Klein-Gordon equation

introduction

• free massive scalar field which describes the spin-0 particles
• in condensed matter physics it describes long wavelength optical phonons
• formulated as a relativistic generalization of Schrodinger equation
• there are real KG equation and complex KG equation
  
  • real case describes electrically neutral particles
  • complex case describes charged particles
• \((\Box + m^2) \psi = 0\) i.e. \((\Box + m^2) \psi =\psi_{tt}-\psi_{xx}-\psi_{yy}-\psi_{zz}+m^2\psi=0\)
• people found 2 problems of KG equations
  
  • negative energy states
  • negative probability density
• correct interpretations of \(\phi\) 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, \(\pi\)-meson
• Thus the Dirac equation comes in to deal with spin-\(1/2\) particles.

 

 

plane wave solutions

• \(u(x,t)=Ae^{i(kx-\omega t)}\)

 

 

 

 

history

• http://www.google.com/search?hl=en&tbs=tl:1&q=

 

 

related items

• special and general relativity
• sine-Gordon equation