Derivation of Klein–Gordon equation

Goals and prior knowledge

The main wave equations of relativistic quantum mechanics are the Klein–Gordon, Dirac, Weyl, and Rarita–Schwinger equations. The principal aim of this chapter is to derive these, by means of the EPI principles (3.16) and (3.18). The derivations generally follow Frieden (1995), or Frieden and Soffer (1995), with numerous additions.

In addition, the Schrödinger wave equation (SWE), which is non-relativistic, is also to be derived. Since rigorous use of EPI gives only relativistic wave equations (Chapter 3), EPI can only derive the SWE in some approximate sense. This is directly carried through in Appendix D, or by taking the well-known non-relativistic limit of the Klein–Gordon equation in Appendix G.

Also, in Appendix D a simple EPI derivation of classical mechanics is obtained. In particular, in the limit h → 0 the EPI extremum condition (3.16) is found to give Newton's second law, and the zero-condition (3.18) gives the virial theorem, of classical mechanics.

Other derived results are Eq. (4.17) – the equivalence of mass, momentum, and energy; the constancy of the Planck constant h (Sec. 4.1.14); the Compton wavelength as an ultimate resolution length (Sec. 4.1.17); and the Heisenberg uncertainty principle (Sec. 4.3).

It is important to state what is assumed a priori. The relativistic properties of space and time coordinates were derived in Sec. 3.5. Hence, these are assumed here. Also, appropriate units for the concepts of mass and energy are assumed to be known, although the basic relation Eq. (4.17) between them is not presumed.