, "quantum"-sized packets called photons...... Click the link for more information.
In quantum mechanics, the EPR paradox is a thought experiment which challenged long-held ideas about the relation between the observed values of physical quantities and the values that can be accounted for by a physical theory.
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action at a distance.
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The debate
Many viewed Popper's experiment as a crucial test of quantum mechanics, and there was a debate on what result an actual realization of the experiment would yield...... Click the link for more information.
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In physics, the Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist Paul Dirac in 1928 and provides a description of elementary spin-½ particles, such as electrons, consistent with both the principles of quantum mechanics and the
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Quantum field theory (QFT) is a theoretical framework for constructing quantum mechanical models of field-like systems, or, equivalently, of many-body systems. It is widely used in particle physics and condensed matter physics.
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In physics the Wightman axioms are an attempt at a mathematically rigorous formulation of quantum field theory. Arthur Wightman formulated the axioms in the early 1950s but they were first published only in 1964, after Haag-Ruelle scattering theory affirmed their significance.
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radiates a gluon. (Time goes left to right, and one space dimension runs from top to bottom.)]]
A Feynman diagram is a tool invented by American physicist Richard Feynman for performing scattering calculations in quantum field theory.
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A Feynman diagram is a tool invented by American physicist Richard Feynman for performing scattering calculations in quantum field theory.
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The Copenhagen interpretation is an interpretation of quantum mechanics formulated by Niels Bohr and Werner Heisenberg while collaborating in Copenhagen around 1927. Bohr and Heisenberg extended the probabilistic interpretation of the wave function, proposed by Max Born.
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The Ensemble Interpretation, or Statistical Interpretation of quantum mechanics, is an interpretation that can be viewed as a minimalist interpretation; it is a quantum mechanical interpretation that claims to make the fewest assumptions associated with the standard
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- ''Hidden variable redirects here. For hidden variables in economics, see latent variable.
In physics, hidden variable theories are espoused by a minority of physicists who argue that the statistical nature of quantum mechanics indicates that quantum
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The transactional interpretation of quantum mechanics (TIQM) is an unusual interpretation of quantum mechanics that describes quantum interactions in terms of a standing wave formed by retarded (forward-in-time) and advanced (backward-in-time) waves.
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The many-worlds interpretation or MWI (also known as relative state formulation, theory of the universal wavefunction, many-universes interpretation, Oxford interpretation or many worlds), is an interpretation of quantum mechanics.
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