Klein-Gordon equation

introduction

in condensed matter physics it describes long wavelength optical phonons
there are real KG equation and complex KG equation
- real case describes electrically neutral particles
- complex case describes charged particles
<math>(\Box + m^2) \psi = 0</math> i.e. <math>(\Box + m^2) \psi =\psi_{tt}-\psi_{xx}-\psi_{yy}-\psi_{zz}+m^2\psi=0</math>
correct interpretations of <math>\phi</math> requires the idea of quantum field rather than the particle wavefunction
- negative probability density -> charge density
Dirac suggested Dirac sea by invoking the exclusion principle and then KG equation only applicable to spinless particles
- for example, <math>\pi</math>-meson
Thus the Dirac equation comes in to deal with spin-<math>1/2</math> particles.