The Klein-Gordon (KG) equation [4,5], one of the fundamental (Poincar´e invariant) equations of quantum field theories (QFT), describes bosons with spin zero that are represented by scalar wave functions. The left side of the equality of ##(5)## is obvious from ##(4)##, however the rest of the terms are still unknown to me. I have tried adding and subtracting terms similar to the rest of the terms so as to produce a commutator and use ##(3)##, but I can't seem to figure out how to get ##(5)## correctly. Learn faster with spaced repetition. This equation is nasty, it is hard to see the relativistic invariance and the square root is difficult to interpret quantum mechanically. of the Klein-Gordon equation, E = ± p2 +m2, since the negative energy solutions have negative probability densities ρ. Prof. M.A. For every solution to the Dirac equation, its components will be a solution of the Klein-Gordon equation, but the converse isn't true: if you form a spinor whose components are solutions of the Klein-Gordon equation, it might not solve the Dirac equation. 2 Second try Strat with the relation 2E = p 2+ m = p02 Using µp →i∂µ =i(∂ ∂x0,−∇) we get (∂ µ∂ µ +m2)Ψ= 0 which is the Klein-Gordon equation… The Dirac equation is one of the two factors, and is conventionally taken to be p m= 0 (31) Making the standard substitution, p !i@ • Start with the Dirac equation (D6) and its Hermitian conjugate (D7) • Consider remembering are Hermitian • Now using the identity: Now, suppose a solution to the Klein-Gordon equation is a free particle with energy Eand mo-mentum p ip= Ne x (11) 1. 5.4 The Dirac Equation The problems with the Klein-Gordon equation led Dirac to search for an alternative relativistic wave equation in … The Dirac equation and the Klein-Gordon (K-G) equation are the most frequently used wave equations for the description of particle dynamics in relativistic quantum mechanics. of coordinates is a Poincar´e invariant equation [3]. The Dirac equation is more restrictive than the Klein-Gordon equation. Dirac Equation: Probability Density and Current. Study The Klein-Gordon Equation, The Maxwell and Proca Equations and The Dirac Equation flashcards from Sophie Wilkinson's class online, or in Brainscape's iPhone or Android app. Thomson Michaelmas 2011 54 • Now consider probability density/current – this is where the perceived problems with the Klein-Gordon equation arose. Whatever these constants The fact that these two equations, for free particles, are constructed using two objects: the four-vector linear The Dirac Equation Dirac in his effort to discover an equation that was free of the problems seen with the Klein Gordon equation tried an equation that was linear in the energy, mass and momentum. He tried the Hamiltonian: H = ⋅ p m = i pi m where H =E (1) with i, being constant objects and E the particle energy. Comparison among Klein-Gordon equation, Dirac equation and Relativistic Stationary Schr¨odinger equation Guang-jiong Ni Department of Physics, Fudan University, Shanghai, 200433, P. R. China Weimin Zhouy and Jun Yanz Department of Physics, New York University, 4 Washington Place, New York, NY, 10003 A particle is always not pure. The Geometrization of Quantum Mechanics proposed in this work is based on the postulate that the quantum probability density can curve the classical s…