KleinGordon Equation
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The KleinGordon Equation is a relativistic generalisation of the Schrordinger Equation to Relativistic Quantum Mechanics. In Quantum Field Theory, it is a relativistic field equation for a spin0 field (KleinGordon Field).
HistoryEdit this section
This equation was actually discovered by Erwin Schrodinger before he discovered the Schrodinger Equation, that bears his name. However, for reasons which we will be discussing later, he soon discarded the equation and decided to use the nonrelativistic energy instead, which resulted in the intrinsically not LorentzInvariant Schrodinger Equation. ;
This equation was named after Oskar Klein and Walter Gordon, who proposed (1926) that it describes relativistic electrons. Vladimir Fock, Johann Kudar, Theophile de Donder and Frans H van den Dungen, and Louis de Broglie also made these claims.
We now know that the Dirac Equation is the actual field equation for these spin1/2 electrons. The KleinGordon Equation however, does describe spin0 particles, such as the Higgs Field/ .
StatementEdit this section
KleinGordon Equation
Derivation of KleinGordon Equation from Schrodinger's Wave Equation 

In the Schrodinger formulation of Quantum Mechanics, quantisation is as follows: Start with the Energy:
Then, quantise as follows:
To be LorentzInvariant (It needs to be relativistic) start with the Special Relativistic Energy formula instead:
The Laplacian cannot be calculated under the square root, so instead, we use
We realise that this cannot be properly calculated (the same problems as before, the square root), so, at least for now, we let , i.e. only free particles.

In a PotentialEdit this section
In a potential, the KleinGordon Equation obviously gets modified as follows:
Free Particle SolutionEdit this section
For the free particle solution described earlier, the solution for the wavefunction remains the same as in the NonRelativistic case:
This is because the additional energy only induces a very fast phase rotation of the state.
However, an additional constraint, called the Dispersion Relation, arises:
\
For massless particles, this becomes .
Gravitational InteractionEdit this section
To see the effect of gravitational interaction, use the form of the KleinGordon Equation:
It is obvious to anyone with any knowledge about General Relativity that when considering gravitational interaction, one does transformations such as . Therefore, in the presence of a gravitational field, the KleinGordon Equation becomes:
Or, with a potential,
IssuesEdit this section
However, the KleinGordon Equation has issues when interpreted as a standard wave equation.
 Since it is secondorder in time (like the classical wave equation from Classical LorentzInvariant EM), probability density is not explicitly always conserved.
 The energy is not forced to be positive.
The Dirac Equation was an attempt to solve this issue/.